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Properties

Label 731.2.a.e
Level $731$
Weight $2$
Character orbit 731.a
Self dual yes
Analytic conductor $5.837$
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] = [731,2,Mod(1,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, 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("731.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.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: \(5.83706438776\)
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: \( x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{16} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{14} + 1) q^{5} - \beta_{17} q^{6} + \beta_{10} q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{16} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{14} + 1) q^{5} - \beta_{17} q^{6} + \beta_{10} q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{5} + 1) q^{9} + (\beta_{18} + \beta_{17} + \cdots + \beta_1) q^{10}+ \cdots + (\beta_{18} + 3 \beta_{17} + \cdots + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9} - 2 q^{10} + 4 q^{11} + 9 q^{12} + 14 q^{13} + 5 q^{14} - 7 q^{15} + 32 q^{16} + 19 q^{17} + 12 q^{18} + 12 q^{19} + 23 q^{20} + 16 q^{21} + 36 q^{22} - q^{23} - 13 q^{24} + 30 q^{25} - 21 q^{26} + 8 q^{27} + 5 q^{28} + 41 q^{29} - 26 q^{30} - 8 q^{31} - 20 q^{32} - 14 q^{33} + 2 q^{34} + 3 q^{35} - 5 q^{36} + 50 q^{37} - 29 q^{38} + 17 q^{39} - 15 q^{40} + 6 q^{41} - q^{42} - 19 q^{43} + 16 q^{44} + 24 q^{45} + 38 q^{46} - 21 q^{47} - 2 q^{48} + 46 q^{49} - 36 q^{50} + 5 q^{51} + 39 q^{52} - 9 q^{53} + 53 q^{54} + 10 q^{55} - 12 q^{56} - 5 q^{57} - 45 q^{58} - 4 q^{59} - 7 q^{60} + 68 q^{61} - 25 q^{62} + 61 q^{63} - 14 q^{64} + 22 q^{65} - 17 q^{66} + 26 q^{68} - 9 q^{69} - 37 q^{70} + 23 q^{71} - 4 q^{72} - q^{73} - 30 q^{74} - 25 q^{75} + 47 q^{76} - 19 q^{77} + 12 q^{78} + 16 q^{79} + 28 q^{80} - 21 q^{81} - 13 q^{82} - 32 q^{83} - 47 q^{84} + 11 q^{85} - 2 q^{86} - 8 q^{87} + 108 q^{88} + 11 q^{89} + 5 q^{90} + 52 q^{91} - 23 q^{92} - 23 q^{93} + 47 q^{94} - 25 q^{95} - 103 q^{96} + 36 q^{97} - 100 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + \cdots + 64 \) : 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
\(\beta_{4}\)\(=\) \( ( - 14507 \nu^{18} + 35963 \nu^{17} + 407182 \nu^{16} - 1107200 \nu^{15} - 4547261 \nu^{14} + \cdots - 1655744 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19398 \nu^{18} + 88717 \nu^{17} + 621578 \nu^{16} - 2632114 \nu^{15} - 7988484 \nu^{14} + \cdots - 2849472 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22069 \nu^{18} - 55896 \nu^{17} - 639438 \nu^{16} + 1743250 \nu^{15} + 7321881 \nu^{14} + \cdots + 8403120 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{18} + 2 \nu^{17} + 30 \nu^{16} - 62 \nu^{15} - 365 \nu^{14} + 786 \nu^{13} + 2295 \nu^{12} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46849 \nu^{18} + 114064 \nu^{17} - 1249518 \nu^{16} - 3525550 \nu^{15} + 13856033 \nu^{14} + \cdots - 16602464 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 46276 \nu^{18} + 139217 \nu^{17} + 1428222 \nu^{16} - 4360874 \nu^{15} - 17616370 \nu^{14} + \cdots - 9733264 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27406 \nu^{18} - 6558 \nu^{17} - 821917 \nu^{16} + 308350 \nu^{15} + 10057360 \nu^{14} + \cdots + 1217600 ) / 500824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76183 \nu^{18} + 21343 \nu^{17} - 2186962 \nu^{16} - 589856 \nu^{15} + 25862933 \nu^{14} + \cdots - 6465728 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 84137 \nu^{18} + 15063 \nu^{17} - 2382182 \nu^{16} - 513220 \nu^{15} + 27801095 \nu^{14} + \cdots - 12742608 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 