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Properties

Label 483.8.a.h
Level $483$
Weight $8$
Character orbit 483.a
Self dual yes
Analytic conductor $150.882$
Analytic rank $0$
Dimension $20$
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] = [483,8,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, 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("483.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(150.881967309\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 2001 x^{18} + 9297 x^{17} + 1659337 x^{16} - 8672053 x^{15} - 738401777 x^{14} + \cdots - 22\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{16}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + 27 q^{3} + (\beta_{2} + \beta_1 + 74) q^{4} + (\beta_{5} - \beta_1 + 54) q^{5} + (27 \beta_1 + 27) q^{6} + 343 q^{7} + (\beta_{3} + 2 \beta_{2} + 80 \beta_1 + 90) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + 27 q^{3} + (\beta_{2} + \beta_1 + 74) q^{4} + (\beta_{5} - \beta_1 + 54) q^{5} + (27 \beta_1 + 27) q^{6} + 343 q^{7} + (\beta_{3} + 2 \beta_{2} + 80 \beta_1 + 90) q^{8} + 729 q^{9} + (4 \beta_{5} + \beta_{4} - \beta_{2} + \cdots - 108) q^{10}+ \cdots + ( - 729 \beta_{6} + 1458 \beta_{5} + \cdots + 366687) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 24 q^{2} + 540 q^{3} + 1486 q^{4} + 1069 q^{5} + 648 q^{6} + 6860 q^{7} + 2127 q^{8} + 14580 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 24 q^{2} + 540 q^{3} + 1486 q^{4} + 1069 q^{5} + 648 q^{6} + 6860 q^{7} + 2127 q^{8} + 14580 q^{9} - 1949 q^{10} + 10073 q^{11} + 40122 q^{12} + 13391 q^{13} + 8232 q^{14} + 28863 q^{15} + 133122 q^{16} + 62626 q^{17} + 17496 q^{18} + 9895 q^{19} + 106064 q^{20} + 185220 q^{21} + 28599 q^{22} - 243340 q^{23} + 57429 q^{24} + 265365 q^{25} + 594400 q^{26} + 393660 q^{27} + 509698 q^{28} + 594658 q^{29} - 52623 q^{30} + 514862 q^{31} + 832720 q^{32} + 271971 q^{33} - 106257 q^{34} + 366667 q^{35} + 1083294 q^{36} + 891864 q^{37} + 680125 q^{38} + 361557 q^{39} + 44594 q^{40} + 296689 q^{41} + 222264 q^{42} - 704949 q^{43} + 2001503 q^{44} + 779301 q^{45} - 292008 q^{46} + 2102453 q^{47} + 3594294 q^{48} + 2352980 q^{49} + 4129604 q^{50} + 1690902 q^{51} + 4416739 q^{52} + 5841486 q^{53} + 472392 q^{54} + 4290005 q^{55} + 729561 q^{56} + 267165 q^{57} + 7165650 q^{58} + 7015980 q^{59} + 2863728 q^{60} + 2474138 q^{61} + 4418145 q^{62} + 5000940 q^{63} + 12695973 q^{64} + 6582462 q^{65} + 772173 q^{66} + 2305855 q^{67} + 10253157 q^{68} - 6570180 q^{69} - 668507 q^{70} + 12287349 q^{71} + 1550583 q^{72} + 9140922 q^{73} - 832604 q^{74} + 7164855 q^{75} + 290029 q^{76} + 3455039 q^{77} + 16048800 q^{78} - 1444882 q^{79} + 2254323 q^{80} + 10628820 q^{81} + 6031922 q^{82} + 4284072 q^{83} + 13761846 q^{84} + 15450581 q^{85} + 19710382 q^{86} + 16055766 q^{87} - 4553328 q^{88} + 36265659 q^{89} - 1420821 q^{90} + 4593113 q^{91} - 18080162 q^{92} + 13901274 q^{93} + 11807737 q^{94} + 35752199 q^{95} + 22483440 q^{96} + 15575692 q^{97} + 2823576 q^{98} + 7343217 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 2001 x^{18} + 9297 x^{17} + 1659337 x^{16} - 8672053 x^{15} - 738401777 x^{14} + \cdots - 22\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 201 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 335\nu + 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!29 \nu^{19} + \cdots + 26\!\cdots\!68 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!23 \nu^{19} + \cdots - 20\!\cdots\!92 ) / 60\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!05 \nu^{19} + \cdots - 12\!\cdots\!32 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!23 \nu^{19} + \cdots + 36\!\cdots\!88 ) / 15\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83\!\cdots\!25 \nu^{19} + \cdots - 21\!\cdots\!24 ) / 60\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42\!\cdots\!45 \nu^{19} + \cdots - 54\!\cdots\!88 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 75\!\cdots\!13 \nu^{19} + \cdots - 46\!\cdots\!00 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!31 \nu^{19} + \cdots + 18\!\cdots\!12 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!11 \nu^{19} + \cdots - 11\!\cdots\!24 ) / 60\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 70\!\cdots\!55 \nu^{19} + \cdots - 63\!\cdots\!68 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 79\!\cdots\!61 \nu^{19} + \cdots - 10\!\cdots\!08 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!97 \nu^{19} + \cdots + 95\!\cdots\!32 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 24\!\cdots\!93 \nu^{19} + \cdots + 56\!\cdots\!72 ) / 24\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 40\!\cdots\!19 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 37\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 13\!\cdots\!49 \nu^{19} + \cdots - 21\!\cdots\!80 ) / 12\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 15\!\cdots\!87 \nu^{19} + \cdots - 25\!