Properties

Label 1617.4.a.s
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $7$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 43x^{5} - 3x^{4} + 486x^{3} + 191x^{2} - 1348x - 1108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{4} - \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 - 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{4} - \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 - 2) q^{8} + 9 q^{9} + (\beta_{5} + \beta_1 + 7) q^{10} - 11 q^{11} + ( - 3 \beta_{2} - 12) q^{12} + ( - 2 \beta_{4} - \beta_{3} + \cdots - 13) q^{13}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 21 q^{3} + 30 q^{4} - 9 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 21 q^{3} + 30 q^{4} - 9 q^{8} + 63 q^{9} + 49 q^{10} - 77 q^{11} - 90 q^{12} - 92 q^{13} + 138 q^{16} + 98 q^{17} - 76 q^{19} - 87 q^{20} + 36 q^{23} + 27 q^{24} - 99 q^{25} - 99 q^{26} - 189 q^{27} + 240 q^{29} - 147 q^{30} - 198 q^{31} + 598 q^{32} + 231 q^{33} + 112 q^{34} + 270 q^{36} - 274 q^{37} - 923 q^{38} + 276 q^{39} - 176 q^{40} - 430 q^{41} + 208 q^{43} - 330 q^{44} + 8 q^{46} - 82 q^{47} - 414 q^{48} + 269 q^{50} - 294 q^{51} - 2033 q^{52} + 1102 q^{53} + 228 q^{57} - 2119 q^{58} - 1140 q^{59} + 261 q^{60} + 452 q^{61} + 336 q^{62} + 1021 q^{64} + 1294 q^{65} + 766 q^{67} - 968 q^{68} - 108 q^{69} + 740 q^{71} - 81 q^{72} - 1538 q^{73} + 2011 q^{74} + 297 q^{75} + 629 q^{76} + 297 q^{78} - 1072 q^{79} - 2181 q^{80} + 567 q^{81} - 1544 q^{82} - 1770 q^{83} + 648 q^{85} + 538 q^{86} - 720 q^{87} + 99 q^{88} - 1560 q^{89} + 441 q^{90} + 1736 q^{92} + 594 q^{93} - 2951 q^{94} + 1764 q^{95} - 1794 q^{96} + 880 q^{97} - 693 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 43x^{5} - 3x^{4} + 486x^{3} + 191x^{2} - 1348x - 1108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 21\nu - 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} - 40\nu^{4} + 39\nu^{3} + 383\nu^{2} - 254\nu - 740 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 42\nu^{4} - 103\nu^{3} - 433\nu^{2} + 596\nu + 1024 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + \nu^{5} + 44\nu^{4} - 39\nu^{3} - 499\nu^{2} + 274\nu + 1164 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 21\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 3\beta_{4} + 29\beta_{2} - 5\beta _1 + 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} + 6\beta_{5} + 3\beta_{4} + 32\beta_{3} - 36\beta_{2} + 506\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 39\beta_{6} + 6\beta_{5} + 135\beta_{4} - 7\beta_{3} + 780\beta_{2} - 259\beta _1 + 5726 \) Copy content Toggle raw display

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
5.03971
3.78151
2.20511
−1.03672
−1.49421
−3.23638
−5.25903
−5.03971 −3.00000 17.3987 −7.48968 15.1191 0 −47.3667 9.00000 37.7458
1.2 −3.78151 −3.00000 6.29982 8.16095 11.3445 0 6.42926 9.00000 −30.8607
1.3 −2.20511 −3.00000 −3.13747 −10.3292 6.61534 0 24.5594 9.00000 22.7770
1.4 1.03672 −3.00000 −6.92521 13.7243 −3.11017 0 −15.4733 9.00000 14.2283
1.5 1.49421 −3.00000 −5.76735 −13.8155 −4.48262 0 −20.5713 9.00000 −20.6432
1.6 3.23638 −3.00000 2.47416 12.6162 −9.70914 0 −17.8837 9.00000 40.8307
1.7 5.25903 −3.00000 19.6574 −2.86707 −15.7771 0 61.3063 9.00000 −15.0780
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( -1 \)
\(11\) \( +1 \)

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 1617.4.a.s 7
7.b odd 2 1 1617.4.a.t yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.4.a.s 7 1.a even 1 1 trivial
1617.4.a.t yes 7 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{7} - 43T_{2}^{5} + 3T_{2}^{4} + 486T_{2}^{3} - 191T_{2}^{2} - 1348T_{2} + 1108 \) Copy content Toggle raw display
\( T_{5}^{7} - 388T_{5}^{5} - 318T_{5}^{4} + 46203T_{5}^{3} + 83766T_{5}^{2} - 1631728T_{5} - 4330032 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 43 T^{5} + \cdots + 1108 \) Copy content Toggle raw display
$3$ \( (T + 3)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 388 T^{5} + \cdots - 4330032 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( (T + 11)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 66442437824 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 50202012672 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 7921780989808 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 62769219239936 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 59490644194504 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 259113809514616 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 179651850598656 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 704756149886544 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 28704942755328 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 50\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 44\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 57\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
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