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Properties

Label 1617.4.a.bd
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: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199466 x^{9} + \cdots - 738304 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{5} + \beta_1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2} + \cdots - 4) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{5} + \beta_1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2} + \cdots - 4) q^{8}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 48 q^{3} + 72 q^{4} - 12 q^{6} - 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 48 q^{3} + 72 q^{4} - 12 q^{6} - 66 q^{8} + 144 q^{9} - 178 q^{10} - 176 q^{11} + 216 q^{12} - 104 q^{13} + 220 q^{16} - 220 q^{17} - 36 q^{18} - 152 q^{19} - 182 q^{20} + 44 q^{22} - 180 q^{23} - 198 q^{24} + 284 q^{25} - 10 q^{26} + 432 q^{27} - 604 q^{29} - 534 q^{30} - 380 q^{31} - 592 q^{32} - 528 q^{33} - 632 q^{34} + 648 q^{36} + 148 q^{37} - 266 q^{38} - 312 q^{39} - 1792 q^{40} - 60 q^{41} + 252 q^{43} - 792 q^{44} - 116 q^{46} - 1468 q^{47} + 660 q^{48} - 850 q^{50} - 660 q^{51} - 310 q^{52} - 1456 q^{53} - 108 q^{54} - 456 q^{57} - 1350 q^{58} - 1312 q^{59} - 546 q^{60} - 2880 q^{61} - 708 q^{62} + 630 q^{64} - 4064 q^{65} + 132 q^{66} + 1220 q^{67} - 4956 q^{68} - 540 q^{69} - 2040 q^{71} - 594 q^{72} - 1628 q^{73} - 3126 q^{74} + 852 q^{75} - 6286 q^{76} - 30 q^{78} - 416 q^{79} + 874 q^{80} + 1296 q^{81} - 3040 q^{82} - 3724 q^{83} + 628 q^{85} - 1608 q^{86} - 1812 q^{87} + 726 q^{88} - 752 q^{89} - 1602 q^{90} - 32 q^{92} - 1140 q^{93} - 610 q^{94} - 912 q^{95} - 1776 q^{96} - 1088 q^{97} - 1584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199466 x^{9} + \cdots - 738304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24386725 \nu^{15} + 511703837 \nu^{14} + 407411307 \nu^{13} - 43182271817 \nu^{12} + \cdots + 242365506733056 ) / 13638029623296 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8675592017 \nu^{15} - 368142774057 \nu^{14} - 191259357567 \nu^{13} + 33079986356389 \nu^{12} + \cdots - 46\!\cdots\!36 ) / 44\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11116152549 \nu^{15} + 55793030093 \nu^{14} + 1117784822795 \nu^{13} + \cdots - 91\!\cdots\!72 ) / 44\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11116152549 \nu^{15} + 55793030093 \nu^{14} + 1117784822795 \nu^{13} + \cdots - 50\!\cdots\!16 ) / 44\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2978675587 \nu^{15} + 689934803 \nu^{14} + 270159417005 \nu^{13} + \cdots + 19\!\cdots\!96 ) / 560863968258048 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26097117323 \nu^{15} - 227967094893 \nu^{14} + 3429090741733 \nu^{13} + \cdots + 13\!\cdots\!76 ) / 44\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23136988571 \nu^{15} + 16893139885 \nu^{14} - 2281492547893 \nu^{13} + \cdots - 65\!\cdots\!92 ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20452910127 \nu^{15} - 16432133993 \nu^{14} + 1982131425633 \nu^{13} + \cdots + 30\!\cdots\!00 ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55080257475 \nu^{15} - 416033083595 \nu^{14} - 4000919116205 \nu^{13} + \cdots + 67\!\cdots\!80 ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16429085063 \nu^{15} - 61887969455 \nu^{14} - 1425014908457 \nu^{13} + \cdots + 27\!\cdots\!52 ) / 640987392294912 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 124480546525 \nu^{15} + 423884464197 \nu^{14} + 11651829308051 \nu^{13} + \cdots - 26\!\cdots\!48 ) / 44\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 131806954357 \nu^{15} + 549377494605 \nu^{14} + 10915371529179 \nu^{13} + \cdots + 25\!\cdots\!60 ) / 44\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 130386402597 \nu^{15} - 648517868837 \nu^{14} - 10855476276667 \nu^{13} + \cdots + 10\!\cdots\!04 ) / 22\!