Properties

Label 1617.4.a.z
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $0$
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: \(0\)
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: \( x^{10} - 4 x^{9} - 45 x^{8} + 168 x^{7} + 651 x^{6} - 2176 x^{5} - 3439 x^{4} + 8716 x^{3} + 7840 x^{2} + \cdots - 4032 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{6} + 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{6} + 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{8} + 9 q^{9} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 1) q^{10}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 30 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 36 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 30 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 36 q^{8} + 90 q^{9} + 44 q^{10} - 110 q^{11} + 78 q^{12} + 82 q^{13} + 60 q^{15} - 10 q^{16} + 164 q^{17} + 36 q^{18} - 76 q^{19} + 356 q^{20} - 44 q^{22} + 140 q^{23} + 108 q^{24} + 472 q^{25} + 360 q^{26} + 270 q^{27} + 40 q^{29} + 132 q^{30} + 24 q^{31} + 112 q^{32} - 330 q^{33} - 262 q^{34} + 234 q^{36} + 412 q^{37} + 16 q^{38} + 246 q^{39} + 828 q^{40} - 228 q^{41} + 530 q^{43} - 286 q^{44} + 180 q^{45} + 1422 q^{46} + 768 q^{47} - 30 q^{48} + 670 q^{50} + 492 q^{51} + 952 q^{52} + 136 q^{53} + 108 q^{54} - 220 q^{55} - 228 q^{57} + 1708 q^{58} + 608 q^{59} + 1068 q^{60} + 1618 q^{61} + 164 q^{62} - 1238 q^{64} + 2764 q^{65} - 132 q^{66} + 1002 q^{67} + 2816 q^{68} + 420 q^{69} + 812 q^{71} + 324 q^{72} + 134 q^{73} + 342 q^{74} + 1416 q^{75} - 450 q^{76} + 1080 q^{78} + 1262 q^{79} - 1316 q^{80} + 810 q^{81} + 982 q^{82} + 1078 q^{83} - 1402 q^{85} + 1692 q^{86} + 120 q^{87} - 396 q^{88} + 2880 q^{89} + 396 q^{90} + 2364 q^{92} + 72 q^{93} - 2362 q^{94} + 2866 q^{95} + 336 q^{96} - 1030 q^{97} - 990 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 45 x^{8} + 168 x^{7} + 651 x^{6} - 2176 x^{5} - 3439 x^{4} + 8716 x^{3} + 7840 x^{2} + \cdots - 4032 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 17\nu + 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{9} + 79 \nu^{8} + 432 \nu^{7} - 3144 \nu^{6} - 2391 \nu^{5} + 35809 \nu^{4} - 24404 \nu^{3} + \cdots + 27840 ) / 384 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11 \nu^{9} - 65 \nu^{8} - 384 \nu^{7} + 2616 \nu^{6} + 2721 \nu^{5} - 30335 \nu^{4} + 13132 \nu^{3} + \cdots - 29568 ) / 192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25 \nu^{9} + 139 \nu^{8} + 888 \nu^{7} - 5544 \nu^{6} - 6747 \nu^{5} + 63397 \nu^{4} - 24044 \nu^{3} + \cdots + 49728 ) / 384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9 \nu^{9} + 51 \nu^{8} + 320 \nu^{7} - 2040 \nu^{6} - 2475 \nu^{5} + 23453 \nu^{4} - 7668 \nu^{3} + \cdots + 19136 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37 \nu^{9} + 211 \nu^{8} + 1308 \nu^{7} - 8448 \nu^{6} - 9807 \nu^{5} + 97405 \nu^{4} + \cdots + 97152 ) / 192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 121 \nu^{9} + 691 \nu^{8} + 4272 \nu^{7} - 27624 \nu^{6} - 31899 \nu^{5} + 317245 \nu^{4} + \cdots + 280512 ) / 384 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 18\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} + 26\beta_{2} + 29\beta _1 + 173 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 8\beta_{7} - 2\beta_{4} + 30\beta_{3} + 44\beta_{2} + 377\beta _1 + 110 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{9} + 26 \beta_{8} - 50 \beta_{7} - 42 \beta_{6} + 42 \beta_{5} - 4 \beta_{4} + 82 \beta_{3} + \cdots + 3466 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 108 \beta_{9} - 10 \beta_{8} - 358 \beta_{7} - 38 \beta_{6} + 14 \beta_{5} - 108 \beta_{4} + 787 \beta_{3} + \cdots + 3815 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 612 \beta_{9} + 543 \beta_{8} - 1775 \beta_{7} - 1407 \beta_{6} + 1271 \beta_{5} - 244 \beta_{4} + \cdots + 74493 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4038 \beta_{9} - 566 \beta_{8} - 11874 \beta_{7} - 2410 \beta_{6} + 786 \beta_{5} - 3766 \beta_{4} + \cdots + 123466 \) 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
−4.49955
−4.05895
−1.71637
−1.70299
−0.373481
1.11872
2.38864
3.53481
4.14909
5.16008
−4.49955 3.00000 12.2460 −9.43139 −13.4987 0 −19.1049 9.00000 42.4370
1.2 −4.05895 3.00000 8.47509 17.0425 −12.1769 0 −1.92836 9.00000 −69.1748
1.3 −1.71637 3.00000 −5.05408 14.8426 −5.14911 0 22.4056 9.00000 −25.4753
1.4 −1.70299 3.00000 −5.09984 −3.64858 −5.10896 0 22.3088 9.00000 6.21348
1.5 −0.373481 3.00000 −7.86051 −15.6838 −1.12044 0 5.92360 9.00000 5.85760
1.6 1.11872 3.00000 −6.74846 −4.70241 3.35616 0 −16.4994 9.00000 −5.26068
1.7 2.38864 3.00000 −2.29442 9.60191 7.16591 0 −24.5896 9.00000 22.9355
1.8 3.53481 3.00000 4.49490 −15.9961 10.6044 0 −12.3899 9.00000 −56.5431
1.9 4.14909 3.00000 9.21491 21.1121 12.4473 0 5.04076 9.00000 87.5957
1.10 5.16008 3.00000 18.6265 6.86319 15.4802 0 54.8334 9.00000 35.4146
\(n\): 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.z 10
7.b odd 2 1 1617.4.a.y 10
7.d odd 6 2 231.4.i.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.b 20 7.d odd 6 2
1617.4.a.y 10 7.b odd 2 1
1617.4.a.z 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{10} - 4 T_{2}^{9} - 45 T_{2}^{8} + 168 T_{2}^{7} + 651 T_{2}^{6} - 2176 T_{2}^{5} - 3439 T_{2}^{4} + \cdots - 4032 \) Copy content Toggle raw display
\( T_{5}^{10} - 20 T_{5}^{9} - 661 T_{5}^{8} + 12588 T_{5}^{7} + 151892 T_{5}^{6} - 2602348 T_{5}^{5} + \cdots - 14287026432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots - 4032 \) Copy content Toggle raw display
$3$ \( (T - 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 14287026432 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 41\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 80\!\cdots\!13 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 41\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 16\!\cdots\!33 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 49\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 71\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 52\!\cdots\!51 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 33\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 95\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 16\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 34\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 54\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 28\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
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