Properties

Label 1617.4.a.x
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 45 x^{8} + 160 x^{7} + 661 x^{6} - 1934 x^{5} - 3519 x^{4} + 6710 x^{3} + 6802 x^{2} + \cdots - 2880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{3} - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 5) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{3} - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 5) q^{8}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 30 q^{3} + 26 q^{4} - 20 q^{5} - 12 q^{6} - 60 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} + 30 q^{3} + 26 q^{4} - 20 q^{5} - 12 q^{6} - 60 q^{8} + 90 q^{9} + 38 q^{10} - 110 q^{11} + 78 q^{12} + 44 q^{13} - 60 q^{15} + 78 q^{16} - 80 q^{17} - 36 q^{18} - 78 q^{19} - 184 q^{20} + 44 q^{22} - 178 q^{23} - 180 q^{24} + 356 q^{25} - 678 q^{26} + 270 q^{27} - 348 q^{29} + 114 q^{30} + 26 q^{31} - 374 q^{32} - 330 q^{33} + 618 q^{34} + 234 q^{36} - 132 q^{37} - 198 q^{38} + 132 q^{39} + 246 q^{40} - 1216 q^{41} - 684 q^{43} - 286 q^{44} - 180 q^{45} - 408 q^{46} - 358 q^{47} + 234 q^{48} - 1806 q^{50} - 240 q^{51} - 154 q^{52} + 428 q^{53} - 108 q^{54} + 220 q^{55} - 234 q^{57} - 24 q^{58} - 90 q^{59} - 552 q^{60} + 30 q^{61} + 84 q^{62} - 1266 q^{64} - 1476 q^{65} + 132 q^{66} - 552 q^{67} + 1020 q^{68} - 534 q^{69} - 614 q^{71} - 540 q^{72} - 92 q^{73} + 2172 q^{74} + 1068 q^{75} + 872 q^{76} - 2034 q^{78} - 2140 q^{79} - 4424 q^{80} + 810 q^{81} + 3054 q^{82} - 1782 q^{83} - 2726 q^{85} - 1366 q^{86} - 1044 q^{87} + 660 q^{88} - 1288 q^{89} + 342 q^{90} - 4284 q^{92} + 78 q^{93} - 7226 q^{94} - 1046 q^{95} - 1122 q^{96} + 284 q^{97} - 990 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 45 x^{8} + 160 x^{7} + 661 x^{6} - 1934 x^{5} - 3519 x^{4} + 6710 x^{3} + 6802 x^{2} + \cdots - 2880 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 403 \nu^{9} - 1681 \nu^{8} - 17472 \nu^{7} + 66496 \nu^{6} + 237199 \nu^{5} - 793331 \nu^{4} + \cdots - 1301952 ) / 36288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59 \nu^{9} - 293 \nu^{8} - 2436 \nu^{7} + 11960 \nu^{6} + 29879 \nu^{5} - 147175 \nu^{4} + \cdots - 217104 ) / 3024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1111 \nu^{9} - 5197 \nu^{8} - 46704 \nu^{7} + 210016 \nu^{6} + 595747 \nu^{5} - 2559431 \nu^{4} + \cdots - 4088640 ) / 36288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1633 \nu^{9} - 7879 \nu^{8} - 67116 \nu^{7} + 317728 \nu^{6} + 821557 \nu^{5} - 3879533 \nu^{4} + \cdots - 7014144 ) / 18144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1885 \nu^{9} + 8887 \nu^{8} + 78456 \nu^{7} - 358048 \nu^{6} - 988129 \nu^{5} + 4360853 \nu^{4} + \cdots + 7243968 ) / 12096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1615 \nu^{9} + 7717 \nu^{8} + 66864 \nu^{7} - 310816 \nu^{6} - 834139 \nu^{5} + 3784655 \nu^{4} + \cdots + 6368736 ) / 6048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6527 \nu^{9} - 30953 \nu^{8} - 270564 \nu^{7} + 1244912 \nu^{6} + 3381899 \nu^{5} - 15125635 \nu^{4} + \cdots - 24958464 ) / 18144 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 19\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + 25\beta_{2} + 31\beta _1 + 181 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} - 3\beta_{6} - 35\beta_{5} + 30\beta_{4} + 26\beta_{3} + 42\beta_{2} + 409\beta _1 + 200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{9} + 36 \beta_{8} - 22 \beta_{7} - 28 \beta_{6} - 76 \beta_{5} + 94 \beta_{4} - 14 \beta_{3} + \cdots + 3780 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 22 \beta_{9} + 48 \beta_{8} - 154 \beta_{7} - 104 \beta_{6} - 1043 \beta_{5} + 807 \beta_{4} + \cdots + 6775 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 629 \beta_{9} + 1171 \beta_{8} - 1134 \beta_{7} - 627 \beta_{6} - 3189 \beta_{5} + 3220 \beta_{4} + \cdots + 84897 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1256 \beta_{9} + 2994 \beta_{8} - 6011 \beta_{7} - 2707 \beta_{6} - 29825 \beta_{5} + 21664 \beta_{4} + \cdots + 210820 \) 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
5.25538
4.48887
3.37565
2.21027
0.760491
−0.737904
−0.758345
−2.17274
−4.02407
−4.39760
−5.25538 3.00000 19.6190 −17.3326 −15.7661 0 −61.0623 9.00000 91.0895
1.2 −4.48887 3.00000 12.1499 8.63270 −13.4666 0 −18.6285 9.00000 −38.7511
1.3 −3.37565 3.00000 3.39502 13.9352 −10.1270 0 15.5448 9.00000 −47.0402
1.4 −2.21027 3.00000 −3.11472 −21.6829 −6.63080 0 24.5665 9.00000 47.9251
1.5 −0.760491 3.00000 −7.42165 4.03718 −2.28147 0 11.7280 9.00000 −3.07024
1.6 0.737904 3.00000 −7.45550 −9.09244 2.21371 0 −11.4047 9.00000 −6.70935
1.7 0.758345 3.00000 −7.42491 −7.17969 2.27503 0 −11.6974 9.00000 −5.44468
1.8 2.17274 3.00000 −3.27921 16.5779 6.51821 0 −24.5068 9.00000 36.0195
1.9 4.02407 3.00000 8.19314 3.47545 12.0722 0 0.777189 9.00000 13.9854
1.10 4.39760 3.00000 11.3389 −11.3707 13.1928 0 14.6831 9.00000 −50.0039
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.x 10
7.b odd 2 1 1617.4.a.w 10
7.c even 3 2 231.4.i.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.c 20 7.c even 3 2
1617.4.a.w 10 7.b odd 2 1
1617.4.a.x 10 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{10} + 4 T_{2}^{9} - 45 T_{2}^{8} - 160 T_{2}^{7} + 661 T_{2}^{6} + 1934 T_{2}^{5} - 3519 T_{2}^{4} + \cdots - 2880 \) Copy content Toggle raw display
\( T_{5}^{10} + 20 T_{5}^{9} - 603 T_{5}^{8} - 11400 T_{5}^{7} + 120828 T_{5}^{6} + 2074836 T_{5}^{5} + \cdots - 7806081024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + \cdots - 2880 \) Copy content Toggle raw display
$3$ \( (T - 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 7806081024 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 17\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 47\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 87\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 15\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 36\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 23\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 31\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 67\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 10\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 53\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
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