134774 \nu^{18} + 35561 \nu^{17} - 3930504 \nu^{16} - 732498 \nu^{15} + 47051744 \nu^{14} + \cdots - 2665232 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 303511 \nu^{18} + 415338 \nu^{17} + 8739918 \nu^{16} - 12671162 \nu^{15} - 101762115 \nu^{14} + \cdots - 37885984 ) / 2003296 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 152090 \nu^{18} - 193703 \nu^{17} - 4486878 \nu^{16} + 6150318 \nu^{15} + 53575804 \nu^{14} + \cdots + 20421328 ) / 1001648 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 305305 \nu^{18} + 224774 \nu^{17} + 8944762 \nu^{16} - 7408198 \nu^{15} - 106577725 \nu^{14} + \cdots - 20921664 ) / 2003296 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 96459 \nu^{18} + 53597 \nu^{17} - 2880178 \nu^{16} - 1214650 \nu^{15} + 35318239 \nu^{14} + \cdots - 4884880 ) / 500824 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 420357 \nu^{18} - 380718 \nu^{17} - 12473386 \nu^{16} + 12686510 \nu^{15} + 150115353 \nu^{14} + \cdots + 52644032 ) / 2003296 \) 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
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{11} + \beta_{9} - 2\beta_{6} - \beta_{4} + 6\beta_{2} - \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - \beta_{16} + \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{18} + \beta_{17} - \beta_{16} + 11 \beta_{15} - \beta_{14} - 3 \beta_{13} - 9 \beta_{11} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{18} - 13 \beta_{17} - 12 \beta_{16} - \beta_{15} + 13 \beta_{14} + 12 \beta_{13} + \beta_{12} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30 \beta_{18} + 15 \beta_{17} - 15 \beta_{16} + 94 \beta_{15} - 17 \beta_{14} - 43 \beta_{13} + \cdots + 607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 33 \beta_{18} - 123 \beta_{17} - 110 \beta_{16} - 19 \beta_{15} + 124 \beta_{14} + 108 \beta_{13} + \cdots - 231 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 320 \beta_{18} + 163 \beta_{17} - 158 \beta_{16} + 737 \beta_{15} - 199 \beta_{14} - 442 \beta_{13} + \cdots + 3967 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 374 \beta_{18} - 1036 \beta_{17} - 911 \beta_{16} - 239 \beta_{15} + 1054 \beta_{14} + 889 \beta_{13} + \cdots - 1912 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2976 \beta_{18} + 1555 \beta_{17} - 1450 \beta_{16} + 5561 \beta_{15} - 1983 \beta_{14} + \cdots + 26575 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3640 \beta_{18} - 8269 \beta_{17} - 7143 \beta_{16} - 2511 \beta_{15} + 8481 \beta_{14} + 7089 \beta_{13} + \cdots - 15060 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 25750 \beta_{18} + 13820 \beta_{17} - 12388 \beta_{16} + 41181 \beta_{15} - 18074 \beta_{14} + \cdots + 181555 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 32737 \beta_{18} - 64208 \beta_{17} - 54118 \beta_{16} - 23874 \beta_{15} + 66307 \beta_{14} + \cdots - 115867 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 213323 \beta_{18} + 117466 \beta_{17} - 101288 \beta_{16} + 302197 \beta_{15} - 155829 \beta_{14} + \cdots + 1260042 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 281000 \beta_{18} - 491243 \beta_{17} - 400716 \beta_{16} - 213278 \beta_{15} + 510343 \beta_{14} + \cdots - 882149 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1717747 \beta_{18} + 968471 \beta_{17} - 803761 \beta_{16} + 2209007 \beta_{15} - 1294572 \beta_{14} + \cdots + 8855947 \) 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
−2.74066
−2.48303
−2.48141
−2.21579
−1.86082
−1.32959
−0.753901
−0.276381
−0.248120
0.148448
0.614204
1.03821
1.32390
1.49769
2.18694
2.20570
2.30539
2.40179
2.66743
−2.74066 1.82797 5.51123 3.35226 −5.00984 3.40276 −9.62308 0.341469 −9.18741
