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Properties

Label 803.2.a.f
Level $803$
Weight $2$
Character orbit 803.a
Self dual yes
Analytic conductor $6.412$
Analytic rank $0$
Dimension $19$
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] = [803,2,Mod(1,803)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(803, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("803.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 803 = 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 803.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: \(6.41198728231\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 7 x^{18} - 6 x^{17} + 141 x^{16} - 155 x^{15} - 1063 x^{14} + 2102 x^{13} + 3638 x^{12} + \cdots - 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{15} q^{5} + \beta_{5} q^{6} + (\beta_{9} + 1) q^{7} + (\beta_{13} + \beta_{12} + \beta_{7} + \cdots + 1) q^{8}+ \cdots + (\beta_{14} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{15} q^{5} + \beta_{5} q^{6} + (\beta_{9} + 1) q^{7} + (\beta_{13} + \beta_{12} + \beta_{7} + \cdots + 1) q^{8}+ \cdots + (\beta_{14} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 7 q^{2} + 3 q^{3} + 23 q^{4} + 2 q^{5} + 16 q^{7} + 18 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 7 q^{2} + 3 q^{3} + 23 q^{4} + 2 q^{5} + 16 q^{7} + 18 q^{8} + 22 q^{9} + 8 q^{10} + 19 q^{11} - 5 q^{12} + 30 q^{13} - 9 q^{14} - 11 q^{15} + 31 q^{16} + 9 q^{17} + 3 q^{18} - q^{19} - 9 q^{20} + 13 q^{21} + 7 q^{22} - 2 q^{23} - 19 q^{24} + 39 q^{25} - 3 q^{26} - 9 q^{27} + 49 q^{28} + 18 q^{29} - 17 q^{30} + 5 q^{31} + 38 q^{32} + 3 q^{33} + 26 q^{34} - 14 q^{35} + 19 q^{36} + 36 q^{37} + 9 q^{38} + 27 q^{39} + 7 q^{40} + 9 q^{41} + 25 q^{42} + 33 q^{43} + 23 q^{44} + 8 q^{45} - 47 q^{46} - 28 q^{47} - 49 q^{48} + 33 q^{49} + 27 q^{50} + 2 q^{51} + 18 q^{52} + 27 q^{53} + 7 q^{54} + 2 q^{55} - 28 q^{56} - 6 q^{57} + 16 q^{58} - 28 q^{59} + 4 q^{61} + 5 q^{62} + 26 q^{63} + 80 q^{64} + 12 q^{65} + 21 q^{67} - 23 q^{68} - 34 q^{69} - 74 q^{70} - 7 q^{71} + 54 q^{72} - 19 q^{73} + 4 q^{74} - 25 q^{75} - 23 q^{76} + 16 q^{77} - 28 q^{78} + 25 q^{79} - 49 q^{80} - 9 q^{81} - 10 q^{82} + 15 q^{83} + 37 q^{84} + 23 q^{85} - 29 q^{86} + q^{87} + 18 q^{88} - 21 q^{89} - 94 q^{90} + 4 q^{91} - 74 q^{92} + 22 q^{93} + 69 q^{94} + 23 q^{95} - 116 q^{96} + 18 q^{97} - 4 q^{98} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 7 x^{18} - 6 x^{17} + 141 x^{16} - 155 x^{15} - 1063 x^{14} + 2102 x^{13} + 3638 x^{12} + \cdots - 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 84887 \nu^{18} - 600421 \nu^{17} - 608814 \nu^{16} + 12767427 \nu^{15} - 12130081 \nu^{14} + \cdots + 140116224 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 167835 \nu^{18} + 1130893 \nu^{17} + 1458386 \nu^{16} - 24189791 \nu^{15} + 18019113 \nu^{14} + \cdots - 242037248 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59597 \nu^{18} - 377991 \nu^{17} - 582202 \nu^{16} + 7869457 \nu^{15} - 4215819 \nu^{14} + \cdots + 36198400 ) / 273632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70700 \nu^{18} - 435303 \nu^{17} - 802191 \nu^{16} + 9386498 \nu^{15} - 3089043 \nu^{14} + \cdots + 49016416 ) / 273632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 291351 \nu^{18} - 1801069 \nu^{17} - 3260070 \nu^{16} + 38751683 \nu^{15} - 13681577 \nu^{14} + \cdots + 198254720 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 462831 \nu^{18} + 2912897 \nu^{17} + 4870306 \nu^{16} - 62169699 \nu^{15} + 28265325 \nu^{14} + \cdots - 392351616 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 875879 \nu^{18} + 5546237 \nu^{17} + 8967446 \nu^{16} - 117606115 \nu^{15} + 57439577 \nu^{14} + \cdots - 690368000 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 230885 \nu^{18} + 1446907 \nu^{17} + 2451878 \nu^{16} - 30855745 \nu^{15} + 13478607 \nu^{14} + \cdots - 183208928 ) / 273632 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 631187 \nu^{18} - 3959285 \nu^{17} - 