\cdots\!92 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + 20\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} + 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 264 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 4 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 2 \beta_{10} + \beta_{9} - 5 \beta_{8} + \cdots + 172 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32 \beta_{15} - 6 \beta_{14} + 42 \beta_{13} + 132 \beta_{12} - 72 \beta_{11} + 93 \beta_{10} + \cdots + 6125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86 \beta_{15} + 200 \beta_{14} + 148 \beta_{13} + 170 \beta_{12} + 34 \beta_{11} + 120 \beta_{10} + \cdots + 6080 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 870 \beta_{15} - 300 \beta_{14} + 1482 \beta_{13} + 4502 \beta_{12} - 2040 \beta_{11} + 3312 \beta_{10} + \cdots + 152607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2870 \beta_{15} + 7144 \beta_{14} + 5520 \beta_{13} + 7188 \beta_{12} + 1972 \beta_{11} + 5580 \beta_{10} + \cdots + 203330 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 23727 \beta_{15} - 10580 \beta_{14} + 49469 \beta_{13} + 141151 \beta_{12} - 54070 \beta_{11} + \cdots + 3969318 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 91212 \beta_{15} + 225584 \beta_{14} + 188183 \beta_{13} + 269451 \beta_{12} + 76760 \beta_{11} + \cdots + 6611554 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 672264 \beta_{15} - 324594 \beta_{14} + 1600284 \beta_{13} + 4274966 \beta_{12} - 1409044 \beta_{11} + \cdots + 106209093 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2900502 \beta_{15} + 6727236 \beta_{14} + 6182820 \beta_{13} + 9467666 \beta_{12} + 2514238 \beta_{11} + \cdots + 211230852 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19794032 \beta_{15} - 9225444 \beta_{14} + 50676348 \beta_{13} + 127515168 \beta_{12} + \cdots + 2900153889 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 92925716 \beta_{15} + 194720512 \beta_{14} + 199422020 \beta_{13} + 319828308 \beta_{12} + \cdots + 6669175096 \) 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
5.50208
5.02984
4.51833
3.65350
3.30535
3.04558
1.20555
0.626276
0.120661
−0.903074
−2.39847
−2.47130
−3.48862
−3.90605
−4.70685
−5.13282
−5.50208 3.00000 22.2728 −1.21430 −16.5062 0 −78.5303 9.00000 6.68115
1.2 −5.02984 3.00000 17.2993 20.5790 −15.0895 0 −46.7737 9.00000 −103.509
1.3 −4.51833 3.00000 12.4153 −3.55084 −13.5550 0 −19.9499 9.00000 16.0439
1.4 −3.65350 3.00000 5.34810 11.9317 −10.9605 0 9.68874 9.00000 −43.5923
1.5 −3.30535 3.00000 2.92536 −9.14781 −9.91606 0 16.7735 9.00000 30.2367
1.6 −3.04558 3.00000 1.27555 −15.0528 −9.13674 0 20.4798 9.00000 45.8446
1.7 −1.20555 3.00000 −6.54664 9.84913 −3.61666 0 17.5368 9.00000 −11.8737
1.8 −0.626276 3.00000 −7.60778 4.84903 −1.87883 0 9.77478 9.00000 −3.03683
1.9 −0.120661 3.00000 −7.98544 18.6516 −0.361983 0 1.92882 9.00000 −2.25052
1.10 0.903074 3.00000 −7.18446 −21.8819 2.70922 0 −13.7127 9.00000 −19.7610
1.11 2.39847 3.00000 −2.24736 1.37648 7.19540 0 −24.5779 9.00000 3.30143
1.12 2.47130 3.00000 −1.89269 4.05950 7.41389 0 −24.4478 9.00000 10.0322
1.13 3.48862 3.00000 4.17050 −0.768490 10.4659 0 −13.3597 9.00000 −2.68097
1.14 3.90605 3.00000 7.25725 7.35832 11.7182 0 −2.90120 9.00000 28.7420
1.15 4.70685 3.00000 14.1544 −15.5088 14.1205 0 28.9678 9.00000 −72.9976
1.16 5.13282 3.00000 18.3458 −11.5297 15.3984 0 53.1030 9.00000 −59.1800
\(n\): 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.bd yes 16
7.b odd 2 1 1617.4.a.bc ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.4.a.bc ✓ 16 7.b odd 2 1
1617.4.a.bd yes 16 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{16} + 4 T_{2}^{15} - 92 T_{2}^{14} - 346 T_{2}^{13} + 3385 T_{2}^{12} + 11756 T_{2}^{11} + \cdots - 738304 \) Copy content Toggle raw display
\( T_{5}^{16} - 1142 T_{5}^{14} + 508 T_{5}^{13} + 477806 T_{5}^{12} - 450684 T_{5}^{11} + \cdots + 16055671813888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{15} + \cdots - 738304 \) Copy content Toggle raw display
$3$ \( (T - 3)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 16055671813888 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T + 11)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots - 51\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 18\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 24\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 57\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 58\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 24\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 58\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 59\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 87\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 26\!\cdots\!52 \) Copy content Toggle raw display
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