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Properties

Label 1083.4.a.s
Level $1083$
Weight $4$
Character orbit 1083.a
Self dual yes
Analytic conductor $63.899$
Analytic rank $0$
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] = [1083,4,Mod(1,1083)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1083, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1083.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1083.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: \(63.8990685362\)
Analytic rank: \(0\)
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: \( x^{18} - 120 x^{16} - 19 x^{15} + 5904 x^{14} + 1731 x^{13} - 153482 x^{12} - 62307 x^{11} + \cdots - 49519296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3}\cdot 19^{3} \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{7} + 1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{9} + 3) q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{7} + 1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{9} + 3) q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1 + 3) q^{8} + 9 q^{9} + ( - \beta_{16} - \beta_{13} + \beta_{8} + \cdots - 3) q^{10}+ \cdots + ( - 9 \beta_{16} + 9 \beta_{7} + \cdots + 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 54 q^{3} + 96 q^{4} + 18 q^{5} + 48 q^{7} + 57 q^{8} + 162 q^{9} - 60 q^{10} + 108 q^{11} - 288 q^{12} - 42 q^{13} + 60 q^{14} - 54 q^{15} + 576 q^{16} + 300 q^{17} + 27 q^{20} - 144 q^{21} - 219 q^{22} + 174 q^{23} - 171 q^{24} + 1068 q^{25} - 72 q^{26} - 486 q^{27} + 867 q^{28} + 168 q^{29} + 180 q^{30} - 1032 q^{31} + 921 q^{32} - 324 q^{33} + 75 q^{34} + 1524 q^{35} + 864 q^{36} + 132 q^{37} + 126 q^{39} - 363 q^{40} + 120 q^{41} - 180 q^{42} + 420 q^{43} + 2328 q^{44} + 162 q^{45} - 2229 q^{46} + 810 q^{47} - 1728 q^{48} + 1122 q^{49} - 1503 q^{50} - 900 q^{51} + 228 q^{52} - 174 q^{53} + 2550 q^{55} + 1119 q^{56} + 756 q^{58} + 474 q^{59} - 81 q^{60} + 1488 q^{61} + 333 q^{62} + 432 q^{63} + 2679 q^{64} - 1716 q^{65} + 657 q^{66} - 3060 q^{67} + 4623 q^{68} - 522 q^{69} - 1383 q^{70} + 1464 q^{71} + 513 q^{72} + 1470 q^{73} - 135 q^{74} - 3204 q^{75} + 1014 q^{77} + 216 q^{78} - 2508 q^{79} - 2049 q^{80} + 1458 q^{81} + 1485 q^{82} + 4764 q^{83} - 2601 q^{84} + 804 q^{85} - 1068 q^{86} - 504 q^{87} - 3012 q^{88} + 1050 q^{89} - 540 q^{90} + 3408 q^{91} + 3306 q^{92} + 3096 q^{93} - 8205 q^{94} - 2763 q^{96} + 2070 q^{97} + 1767 q^{98} + 972 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 120 x^{16} - 19 x^{15} + 5904 x^{14} + 1731 x^{13} - 153482 x^{12} - 62307 x^{11} + \cdots - 49519296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!01 \nu^{17} + \cdots + 94\!\cdots\!40 ) / 79\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!47 \nu^{17} + \cdots + 66\!\cdots\!36 ) / 79\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!47 \nu^{17} + \cdots - 69\!\cdots\!20 ) / 79\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!47 \nu^{17} + \cdots + 98\!\cdots\!56 ) / 24\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!05 \nu^{17} + \cdots + 92\!\cdots\!08 ) / 24\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 84\!\cdots\!14 \nu^{17} + \cdots + 20\!\cdots\!56 ) / 80\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!18 \nu^{17} + \cdots - 56\!\cdots\!04 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!26 \nu^{17} + \cdots - 13\!\cdots\!88 ) / 26\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!41 \nu^{17} + \cdots + 18\!\cdots\!72 ) / 26\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!54 \nu^{17} + \cdots + 60\!\cdots\!96 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 51\!\cdots\!94 \nu^{17} + \cdots - 15\!\cdots\!76 ) / 24\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!33 \nu^{17} + \cdots + 68\!\cdots\!16 ) / 80\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!09 \nu^{17} + \cdots - 10\!\cdots\!60 ) / 80\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 35\!\cdots\!55 \nu^{17} + \cdots + 11\!\cdots\!28 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{17} + \cdots + 72\!\cdots\!96 ) / 24\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 22\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 278 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{17} - 3 \beta_{15} + \beta_{13} - 2 \beta_{11} - 9 \beta_{10} - 5 \beta_{9} - 4 \beta_{8} + \cdots + 145 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{17} + \beta_{16} + 47 \beta_{14} + 47 \beta_{13} + 40 \beta_{12} - 52 \beta_{11} + 30 \beta_{10} + \cdots + 6774 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 92 \beta_{17} + 22 \beta_{16} - 140 \beta_{15} - 25 \beta_{14} + 97 \beta_{13} + 3 \beta_{12} + \cdots + 5926 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 166 \beta_{17} + 82 \beta_{16} - 25 \beta_{15} + 1639 \beta_{14} + 1742 \beta_{13} + 1257 \beta_{12} + \cdots + 