6640122 \nu^{16} + 84107039 \nu^{15} - 37450113 \nu^{14} + \cdots + 471848960 ) / 547264 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 714591 \nu^{18} - 4540367 \nu^{17} - 7181360 \nu^{16} + 95853575 \nu^{15} - 49050115 \nu^{14} + \cdots + 554906624 ) / 547264 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1437733 \nu^{18} + 9140591 \nu^{17} + 14414026 \nu^{16} - 192912841 \nu^{15} + \cdots - 1109813248 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2184271 \nu^{18} - 13873425 \nu^{17} - 21953730 \nu^{16} + 292730483 \nu^{15} - 149360925 \nu^{14} + \cdots + 1663557376 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1192007 \nu^{18} + 7553417 \nu^{17} + 12188418 \nu^{16} - 160100107 \nu^{15} + 78081077 \nu^{14} + \cdots - 910055296 ) / 547264 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 303034 \nu^{18} - 1919603 \nu^{17} - 3098231 \nu^{16} + 40697850 \nu^{15} - 19946419 \nu^{14} + \cdots + 240354096 ) / 136816 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 2488971 \nu^{18} + 15908849 \nu^{17} + 24515254 \nu^{16} - 335217719 \nu^{15} + \cdots - 1977241856 ) / 1094528 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 4826433 \nu^{18} + 30633083 \nu^{17} + 48979834 \nu^{16} - 648833253 \nu^{15} + \cdots - 3845743104 ) / 1094528 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{7} - \beta_{6} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} + \beta_{13} - \beta_{12} - \beta_{8} - \beta_{3} + 8\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{13} + 8 \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} + 9 \beta_{7} - 9 \beta_{6} - \beta_{3} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{18} - 10 \beta_{17} - 2 \beta_{16} - \beta_{15} + 11 \beta_{13} - 11 \beta_{12} - \beta_{11} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{18} - 2 \beta_{16} - \beta_{15} - \beta_{14} + 59 \beta_{13} + 55 \beta_{12} - 13 \beta_{11} + \cdots + 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{18} - 79 \beta_{17} - 29 \beta_{16} - 14 \beta_{15} + 3 \beta_{14} + 96 \beta_{13} + \cdots + 582 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 32 \beta_{18} - 3 \beta_{17} - 31 \beta_{16} - 17 \beta_{15} - 14 \beta_{14} + 437 \beta_{13} + \cdots + 632 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 139 \beta_{18} - 588 \beta_{17} - 295 \beta_{16} - 140 \beta_{15} + 51 \beta_{14} + 786 \beta_{13} + \cdots + 3913 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 343 \beta_{18} - 67 \beta_{17} - 329 \beta_{16} - 195 \beta_{15} - 135 \beta_{14} + 3264 \beta_{13} + \cdots + 4914 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1193 \beta_{18} - 4311 \beta_{17} - 2601 \beta_{16} - 1233 \beta_{15} + 576 \beta_{14} + 6279 \beta_{13} + \cdots + 26971 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3079 \beta_{18} - 959 \beta_{17} - 2995 \beta_{16} - 1899 \beta_{15} - 1112 \beta_{14} + 24483 \beta_{13} + \cdots + 38275 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 9392 \beta_{18} - 31564 \beta_{17} - 21287 \beta_{16} - 10206 \beta_{15} + 5483 \beta_{14} + \cdots + 189338 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 24892 \beta_{18} - 11231 \beta_{17} - 25210 \beta_{16} - 16971 \beta_{15} - 8339 \beta_{14} + \cdots + 298402 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 69392 \beta_{18} - 231796 \beta_{17} - 166727 \beta_{16} - 81584 \beta_{15} + 47757 \beta_{14} + \cdots + 1348673 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 186735 \beta_{18} - 117362 \beta_{17} - 202931 \beta_{16} - 144150 \beta_{15} - 58142 \beta_{14} + \cdots + 2325322 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 484726 \beta_{18} - 1709479 \beta_{17} - 1269871 \beta_{16} - 638232 \beta_{15} + 395398 \beta_{14} + \cdots + 9722877 \) 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
−2.58011
−2.45371
−1.71773
−1.46137
−1.43644
−1.08225
−0.762223
−0.338670
0.629607
0.660518
0.693824
1.11967
1.49304
1.76634
2.17878
