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Properties

Label 5635.2.a.bn
Level $5635$
Weight $2$
Character orbit 5635.a
Self dual yes
Analytic conductor $44.996$
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] = [5635,2,Mod(1,5635)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5635, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5635.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5635 = 5 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5635.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: \(44.9957015390\)
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: \( x^{17} - 5 x^{16} - 16 x^{15} + 110 x^{14} + 59 x^{13} - 954 x^{12} + 316 x^{11} + 4142 x^{10} + \cdots + 156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{6} - \beta_{4}) q^{6} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + (\beta_{12} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{6} - \beta_{4}) q^{6} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + (\beta_{12} + \beta_1 + 1) q^{9} - \beta_1 q^{10} + \beta_{13} q^{11} + ( - \beta_{10} + \beta_{6} + \cdots + \beta_{3}) q^{12}+ \cdots + (\beta_{16} + \beta_{15} + 2 \beta_{14} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 5 q^{2} + 2 q^{3} + 23 q^{4} - 17 q^{5} - 4 q^{6} + 15 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 5 q^{2} + 2 q^{3} + 23 q^{4} - 17 q^{5} - 4 q^{6} + 15 q^{8} + 25 q^{9} - 5 q^{10} + 7 q^{11} - 6 q^{12} - q^{13} - 2 q^{15} + 27 q^{16} + 3 q^{17} + 53 q^{18} - 7 q^{19} - 23 q^{20} + 15 q^{22} - 17 q^{23} - 4 q^{24} + 17 q^{25} - 6 q^{26} + 5 q^{27} + 29 q^{29} + 4 q^{30} - 19 q^{31} + 50 q^{32} + 16 q^{33} - 2 q^{34} + 35 q^{36} + 14 q^{37} + 22 q^{38} + 4 q^{39} - 15 q^{40} - 18 q^{41} - 12 q^{43} + 45 q^{44} - 25 q^{45} - 5 q^{46} + 23 q^{47} - 52 q^{48} + 5 q^{50} + 7 q^{51} + 31 q^{52} + 56 q^{53} + 16 q^{54} - 7 q^{55} + 12 q^{57} + 12 q^{58} - 2 q^{59} + 6 q^{60} - 26 q^{61} - 6 q^{62} + 49 q^{64} + q^{65} + 69 q^{66} + 7 q^{67} + 9 q^{68} - 2 q^{69} + 37 q^{71} + 97 q^{72} + 19 q^{73} + 20 q^{74} + 2 q^{75} - 53 q^{76} - 25 q^{78} + 12 q^{79} - 27 q^{80} + 49 q^{81} + 8 q^{82} - 11 q^{83} - 3 q^{85} - 8 q^{86} + 24 q^{87} + 24 q^{88} - 2 q^{89} - 53 q^{90} - 23 q^{92} + 53 q^{93} + 2 q^{94} + 7 q^{95} - 25 q^{96} + 18 q^{97} - 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 5 x^{16} - 16 x^{15} + 110 x^{14} + 59 x^{13} - 954 x^{12} + 316 x^{11} + 4142 x^{10} + \cdots + 156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7420298 \nu^{16} + 25097342 \nu^{15} - 293202561 \nu^{14} - 599183355 \nu^{13} + \cdots + 5486569268 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33001802 \nu^{16} - 147837751 \nu^{15} - 563373121 \nu^{14} + 3170081064 \nu^{13} + \cdots - 14039402550 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 54778534 \nu^{16} + 199575135 \nu^{15} + 1122213574 \nu^{14} - 4399384786 \nu^{13} + \cdots + 6644135756 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 70060368 \nu^{16} - 215294283 \nu^{15} - 1564298890 \nu^{14} + 4760178306 \nu^{13} + \cdots + 2710736326 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 143271287 \nu^{16} - 466817293 \nu^{15} - 3136671528 \nu^{14} + 10553760080 \nu^{13} + \cdots - 21550281850 ) / 2211574786 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 92478741 \nu^{16} + 443789465 \nu^{15} + 1603221953 \nu^{14} - 9916818919 \nu^{13} + \cdots + 36875354569 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 92478741 \nu^{16} - 443789465 \nu^{15} - 1603221953 \nu^{14} + 9916818919 \nu^{13} + \cdots - 30240630211 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 220471119 \nu^{16} + 699502767 \nu^{15} + 4764149244 \nu^{14} - 15451644370 \nu^{13} + \cdots + 25446564468 ) / 2211574786 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 128081555 \nu^{16} + 433415142 \nu^{15} + 2793240109 \nu^{14} - 9813161354 \nu^{13} + \cdots + 16702209396 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 131895688 \nu^{16} - 538348227 \nu^{15} - 2576320581 \nu^{14} + 12054347113 \nu^{13} + \cdots - 34121104944 ) / 1105787393 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 280685567 \nu^{16} + 899564693 \nu^{15} + 6087877698 \nu^{14} - 19845993818 \nu^{13} + \cdots + 21809070440 ) / 2211574786 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 416197905 \nu^{16} + 1509145007 \nu^{15} + 8651411102 \nu^{14} - 33695518162 \nu^{13} + \cdots + 43620404902 ) / 2211574786 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 223454874 \nu^{16} + 925011395 \nu^{15} + 4297970956 \nu^{14} - 20702886613 \nu^{13} + \cdots + 60589414385 ) / 1105787393 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + \beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{5} + 10\beta_{3} + 10\beta_{2} + 29\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + 10 \beta_{10} + 9 \beta_{9} - \beta_{8} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{16} + 11 \beta_{14} - 3 \beta_{13} - 13 \beta_{12} - 11 \beta_{11} + 14 \beta_{10} - \beta_{9} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{15} - 28 \beta_{14} - 18 \beta_{13} - 15 \beta_{12} + 14 \beta_{11} + 80 \beta_{10} + \cdots + 561 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 105 \beta_{16} + \beta_{15} + 89 \beta_{14} - 51 \beta_{13} - 125 \beta_{12} - 90 \beta_{11} + \cdots + 306 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2 \beta_{16} - 31 \beta_{15} - 274 \beta_{14} - 211 \beta_{13} - 160 \beta_{12} + 137 \beta_{11} + \cdots + 3751 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 820 \beta_{16} + 14 \beta_{15} + 642 \beta_{14} - 587 \beta_{13} - 1069 \beta_{12} - 658 \beta_{11} + \cdots + 2842 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 54 \beta_{16} - 322 \beta_{15} - 2321 \beta_{14} - 2073 \beta_{13} - 1488 \beta_{12} + 1161 \beta_{11} + \cdots + 25814 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6096 \beta_{16} + 128 \beta_{15} + 4374 \beta_{14} - 5741 \beta_{13} - 8620 \beta_{12} - 4553 \beta_{11} + \cdots + 24689 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 869 \beta_{16} - 2822 \beta_{15} - 18247 \beta_{14} - 18602 \beta_{13} - 12886 \beta_{12} + \cdots + 181001 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 44358 \beta_{16} + 995 \beta_{15} + 28801 \beta_{14} - 51528 \beta_{13} - 67242 \beta_{12} + \cdots + 206147 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11002 \beta_{16} - 22509 \beta_{15} - 137316 \beta_{14} - 158461 \beta_{13} - 107001 \beta_{12} + \cdots + 1285813 \) 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
−2.57122
−2.30596
−2.01415
−1.63621
−1.51514
−1.00397
−0.402150
0.209722
0.557347
0.667648
1.30340
1.73054
1.80786
2.15093
2.52168
2.72632
2.77336
−2.57122 −0.983035 4.61119 −1.00000 2.52760 0 −6.71396 −2.03364 2.57122
1.2 −2.30596 1.10059 3.31745 −1.00000 −2.53792 0 −3.03798 −1.78870 2.30596
1.3 −2.01415 −1.34509 2.05681 −1.00000 2.70921 0 −0.114431 −1.19074 2.01415
1.4 −1.63621 0.856023 0.677193 −1.00000 −1.40064 0 2.16439 −2.26723 1.63621
1.5 −1.51514 3.03647 0.295642 −1.00000 −4.60066 0 2.58234 6.22012 1.51514