176168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3228 \beta_{17} + 1431 \beta_{16} - 4911 \beta_{15} - 1430 \beta_{14} + 5171 \beta_{13} + \cdots + 221157 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8212 \beta_{17} + 4279 \beta_{16} - 2139 \beta_{15} + 51668 \beta_{14} + 59823 \beta_{13} + \cdots + 4770393 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 102832 \beta_{17} + 64233 \beta_{16} - 156363 \beta_{15} - 55001 \beta_{14} + 220802 \beta_{13} + \cdots + 7777308 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 329758 \beta_{17} + 183953 \beta_{16} - 115170 \beta_{15} + 1562280 \beta_{14} + 1989556 \beta_{13} + \cdots + 132693149 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3117058 \beta_{17} + 2483046 \beta_{16} - 4776417 \beta_{15} - 1772043 \beta_{14} + 8448270 \beta_{13} + \cdots + 263229628 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11897272 \beta_{17} + 7160129 \beta_{16} - 5047379 \beta_{15} + 46399131 \beta_{14} + 65146944 \beta_{13} + \cdots + 3761484166 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 91502282 \beta_{17} + 88873811 \beta_{16} - 143127132 \beta_{15} - 51154201 \beta_{14} + \cdots + 8690879108 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 403493632 \beta_{17} + 263388452 \beta_{16} - 197454492 \beta_{15} + 1368429382 \beta_{14} + \cdots + 108138700561 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2621774284 \beta_{17} + 3041265206 \beta_{16} - 4251017516 \beta_{15} - 1351155886 \beta_{14} + \cdots + 282261884635 \) 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
−5.28330
−4.94647
−4.66881
−4.11315
−3.74685
−2.55293
−1.42126
−1.28146
−1.21690
0.494961
1.10590
1.38322
3.36651
3.50909
3.61788
4.77547
5.38386
5.59425
−5.28330 −3.00000 19.9133 −9.77721 15.8499 −16.0263 −62.9416 9.00000 51.6560
1.2 −4.94647 −3.00000 16.4675 19.4353 14.8394 9.24913 −41.8844 9.00000 −96.1362
1.3 −4.66881 −3.00000 13.7978 4.88826 14.0064 26.8280 −27.0688 9.00000 −22.8224
1.4 −4.11315 −3.00000 8.91801 −20.0476 12.3395 −5.82920 −3.77591 9.00000 82.4587
1.5 −3.74685 −3.00000 6.03885 21.0439 11.2405 27.2729 7.34811 9.00000 −78.8482
1.6 −2.55293 −3.00000 −1.48257 −13.7527 7.65878 −15.6351 24.2083 9.00000 35.1095
1.7 −1.42126 −3.00000 −5.98001 0.335276 4.26379 14.8860 19.8693 9.00000 −0.476515
1.8 −1.28146 −3.00000 −6.35786 1.40131 3.84438 −27.7591 18.3990 9.00000 −1.79572
1.9 −1.21690 −3.00000 −6.51916 20.4205 3.65070 −21.8031 17.6684 9.00000 −24.8497
1.10 0.494961 −3.00000 −7.75501 −6.60281 −1.48488 30.9819 −7.79812 9.00000 −3.26814
1.11 1.10590 −3.00000 −6.77700 −7.95241 −3.31769 12.5752 −16.3418 9.00000 −8.79454
1.12 1.38322 −3.00000 −6.08672 10.9668 −4.14965 −3.37566 −19.4850 9.00000 15.1694
1.13 3.36651 −3.00000 3.33337 −9.34862 −10.0995 −20.4088 −15.7102 9.00000 −31.4722
1.14 3.50909 −3.00000 4.31372 −14.5964 −10.5273 −16.0084 −12.9355 9.00000 −51.2202
1.15 3.61788 −3.00000 5.08902 15.1471 −10.8536 24.6607 −10.5316 9.00000 54.8005
1.16 4.77547 −3.00000 14.8051 17.5954 −14.3264 −4.31903 32.4975 9.00000 84.0265
1.17 5.38386 −3.00000 20.9860 5.35339 −16.1516 36.1048 69.9146 9.00000 28.8219
1.18 5.59425 −3.00000 23.2956 −16.5096 −16.7828 −3.39393 85.5677 9.00000 −92.3588
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(19\) \(1\)

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 1083.4.a.s 18
19.b odd 2 1 1083.4.a.t 18
19.e even 9 2 57.4.i.b ✓ 36
57.l odd 18 2 171.4.u.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.4.i.b ✓ 36 19.e even 9 2
171.4.u.c 36 57.l odd 18 2
1083.4.a.s 18 1.a even 1 1 trivial
1083.4.a.t 18 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 120 T_{2}^{16} - 19 T_{2}^{15} + 5904 T_{2}^{14} + 1731 T_{2}^{13} - 153482 T_{2}^{12} + \cdots - 49519296 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1083))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 120 T^{16} + \cdots - 49519296 \) Copy content Toggle raw display
$3$ \( (T + 3)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 95\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 49\!\cdots\!41 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 49\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 81\!\cdots\!19 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 11\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 17\!\cdots\!57 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 48\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 57\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 82\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 11\!\cdots\!57 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 32\!\cdots\!79 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 41\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 60\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 37\!\cdots\!87 \) Copy content Toggle raw display
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