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Properties

Label 3645.2.a.g
Level $3645$
Weight $2$
Character orbit 3645.a
Self dual yes
Analytic conductor $29.105$
Analytic rank $1$
Dimension $15$
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] = [3645,2,Mod(1,3645)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3645, 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("3645.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3645 = 3^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3645.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: \(29.1054715368\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 15 x^{13} + 47 x^{12} + 84 x^{11} - 279 x^{10} - 219 x^{9} + 783 x^{8} + 279 x^{7} + \cdots + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 135)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 15 x^{13} + 47 x^{12} + 84 x^{11} - 279 x^{10} - 219 x^{9} + 783 x^{8} + 279 x^{7} + \cdots + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 2 \nu^{13} - 17 \nu^{12} + 30 \nu^{11} + 114 \nu^{10} - 165 \nu^{9} - 384 \nu^{8} + 399 \nu^{7} + \cdots + 6 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{14} - 6 \nu^{13} - 29 \nu^{12} + 93 \nu^{11} + 150 \nu^{10} - 543 \nu^{9} - 315 \nu^{8} + \cdots + 42 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4 \nu^{13} + 7 \nu^{12} + 66 \nu^{11} - 99 \nu^{10} - 420 \nu^{9} + 501 \nu^{8} + 1290 \nu^{7} + \cdots + 15 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{13} + 6 \nu^{12} + 28 \nu^{11} - 90 \nu^{10} - 136 \nu^{9} + 499 \nu^{8} + 246 \nu^{7} + \cdots + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{14} + 14 \nu^{13} + 75 \nu^{12} - 213 \nu^{11} - 420 \nu^{10} + 1209 \nu^{9} + 1089 \nu^{8} + \cdots + 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{14} + 7 \nu^{13} + 53 \nu^{12} - 117 \nu^{11} - 372 \nu^{10} + 753 \nu^{9} + 1329 \nu^{8} + \cdots + 57 ) / 9 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6 \nu^{14} + 22 \nu^{13} + 77 \nu^{12} - 336 \nu^{11} - 309 \nu^{10} + 1917 \nu^{9} + 237 \nu^{8} + \cdots - 15 ) / 9 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{14} + 24 \nu^{13} + 94 \nu^{12} - 366 \nu^{11} - 423 \nu^{10} + 2082 \nu^{9} + 621 \nu^{8} + \cdots - 3 ) / 9 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5 \nu^{14} - 14 \nu^{13} - 81 \nu^{12} + 225 \nu^{11} + 516 \nu^{10} - 1380 \nu^{9} - 1680 \nu^{8} + \cdots - 99 ) / 9 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12 \nu^{14} + 35 \nu^{13} + 178 \nu^{12} - 537 \nu^{11} - 978 \nu^{10} + 3087 \nu^{9} + 2463 \nu^{8} + \cdots + 60 ) / 9 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{14} - 9 \nu^{13} - 46 \nu^{12} + 142 \nu^{11} + 270 \nu^{10} - 850 \nu^{9} - 782 \nu^{8} + \cdots - 36 ) / 3 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 2\beta_{12} - \beta_{9} - \beta_{5} + 8\beta_{3} + 20\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - \beta_{12} + 11 \beta_{11} - 10 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{14} + \beta_{13} - 22 \beta_{12} + 3 \beta_{11} - 4 \beta_{10} - 12 \beta_{9} - 2 \beta_{8} + \cdots + 68 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{14} + 2 \beta_{13} - 19 \beta_{12} + 88 \beta_{11} - 77 \beta_{10} - 16 \beta_{9} - 15 \beta_{8} + \cdots + 405 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 95 \beta_{14} + 17 \beta_{13} - 183 \beta_{12} + 47 \beta_{11} - 60 \beta_{10} - 106 \beta_{9} + \cdots + 484 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 172 \beta_{14} + 37 \beta_{13} - 219 \beta_{12} + 627 \beta_{11} - 548 \beta_{10} - 167 \beta_{9} + \cdots + 2345 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 757 \beta_{14} + 192 \beta_{13} - 1384 \beta_{12} + 492 \beta_{11} - 614 \beta_{10} - 831 \beta_{9} + \cdots + 3348 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1563 \beta_{14} + 441 \beta_{13} - 2055 \beta_{12} + 4242 \beta_{11} - 3789 \beta_{10} - 1464 \beta_{9} + \cdots + 14014 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5793 \beta_{14} + 1812 \beta_{13} - 10035 \beta_{12} + 4339 \beta_{11} - 5362 \beta_{10} - 6132 \beta_{9} + \cdots + 22823 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12967 \beta_{14} + 4317 \beta_{13} - 17306 \beta_{12} + 28017 \beta_{11} - 25908 \beta_{10} + \cdots + 85806 \) 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
