Properties

Label 1617.4.a.be
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{8} - \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 4) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{8} - \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 4) q^{8} + 9 q^{9} + ( - \beta_{9} - \beta_{5} - \beta_{2} + \cdots - 11) q^{10}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4027995121 \nu^{15} + 7255221737 \nu^{14} + 388992582015 \nu^{13} + \cdots + 96\!\cdots\!16 ) / 60298062483456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54262353 \nu^{15} + 78635421 \nu^{14} + 5319432119 \nu^{13} - 5694929753 \nu^{12} + \cdots + 214041015049216 ) / 591157475328 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 381781615 \nu^{15} - 628293551 \nu^{14} - 36820853553 \nu^{13} + 46538244979 \nu^{12} + \cdots - 12\!\cdots\!48 ) / 3546944851968 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2795704 \nu^{15} - 5618357 \nu^{14} - 267400230 \nu^{13} + 431149045 \nu^{12} + \cdots - 6619938566656 ) / 22909598208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8023224827 \nu^{15} + 15476858995 \nu^{14} + 768283063029 \nu^{13} + \cdots + 21\!\cdots\!48 ) / 60298062483456 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11081280307 \nu^{15} + 22125566003 \nu^{14} + 1060822477005 \nu^{13} + \cdots + 22\!\cdots\!28 ) / 60298062483456 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5564459 \nu^{15} + 10195462 \nu^{14} + 536164539 \nu^{13} - 767526818 \nu^{12} + \cdots + 15364666790912 ) / 22909598208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3388235695 \nu^{15} + 6179746815 \nu^{14} + 325732912737 \nu^{13} + \cdots + 85\!\cdots\!24 ) / 10049677080576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7108831141 \nu^{15} - 14380308693 \nu^{14} - 678407064507 \nu^{13} + 1101143360129 \nu^{12} + \cdots - 18\!\cdots\!88 ) / 20099354161152 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1794240773 \nu^{15} - 3380120773 \nu^{14} - 172310431995 \nu^{13} + 256252076273 \nu^{12} + \cdots - 46\!\cdots\!68 ) / 3546944851968 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5083956 \nu^{15} + 9146495 \nu^{14} + 489497542 \nu^{13} - 687784703 \nu^{12} + \cdots + 14075467325440 ) / 7636532736 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7481000833 \nu^{15} + 14288645669 \nu^{14} + 717580634331 \nu^{13} + \cdots + 18\!\cdots\!88 ) / 7537257810432 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{11} - 2\beta_{7} - \beta_{4} + \beta_{3} + 28\beta_{2} + 4\beta _1 + 262 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots + 160 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{15} + 42 \beta_{14} + 46 \beta_{13} + 9 \beta_{12} - 39 \beta_{11} - 5 \beta_{10} - 2 \beta_{9} + \cdots + 6025 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 34 \beta_{15} - 48 \beta_{14} + 108 \beta_{13} - 48 \beta_{12} + 40 \beta_{11} + 54 \beta_{10} + \cdots + 5328 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 152 \beta_{15} + 1330 \beta_{14} + 1578 \beta_{13} + 460 \beta_{12} - 1194 \beta_{11} - 280 \beta_{10} + \cdots + 146915 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 878 \beta_{15} - 1720 \beta_{14} + 4142 \beta_{13} - 1734 \beta_{12} + 1324 \beta_{11} + \cdots + 166126 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5764 \beta_{15} + 38325 \beta_{14} + 49027 \beta_{13} + 16872 \beta_{12} - 33727 \beta_{11} + \cdots + 3706840 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20385 \beta_{15} - 55341 \beta_{14} + 139060 \beta_{13} - 55393 \beta_{12} + 41715 \beta_{11} + \cdots + 5016730 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 196083 \beta_{15} + 1060108 \beta_{14} + 1456614 \beta_{13} + 545505 \beta_{12} - 919803 \beta_{11} + \cdots + 95706153 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 439416 \beta_{15} - 1688840 \beta_{14} + 4367788 \beta_{13} - 1646094 \beta_{12} + 1278030 \beta_{11} + \cdots + 148742444 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6292984 \beta_{15} + 28715340 \beta_{14} + 42250632 \beta_{13} + 16571144 \beta_{12} - 24657684 \beta_{11} + \cdots + 2512150465 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 8681880 \beta_{15} - 50002556 \beta_{14} + 132107596 \beta_{13} - 46611680 \beta_{12} + \cdots + 4358353936 \) 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
−5.15488
−4.48532
−4.31445
−3.31007
−2.57449
−1.86830
−1.36369
−0.347888
1.06618
2.10744
2.12238
2.95461
4.13448
4.50124
5.23364
5.29912
−5.15488 −3.00000 18.5728 4.10022 15.4646 0 −54.5014 9.00000 −21.1361
1.2 −4.48532 −3.00000 12.1181 17.6644 13.4559 0 −18.4708 9.00000 −79.2305
1.3 −4.31445 −3.00000 10.6145 −14.6492 12.9433 0 −11.2800 9.00000 63.2031
1.4 −3.31007 −3.00000 2.95655 −1.76442 9.93020 0 16.6942 9.00000 5.84035
1.5 −2.57449 −3.00000 −1.37200 −19.4253 7.72347 0 24.1281 9.00000 50.0102
1.6 −1.86830 −3.00000 −4.50945 9.74429 5.60490 0 23.3714 9.00000 −18.2053
1.7 −1.36369 −3.00000 −6.14035 14.9279 4.09107 0 19.2830 9.00000 −20.3570
1.8 −0.347888 −3.00000 −7.87897 −10.0796 1.04366 0 5.52411 9.00000 3.50656
1.9 1.06618 −3.00000 −6.86326 3.08635 −3.19854 0 −15.8469 9.00000 3.29060
1.10 2.10744 −3.00000 −3.55870 5.76870 −6.32232 0 −24.3593 9.00000 12.1572
1.11 2.12238 −3.00000 −3.49551 −16.9491 −6.36714 0 −24.3978 9.00000 −35.9724
1.12 2.95461 −3.00000 0.729713 −19.4605 −8.86383 0 −21.4809 9.00000 −57.4982
1.13 4.13448 −3.00000 9.09391 10.9590 −12.4034 0 4.52277 9.00000 45.3099
1.14 4.50124 −3.00000 12.2612 3.51724 −13.5037 0 19.1806 9.00000 15.8320
1.15 5.23364 −3.00000 19.3910 −10.0477 −15.7009 0 59.6162 9.00000 −52.5859
1.16 5.29912 −3.00000 20.0806 −17.3924 −15.8973 0 64.0166 9.00000 −92.1644
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.be 16
7.b odd 2 1 1617.4.a.bf yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.4.a.be 16 1.a even 1 1 trivial
1617.4.a.bf yes 16 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{16} - 4 T_{2}^{15} - 92 T_{2}^{14} + 346 T_{2}^{13} + 3385 T_{2}^{12} - 11756 T_{2}^{11} + \cdots + 5479424 \) Copy content Toggle raw display
\( T_{5}^{16} + 40 T_{5}^{15} - 494 T_{5}^{14} - 34892 T_{5}^{13} - 17890 T_{5}^{12} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 5479424 \) Copy content Toggle raw display
$3$ \( (T + 3)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T - 11)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 59\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 38\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 65\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 18\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 99\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 40\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 97\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 49\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 59\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 12\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 30\!\cdots\!32 \) Copy content Toggle raw display
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