1.6 −1.00397 0.830360 −0.992052 −1.00000 −0.833654 0 3.00392 −2.31050 1.00397
1.7 −0.402150 −2.89693 −1.83828 −1.00000 1.16500 0 1.54356 5.39221 0.402150
1.8 0.209722 1.57267 −1.95602 −1.00000 0.329825 0 −0.829665 −0.526694 −0.209722
1.9 0.557347 1.95515 −1.68936 −1.00000 1.08970 0 −2.05626 0.822626 −0.557347
1.10 0.667648 −2.66501 −1.55425 −1.00000 −1.77929 0 −2.37299 4.10225 −0.667648
1.11 1.30340 0.729037 −0.301150 −1.00000 0.950226 0 −2.99932 −2.46851 −1.30340
1.12 1.73054 3.25621 0.994760 −1.00000 5.63499 0 −1.73961 7.60289 −1.73054
1.13 1.80786 −1.59892 1.26837 −1.00000 −2.89063 0 −1.32268 −0.443446 −1.80786
1.14 2.15093 −2.55816 2.62650 −1.00000 −5.50243 0 1.34755 3.54420 −2.15093
1.15 2.52168 3.22288 4.35887 −1.00000 8.12707 0 5.94831 7.38694 −2.52168
1.16 2.72632 0.447183 5.43281 −1.00000 1.21916 0 9.35892 −2.80003 −2.72632
1.17 2.77336 −2.95943 5.69151 −1.00000 −8.20756 0 10.2379 5.75824 −2.77336
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 5635.2.a.bn 17
7.b odd 2 1 5635.2.a.bm 17
7.d odd 6 2 805.2.i.f ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.i.f ✓ 34 7.d odd 6 2
5635.2.a.bm 17 7.b odd 2 1
5635.2.a.bn 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5635))\):

\( T_{2}^{17} - 5 T_{2}^{16} - 16 T_{2}^{15} + 110 T_{2}^{14} + 59 T_{2}^{13} - 954 T_{2}^{12} + 316 T_{2}^{11} + \cdots + 156 \) Copy content Toggle raw display
\( T_{3}^{17} - 2 T_{3}^{16} - 36 T_{3}^{15} + 69 T_{3}^{14} + 513 T_{3}^{13} - 964 T_{3}^{12} + \cdots + 3088 \) Copy content Toggle raw display
\( T_{11}^{17} - 7 T_{11}^{16} - 93 T_{11}^{15} + 704 T_{11}^{14} + 3255 T_{11}^{13} - 28182 T_{11}^{12} + \cdots - 3202272 \) Copy content Toggle raw display
\( T_{17}^{17} - 3 T_{17}^{16} - 185 T_{17}^{15} + 558 T_{17}^{14} + 13543 T_{17}^{13} - 42793 T_{17}^{12} + \cdots - 1351715880 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 5 T^{16} + \cdots + 156 \) Copy content Toggle raw display
$3$ \( T^{17} - 2 T^{16} + \cdots + 3088 \) Copy content Toggle raw display
$5$ \( (T + 1)^{17} \) Copy content Toggle raw display
$7$ \( T^{17} \) Copy content Toggle raw display
$11$ \( T^{17} - 7 T^{16} + \cdots - 3202272 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 354340736 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 1351715880 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 9955995968 \) Copy content Toggle raw display
$23$ \( (T + 1)^{17} \) Copy content Toggle raw display
$29$ \( T^{17} - 29 T^{16} + \cdots + 4035078 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 324536864 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 2901304694 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 1020817620 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 25252994501 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 113215488 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 77020535496 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 368034379692 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 271625255680 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 107401701040 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 43335728890428 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 243734368 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 1119892535296 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 29768910812340 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 34\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 9346392143345 \) Copy content Toggle raw display
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