2.28974
2.47288
2.71989
2.80821
−2.58011 1.25240 4.65697 0.123243 −3.23133 4.27162 −6.85527 −1.43149 −0.317980
1.2 −2.45371 −1.68061 4.02071 −3.53320 4.12374 1.81968 −4.95824 −0.175547 8.66947
1.3 −1.71773 3.19204 0.950599 −0.199483 −5.48306 3.17444 1.80259 7.18910 0.342658
1.4 −1.46137 1.02561 0.135601 3.29616 −1.49880 −1.05535 2.72458 −1.94812 −4.81691
1.5 −1.43644 −3.33808 0.0633468 3.15458 4.79494 1.63687 2.78188 8.14280 −4.53134
1.6 −1.08225 −1.57543 −0.828743 −0.128022 1.70500 4.63180 3.06140 −0.518020 0.138552
1.7 −0.762223 1.04338 −1.41902 −4.09702 −0.795286 −4.89750 2.60605 −1.91136 3.12284
1.8 −0.338670 2.48078 −1.88530 −1.11324 −0.840166 −0.256649 1.31583 3.15428 0.377022
1.9 0.629607 −2.34536 −1.60360 −0.481661 −1.47666 −2.22070 −2.26885 2.50073 −0.303257
1.10 0.660518 −0.662050 −1.56372 −3.20563 −0.437296 0.803744 −2.35390 −2.56169 −2.11738
1.11 0.693824 2.06808 −1.51861 3.85866 1.43488 0.209786 −2.44129 1.27695 2.67723
1.12 1.11967 0.697317 −0.746338 2.24336 0.780765 4.15002 −3.07499 −2.51375 2.51183
1.13 1.49304 −2.42026 0.229162 3.64648 −3.61354 −2.99455 −2.64393 2.85767 5.44433
1.14 1.76634 2.72683 1.11997 0.601503 4.81651 −0.644194 −1.55444 4.43560 1.06246
1.15 2.17878 2.20758 2.74709 −4.20488 4.80983 4.68335 1.62774 1.87339 −9.16151
1.16 2.28974 −0.237496 3.24290 2.59482 −0.543804 3.53894 2.84591 −2.94360 5.94145
1.17 2.47288 2.26416 4.11514 −0.926336 5.59900 0.227646 5.23050 2.12643 −2.29072
1.18 2.71989 −2.74549 5.39781 −2.38124 −7.46742 2.43950 9.24167 4.53769 −6.47672
1.19 2.80821 −0.953388 5.88603 2.75192 −2.67731 −3.51846 10.9128 −2.09105 7.72797
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(73\) \(1\)

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 803.2.a.f ✓ 19
3.b odd 2 1 7227.2.a.ba 19
11.b odd 2 1 8833.2.a.k 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
803.2.a.f ✓ 19 1.a even 1 1 trivial
7227.2.a.ba 19 3.b odd 2 1
8833.2.a.k 19 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{19} - 7 T_{2}^{18} - 6 T_{2}^{17} + 141 T_{2}^{16} - 155 T_{2}^{15} - 1063 T_{2}^{14} + \cdots - 512 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(803))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 7 T^{18} + \cdots - 512 \) Copy content Toggle raw display
$3$ \( T^{19} - 3 T^{18} + \cdots + 4307 \) Copy content Toggle raw display
$5$ \( T^{19} - 2 T^{18} + \cdots - 1024 \) Copy content Toggle raw display
$7$ \( T^{19} - 16 T^{18} + \cdots + 24091 \) Copy content Toggle raw display
$11$ \( (T - 1)^{19} \) Copy content Toggle raw display
$13$ \( T^{19} - 30 T^{18} + \cdots - 2330703 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots - 123261791 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 51550886400 \) Copy content Toggle raw display
$23$ \( T^{19} + 2 T^{18} + \cdots - 65891993 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 50058401874 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 270042112 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 651642399846 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots - 54005334016 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 322886400 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 208410214400 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 91800970752 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots - 6274807572992 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 21767472691200 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots - 12683222203961 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots + 800600912359176 \) Copy content Toggle raw display
$73$ \( (T + 1)^{19} \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 58\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 179646809381811 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 10\!\cdots\!82 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 75\!\cdots\!02 \) Copy content Toggle raw display
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