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Properties

Label 813.2.a.f
Level $813$
Weight $2$
Character orbit 813.a
Self dual yes
Analytic conductor $6.492$
Analytic rank $0$
Dimension $17$
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] = [813,2,Mod(1,813)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(813, 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("813.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 813 = 3 \cdot 271 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 813.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: \(6.49183768433\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - x^{16} - 27 x^{15} + 25 x^{14} + 293 x^{13} - 248 x^{12} - 1633 x^{11} + 1240 x^{10} + 4954 x^{9} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{9} q^{5} - \beta_1 q^{6} + ( - \beta_{16} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_{9} q^{5} - \beta_1 q^{6} + ( - \beta_{16} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + (\beta_{16} - \beta_{14} + \cdots + \beta_{6}) q^{10}+ \cdots - \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - q^{2} + 17 q^{3} + 21 q^{4} + 4 q^{5} - q^{6} + 16 q^{7} - 3 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - q^{2} + 17 q^{3} + 21 q^{4} + 4 q^{5} - q^{6} + 16 q^{7} - 3 q^{8} + 17 q^{9} + 5 q^{11} + 21 q^{12} + 17 q^{13} - 3 q^{14} + 4 q^{15} + 33 q^{16} - q^{18} + 31 q^{19} + 3 q^{20} + 16 q^{21} - 8 q^{22} + 6 q^{23} - 3 q^{24} + 37 q^{25} - 13 q^{26} + 17 q^{27} + 24 q^{28} - 4 q^{29} + 24 q^{31} - 12 q^{32} + 5 q^{33} - 5 q^{34} - 3 q^{35} + 21 q^{36} + 14 q^{37} + 4 q^{38} + 17 q^{39} - 13 q^{40} - 18 q^{41} - 3 q^{42} + 29 q^{43} - 35 q^{44} + 4 q^{45} - 25 q^{46} + 11 q^{47} + 33 q^{48} + 49 q^{49} - 51 q^{50} + 28 q^{52} - q^{53} - q^{54} - 20 q^{55} - 53 q^{56} + 31 q^{57} + 16 q^{58} - 6 q^{59} + 3 q^{60} + 5 q^{61} - 28 q^{62} + 16 q^{63} + 11 q^{64} - 26 q^{65} - 8 q^{66} + 57 q^{67} - 40 q^{68} + 6 q^{69} - 19 q^{70} - 5 q^{71} - 3 q^{72} + 40 q^{73} - 40 q^{74} + 37 q^{75} + 29 q^{76} - 13 q^{78} - 2 q^{79} + 30 q^{80} + 17 q^{81} - 7 q^{82} + 31 q^{83} + 24 q^{84} - 13 q^{85} - 35 q^{86} - 4 q^{87} - 26 q^{88} - 18 q^{89} + 30 q^{91} + 7 q^{92} + 24 q^{93} - 26 q^{94} + 9 q^{95} - 12 q^{96} + 42 q^{97} - 14 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - x^{16} - 27 x^{15} + 25 x^{14} + 293 x^{13} - 248 x^{12} - 1633 x^{11} + 1240 x^{10} + 4954 x^{9} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1893168 \nu^{16} - 98962319 \nu^{15} - 81881109 \nu^{14} + 2787191522 \nu^{13} + \cdots + 28515865015 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8675814 \nu^{16} - 200963028 \nu^{15} + 581960257 \nu^{14} + 5006522277 \nu^{13} + \cdots - 36182805798 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 80873383 \nu^{16} + 224652302 \nu^{15} + 2194290879 \nu^{14} - 5619520146 \nu^{13} + \cdots - 35068410979 ) / 12513613558 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 125679796 \nu^{16} + 258477110 \nu^{15} + 3352811842 \nu^{14} - 6386163447 \nu^{13} + \cdots - 17409675180 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 331314501 \nu^{16} + 526926640 \nu^{15} + 8485350097 \nu^{14} - 13638971502 \nu^{13} + \cdots + 22651340447 ) / 12513613558 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 320061638 \nu^{16} + 468453434 \nu^{15} + 8420103566 \nu^{14} - 11676576833 \nu^{13} + \cdots - 12436302586 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 690320369 \nu^{16} + 1039656242 \nu^{15} + 18349667605 \nu^{14} - 25689834950 \nu^{13} + \cdots - 43542892359 ) / 12513613558 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 347401356 \nu^{16} - 271093502 \nu^{15} - 9182994013 \nu^{14} + 6583535235 \nu^{13} + \cdots + 32199726138 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 435459821 \nu^{16} + 359300070 \nu^{15} + 11949618521 \nu^{14} - 9001249609 \nu^{13} + \cdots + 3809904185 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 525853844 \nu^{16} - 218060929 \nu^{15} - 14428140844 \nu^{14} + 5318594883 \nu^{13} + \cdots + 52623031145 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 574198399 \nu^{16} - 458285493 \nu^{15} - 15416962028 \nu^{14} + 10832362133 \nu^{13} + \cdots + 29526733241 ) / 6256806779 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1244792961 \nu^{16} - 1484315254 \nu^{15} - 33515767465 \nu^{14} + 38195011146 \nu^{13} + \cdots + 28024382303 ) / 12513613558 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 631900545 \nu^{16} - 612514304 \nu^{15} - 17081746633 \nu^{14} + 15313804571 \nu^{13} + \cdots + 36055330145 ) / 6256806779 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{16} + \beta_{14} + \beta_{13} - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + \cdots + 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{16} - 10 \beta_{15} + 11 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 10 \beta_{11} + \cdots + 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{16} + 15 \beta_{14} + 11 \beta_{13} - 13 \beta_{11} + \beta_{10} + 15 \beta_{9} - 12 \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 79 \beta_{16} - 82 \beta_{15} + 98 \beta_{14} - 30 \beta_{13} - 94 \beta_{12} - 83 \beta_{11} + \cdots + 717 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 83 \beta_{16} - 4 \beta_{15} + 159 \beta_{14} + 87 \beta_{13} - \beta_{12} - 127 \beta_{11} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 580 \beta_{16} - 632 \beta_{15} + 815 \beta_{14} - 319 \beta_{13} - 738 \beta_{12} - 657 \beta_{11} + \cdots + 4885 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 520 \beta_{16} - 81 \beta_{15} + 1468 \beta_{14} + 599 \beta_{13} - 19 \beta_{12} - 1116 \beta_{11} + \cdots - 118 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4134 \beta_{16} - 4742 \beta_{15} + 6556 \beta_{14} - 2953 \beta_{13} - 5579 \beta_{12} + \cdots + 33898 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2800 \beta_{16} - 1049 \beta_{15} + 12626 \beta_{14} + 3789 \beta_{13} - 241 \beta_{12} - 9298 \beta_{11} + \cdots - 911 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 29081 \beta_{16} - 35122 \beta_{15} + 51708 \beta_{14} - 25421 \beta_{13} - 41380 \beta_{12} + \cdots + 238491 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 12270 \beta_{16} - 11137 \beta_{15} + 104184 \beta_{14} + 22351 \beta_{13} - 2568 \beta_{12} + \cdots - 6248 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 203424 \beta_{16} - 258539 \beta_{15} + 402553 \beta_{14} - 209538 \beta_{13} - 303904 \beta_{12} + \cdots + 1695742 \) 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
2.73973
2.49540
2.43164
2.07811
1.51316
1.28485
0.881418
0.337403
0.0103971
−0.152174
−0.700666
−1.04136
−1.38944
−2.16221
−2.24889
−2.37665
−2.70073
−2.73973 1.00000 5.50610 4.16677 −2.73973 0.562094 −9.60576 1.00000 −11.4158
1.2 −2.49540 1.00000 4.22701 −1.66349 −2.49540 5.11222 −5.55727 1.00000 4.15107
1.3 −2.43164 1.00000 3.91287 −0.932588 −2.43164 0.704758 −4.65142 1.00000 2.26772
1.4 −2.07811 1.00000 2.31856 −4.28232 −2.07811 −1.83220 −0.662005 1.00000 8.89914
1.5 −1.51316 1.00000 0.289666 1.21193 −1.51316 3.83606 2.58802 1.00000 −1.83385
1.6 −1.28485 1.00000 −0.349159 2.54122 −1.28485 −0.326384 3.01832 1.00000 −3.26509
1.7 −0.881418 1.00000 −1.22310 3.32686 −0.881418 −3.69292 2.84090 1.00000 −2.93236
1.8 −0.337403 1.00000 −1.88616 −2.59646 −0.337403 3.05187 1.31120 1.00000 0.876055
1.9 −0.0103971 1.00000 −1.99989 −2.33694 −0.0103971 −5.17129 0.0415871 1.00000 0.0242973
1.10 0.152174 1.00000 −1.97684 3.95654 0.152174 4.85804 −0.605173 1.00000 0.602084
1.11 0.700666 1.00000 −1.50907 −3.24695 0.700666 3.56641 −2.45868 1.00000 −2.27503
1.12 1.04136 1.00000 −0.915576 1.81827 1.04136 1.78022 −3.03615 1.00000 1.89347
1.13 1.38944 1.00000 −0.0694584 1.68401 1.38944 −0.252381 −2.87539 1.00000 2.33983
1.14 2.16221 1.00000 2.67517 1.34505 2.16221 4.16282 1.45986 1.00000 2.90828
1.15 2.24889 1.00000 3.05750 −2.97568 2.24889 2.94943 2.37820 1.00000 −6.69197
1.16 2.37665 1.00000 3.64844 2.79398 2.37665 −2.68520 3.91776 1.00000 6.64029
1.17 2.70073 1.00000 5.29393 −0.810209 2.70073 −0.623539 8.89601 1.00000 −2.18815
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(271\) \(1\)

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 813.2.a.f ✓ 17
3.b odd 2 1 2439.2.a.i 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
813.2.a.f ✓ 17 1.a even 1 1 trivial
2439.2.a.i 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} + T_{2}^{16} - 27 T_{2}^{15} - 25 T_{2}^{14} + 293 T_{2}^{13} + 248 T_{2}^{12} - 1633 T_{2}^{11} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(813))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + T^{16} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} - 4 T^{16} + \cdots - 613352 \) Copy content Toggle raw display
$7$ \( T^{17} - 16 T^{16} + \cdots + 43328 \) Copy content Toggle raw display
$11$ \( T^{17} - 5 T^{16} + \cdots - 91252 \) Copy content Toggle raw display
$13$ \( T^{17} - 17 T^{16} + \cdots + 4284416 \) Copy content Toggle raw display
$17$ \( T^{17} - 157 T^{15} + \cdots - 3281192 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 196481024 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 7986774016 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 3189192704 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 100140439552 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 1985195692334 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 185961572272 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 52997820416 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 345112576 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 58120749056 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 567748918934 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 283661199232 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 364814434304 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 22010713358336 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 459130371412892 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 841976137008784 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 39894810935296 \) Copy content Toggle raw display
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