\cdots\!32 ) / 12\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 201 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 336\beta _1 - 258 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 4\beta_{5} - 4\beta_{3} + 463\beta_{2} - 710\beta _1 + 67606 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{19} + 7 \beta_{18} + 10 \beta_{17} - 6 \beta_{16} - 6 \beta_{15} + \beta_{14} + \cdots - 169314 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{19} + 12 \beta_{18} - 44 \beta_{17} + 10 \beta_{16} + 20 \beta_{15} - 4 \beta_{14} + \cdots + 26251791 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2938 \beta_{19} + 5239 \beta_{18} + 8530 \beta_{17} - 4782 \beta_{16} - 5520 \beta_{15} + \cdots - 101003441 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10582 \beta_{19} + 2953 \beta_{18} - 44868 \beta_{17} + 13878 \beta_{16} + 28154 \beta_{15} + \cdots + 10935921053 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1614196 \beta_{19} + 3005683 \beta_{18} + 5262154 \beta_{17} - 2784288 \beta_{16} + \cdots - 58043382338 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9466454 \beta_{19} - 2758203 \beta_{18} - 31680854 \beta_{17} + 10667064 \beta_{16} + \cdots + 4743326824650 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 797881296 \beta_{19} + 1583211598 \beta_{18} + 2864949078 \beta_{17} - 1428944468 \beta_{16} + \cdots - 32324751581392 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6150434070 \beta_{19} - 4031391643 \beta_{18} - 19307572756 \beta_{17} + 6416259696 \beta_{16} + \cdots + 21\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 373215243094 \beta_{19} + 803721647378 \beta_{18} + 1465496573860 \beta_{17} - 684974015758 \beta_{16} + \cdots - 17\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3408469496940 \beta_{19} - 3383965995962 \beta_{18} - 10923247097882 \beta_{17} + 3362521724126 \beta_{16} + \cdots + 95\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 168157838829696 \beta_{19} + 400368150133650 \beta_{18} + 723759472053958 \beta_{17} + \cdots - 93\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 17\!\cdots\!32 \beta_{19} + \cdots + 44\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 73\!\cdots\!24 \beta_{19} + \cdots - 48\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 79\!\cdots\!64 \beta_{19} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 31\!\cdots\!86 \beta_{19} + \cdots - 25\!\cdots\!43 \) 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
−22.3071
−21.7238
−18.2300
−16.6117
−13.8408
−12.6171
−10.0528
−4.90972
−0.917971
0.206681
1.23696
5.08154
8.23051
8.88530
12.4238
12.5017
16.6745
18.8751
20.5409
20.5538
−21.3071 27.0000 325.992 362.175 −575.292 343.000 −4218.64 729.000 −7716.90
1.2 −20.7238 27.0000 301.474 −229.946 −559.542 343.000 −3595.04 729.000 4765.34
1.3 −17.2300 27.0000 168.872 35.0910 −465.209 343.000 −704.224 729.000 −604.618
1.4 −15.6117 27.0000 115.726 147.812 −421.517 343.000 191.613 729.000 −2307.61
1.5 −12.8408 27.0000 36.8849 −338.952 −346.700 343.000 1169.99 729.000 4352.40
1.6 −11.6171 27.0000 6.95678 349.575 −313.661 343.000 1406.17 729.000 −4061.04
1.7 −9.05277 27.0000 −46.0473 436.940 −244.425 343.000 1575.61 729.000 −3955.52
1.8 −3.90972 27.0000 −112.714 −206.750 −105.562 343.000 941.124 729.000 808.334
1.9 0.0820291 27.0000 −127.993 140.689 2.21479 343.000 −20.9989 729.000 11.5406
1.10 1.20668 27.0000 −126.544 −294.427 32.5804 343.000 −307.153 729.000 −355.280
1.11 2.23696 27.0000 −122.996 12.6190 60.3978 343.000 −561.467 729.000 28.2282
1.12 6.08154 27.0000 −91.0149 440.690 164.202 343.000 −1331.95 729.000 2680.07
1.13 9.23051 27.0000 −42.7976 406.225 249.224 343.000 −1576.55 729.000 3749.67
1.14 9.88530 27.0000 −30.2809 −299.211 266.903 343.000 −1564.65 729.000 −2957.79
1.15 13.4238 27.0000 52.1992 −246.120 362.443 343.000 −1017.54 729.000 −3303.87
1.16 13.5017 27.0000 54.2958 −165.417 364.546 343.000 −995.132 729.000 −2233.41
1.17 17.6745 27.0000 184.388 328.728 477.211 343.000 996.629 729.000 5810.11
1.18 19.8751 27.0000 267.021 443.021 536.629 343.000 2763.06 729.000 8805.11
1.19 21.5409 27.0000 336.012 −418.533 581.605 343.000 4480.78 729.000 −9015.59
1.20 21.5538 27.0000 336.565 164.789 581.952 343.000 4495.37 729.000 3551.82
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 483.8.a.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.8.a.h ✓ 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 24 T_{2}^{19} - 1735 T_{2}^{18} + 43491 T_{2}^{17} + 1203856 T_{2}^{16} + \cdots + 22\!\cdots\!60 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(483))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 22\!\cdots\!60 \) Copy content Toggle raw display
$3$ \( (T - 27)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 343)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots - 30\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 37\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T + 12167)^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 49\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 20\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 21\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 96\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 24\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 58\!\cdots\!28 \) Copy content Toggle raw display
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