2.60146
2.36967
2.11967
1.63308
1.47820
1.30569
0.300859
0.243507
−0.174249
−0.694378
−0.748857
−1.29880
−1.84992
−1.98404
−2.30187
−2.60146 0 4.76758 1.00000 0 −4.24584 −7.19975 0 −2.60146
1.2 −2.36967 0 3.61532 1.00000 0 −0.197267 −3.82778 0 −2.36967
1.3 −2.11967 0 2.49298 1.00000 0 −0.0632009 −1.04496 0 −2.11967
1.4 −1.63308 0 0.666944 1.00000 0 −3.31953 2.17698 0 −1.63308
1.5 −1.47820 0 0.185067 1.00000 0 3.14304 2.68283 0 −1.47820
1.6 −1.30569 0 −0.295165 1.00000 0 0.875317 2.99678 0 −1.30569
1.7 −0.300859 0 −1.90948 1.00000 0 −4.28937 1.17620 0 −0.300859
1.8 −0.243507 0 −1.94070 1.00000 0 3.96723 0.959589 0 −0.243507
1.9 0.174249 0 −1.96964 1.00000 0 −1.74201 −0.691706 0 0.174249
1.10 0.694378 0 −1.51784 1.00000 0 −0.291597 −2.44271 0 0.694378
1.11 0.748857 0 −1.43921 1.00000 0 −0.530152 −2.57548 0 0.748857
1.12 1.29880 0 −0.313113 1.00000 0 −1.51756 −3.00428 0 1.29880
1.13 1.84992 0 1.42221 1.00000 0 −1.13884 −1.06887 0 1.84992
1.14 1.98404 0 1.93643 1.00000 0 1.64269 −0.126117 0 1.98404
1.15 2.30187 0 3.29862 1.00000 0 −4.29291 2.98925 0 2.30187
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

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 3645.2.a.g 15
3.b odd 2 1 3645.2.a.h 15
27.e even 9 2 405.2.k.a 30
27.f odd 18 2 135.2.k.a ✓ 30
135.n odd 18 2 675.2.l.d 30
135.q even 36 4 675.2.u.c 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.a ✓ 30 27.f odd 18 2
405.2.k.a 30 27.e even 9 2
675.2.l.d 30 135.n odd 18 2
675.2.u.c 60 135.q even 36 4
3645.2.a.g 15 1.a even 1 1 trivial
3645.2.a.h 15 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 3 T_{2}^{14} - 15 T_{2}^{13} - 47 T_{2}^{12} + 84 T_{2}^{11} + 279 T_{2}^{10} - 219 T_{2}^{9} + \cdots - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3645))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 3 T^{14} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{15} \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 12 T^{14} + \cdots + 27 \) Copy content Toggle raw display
$11$ \( T^{15} - 78 T^{13} + \cdots - 51111 \) Copy content Toggle raw display
$13$ \( T^{15} + 12 T^{14} + \cdots + 75853 \) Copy content Toggle raw display
$17$ \( T^{15} + 12 T^{14} + \cdots + 88719 \) Copy content Toggle raw display
$19$ \( T^{15} + 24 T^{14} + \cdots - 7019 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 208681353 \) Copy content Toggle raw display
$29$ \( T^{15} - 3 T^{14} + \cdots + 70766457 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 2774253969 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 360965557 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 7229668599 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 2166924437 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 1262687781 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 30327689193 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 4279835777463 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 1132793927 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 894784563 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 462163501167 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 72125996039 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 59845453641 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 1117006209153 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 446759170029 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 33134907354553 \) Copy content Toggle raw display
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