1.2 −2.48303 1.10293 4.16544 −3.73374 −2.73861 −2.89905 −5.37685 −1.78354 9.27098
1.3 −2.48141 −0.945917 4.15740 3.33601 2.34721 −4.81122 −5.35339 −2.10524 −8.27801
1.4 −2.21579 −2.91752 2.90974 −0.513978 6.46463 5.13398 −2.01580 5.51193 1.13887
1.5 −1.86082 2.21039 1.46266 1.15323 −4.11315 2.59124 0.999890 1.88583 −2.14595
1.6 −1.32959 −1.38064 −0.232191 −0.595657 1.83568 −3.72326 2.96790 −1.09384 0.791980
1.7 −0.753901 2.87509 −1.43163 3.13521 −2.16753 0.322864 2.58711 5.26614 −2.36364
1.8 −0.276381 2.26539 −1.92361 −3.42185 −0.626111 1.42090 1.08441 2.13201 0.945732
1.9 −0.248120 −2.62183 −1.93844 3.83673 0.650531 2.64030 0.977206 3.87401 −0.951972
1.10 0.148448 0.359282 −1.97796 2.35987 0.0533348 −1.55967 −0.590522 −2.87092 0.350319
1.11 0.614204 −2.75618 −1.62275 −1.26314 −1.69286 −3.83612 −2.22511 4.59652 −0.775829
1.12 1.03821 −1.17568 −0.922121 −2.88687 −1.22060 3.15156 −3.03377 −1.61778 −2.99717
1.13 1.32390 3.32287 −0.247283 −0.817922 4.39916 5.14151 −2.97518 8.04147 −1.08285
1.14 1.49769 1.88586 0.243066 2.23917 2.82443 1.15776 −2.63134 0.556465 3.35357
1.15 2.18694 −2.61834 2.78271 4.32234 −5.72616 −1.52237 1.71175 3.85571 9.45271
1.16 2.20570 2.69039 2.86513 0.920247 5.93420 −2.62293 1.90823 4.23819 2.02979
1.17 2.30539 −0.0652591 3.31484 2.38814 −0.150448 2.55322 3.03122 −2.99574 5.50561
1.18 2.40179 2.16240 3.76858 −1.65227 5.19362 −1.91106 4.24776 1.67596 −3.96840
1.19 2.66743 −1.22120 5.11519 −1.15779 −3.25748 2.36958 8.30956 −1.50866 −3.08834
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(43\) \(1\)

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 731.2.a.e ✓ 19
3.b odd 2 1 6579.2.a.t 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.a.e ✓ 19 1.a even 1 1 trivial
6579.2.a.t 19 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{19} - 2 T_{2}^{18} - 30 T_{2}^{17} + 62 T_{2}^{16} + 365 T_{2}^{15} - 786 T_{2}^{14} - 2295 T_{2}^{13} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 2 T^{18} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{19} - 5 T^{18} + \cdots + 2568 \) Copy content Toggle raw display
$5$ \( T^{19} - 11 T^{18} + \cdots + 173786 \) Copy content Toggle raw display
$7$ \( T^{19} - 7 T^{18} + \cdots - 14755776 \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots - 195305472 \) Copy content Toggle raw display
$13$ \( T^{19} - 14 T^{18} + \cdots + 4258102 \) Copy content Toggle raw display
$17$ \( (T - 1)^{19} \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots + 328708096 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots - 4303650816 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots + 385689600 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 1953564442624 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 52382209194 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 296642936832 \) Copy content Toggle raw display
$43$ \( (T + 1)^{19} \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 27160771792032 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 135698658 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 152606411509644 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 21\!\cdots\!26 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 27094271124876 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 155973305804 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 89\!\cdots\!14 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 39856288075776 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 287953675684608 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 442234497024 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 110824588220416 \) Copy content Toggle raw display
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