Properties

Label 17.10.a.a
Level $17$
Weight $10$
Character orbit 17.a
Self dual yes
Analytic conductor $8.756$
Analytic rank $1$
Dimension $5$
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] = [17,10,Mod(1,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 17.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: \(8.75560921479\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 7) q^{2} + (\beta_{4} + 2 \beta_1 - 48) q^{3} + ( - 3 \beta_{4} + \beta_{3} - 17 \beta_1 + 177) q^{4} + ( - 6 \beta_{4} - 6 \beta_{3} + \beta_{2} - 26 \beta_1 + 309) q^{5} + ( - 12 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} - 154 \beta_1 + 1574) q^{6} + (13 \beta_{4} + 12 \beta_{3} + 13 \beta_{2} - 44 \beta_1 - 2625) q^{7} + (67 \beta_{4} - 9 \beta_{3} + 28 \beta_{2} + 179 \beta_1 - 8547) q^{8} + ( - 14 \beta_{4} + 50 \beta_{3} - 73 \beta_{2} - 606 \beta_1 + 2404) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 7) q^{2} + (\beta_{4} + 2 \beta_1 - 48) q^{3} + ( - 3 \beta_{4} + \beta_{3} - 17 \beta_1 + 177) q^{4} + ( - 6 \beta_{4} - 6 \beta_{3} + \beta_{2} - 26 \beta_1 + 309) q^{5} + ( - 12 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} - 154 \beta_1 + 1574) q^{6} + (13 \beta_{4} + 12 \beta_{3} + 13 \beta_{2} - 44 \beta_1 - 2625) q^{7} + (67 \beta_{4} - 9 \beta_{3} + 28 \beta_{2} + 179 \beta_1 - 8547) q^{8} + ( - 14 \beta_{4} + 50 \beta_{3} - 73 \beta_{2} - 606 \beta_1 + 2404) q^{9} + (104 \beta_{4} - 104 \beta_{3} + 560 \beta_1 - 18048) q^{10} + ( - 139 \beta_{4} + 48 \beta_{3} - 54 \beta_{2} - 138 \beta_1 - 13582) q^{11} + (190 \beta_{4} - 266 \beta_{3} + 304 \beta_{2} + 3538 \beta_1 - 82450) q^{12} + ( - 546 \beta_{4} + 198 \beta_{3} - 37 \beta_{2} - 150 \beta_1 - 31799) q^{13} + ( - 256 \beta_{4} + 384 \beta_{3} - 368 \beta_{2} - 2388 \beta_1 - 16212) q^{14} + ( - 354 \beta_{4} + 56 \beta_{3} + 548 \beta_{2} + 4156 \beta_1 - 139040) q^{15} + ( - \beta_{4} + 179 \beta_{3} - 1244 \beta_{2} - 8421 \beta_1 + 73093) q^{16} - 83521 q^{17} + (3408 \beta_{4} - 1248 \beta_{3} + 2064 \beta_{2} + 14625 \beta_1 - 387255) q^{18} + (858 \beta_{4} - 1832 \beta_{3} + 234 \beta_{2} + 4968 \beta_1 - 75406) q^{19} + (976 \beta_{4} + 2384 \beta_{3} - 1760 \beta_{2} - 28224 \beta_1 + 336512) q^{20} + ( - 3310 \beta_{4} - 754 \beta_{3} - 685 \beta_{2} + \cdots + 361375) q^{21}+ \cdots + ( - 2435865 \beta_{4} - 516216 \beta_{3} + \cdots + 413521590) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 33 q^{2} - 236 q^{3} + 853 q^{4} + 1480 q^{5} + 7578 q^{6} - 13202 q^{7} - 42423 q^{8} + 10981 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 33 q^{2} - 236 q^{3} + 853 q^{4} + 1480 q^{5} + 7578 q^{6} - 13202 q^{7} - 42423 q^{8} + 10981 q^{9} - 89328 q^{10} - 68036 q^{11} - 406010 q^{12} - 158862 q^{13} - 84700 q^{14} - 687324 q^{15} + 350225 q^{16} - 417605 q^{17} - 1911585 q^{18} - 370992 q^{19} + 1632640 q^{20} + 1783880 q^{21} + 122290 q^{22} + 1645870 q^{23} + 9678702 q^{24} + 3270239 q^{25} + 734846 q^{26} - 2998268 q^{27} + 183372 q^{28} + 3668616 q^{29} + 17048544 q^{30} - 7262362 q^{31} - 5605919 q^{32} - 11334900 q^{33} + 2756193 q^{34} - 26503988 q^{35} + 49782133 q^{36} - 31420708 q^{37} + 18513700 q^{38} - 42449884 q^{39} - 53930464 q^{40} - 7996938 q^{41} - 44519496 q^{42} - 56908268 q^{43} + 43323054 q^{44} + 12799536 q^{45} - 32063472 q^{46} - 16903336 q^{47} - 102794498 q^{48} - 11784059 q^{49} + 85921093 q^{50} + 19710956 q^{51} + 173619082 q^{52} - 83362982 q^{53} + 386329164 q^{54} + 6363364 q^{55} + 317409372 q^{56} + 136615904 q^{57} + 64577488 q^{58} - 37946604 q^{59} - 223158912 q^{60} - 77685452 q^{61} + 324855300 q^{62} - 191945278 q^{63} + 131623105 q^{64} - 40321288 q^{65} + 298037676 q^{66} - 304503600 q^{67} - 71243413 q^{68} - 333409272 q^{69} - 122787392 q^{70} - 476602922 q^{71} - 1301701911 q^{72} - 289980486 q^{73} + 262289012 q^{74} - 153685772 q^{75} - 1031276084 q^{76} - 143385648 q^{77} + 691646196 q^{78} - 828240610 q^{79} + 912750944 q^{80} + 891328609 q^{81} - 1109615654 q^{82} + 194681148 q^{83} + 1541719592 q^{84} - 123611080 q^{85} + 1164707144 q^{86} + 158149884 q^{87} - 1017979978 q^{88} + 376848106 q^{89} - 2240087472 q^{90} + 194543664 q^{91} + 2506713088 q^{92} + 3494835920 q^{93} - 2244811104 q^{94} + 1498679864 q^{95} + 2935047582 q^{96} + 692035246 q^{97} + 871744055 q^{98} + 2027106408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{4} + 207\nu^{3} + 6301\nu^{2} - 209023\nu - 509736 ) / 8096 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{4} + 345\nu^{3} - 24845\nu^{2} - 323145\nu + 3450824 ) / 16192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{4} + 115\nu^{3} - 13679\nu^{2} - 123907\nu + 4604568 ) / 16192 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -3\beta_{4} + \beta_{3} - 3\beta _1 + 640 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{4} + 12\beta_{3} + 28\beta_{2} + 993\beta _1 - 1932 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3615\beta_{4} + 1757\beta_{3} - 460\beta_{2} - 4475\beta _1 + 624596 \) 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
−35.0613
−21.1654
5.77274
18.8209
33.6330
−42.0613 −256.773 1257.15 1407.25 10800.2 −5138.64 −31342.2 46249.6 −59190.7
1.2 −28.1654 −3.02373 281.287 762.851 85.1644 5573.11 6498.11 −19673.9 −21486.0
1.3 −1.22726 177.437 −510.494 −1620.18 −217.762 −1834.42 1254.87 11801.0 1988.39
1.4 11.8209 −67.6654 −372.266 2390.67 −799.867 −11355.8 −10452.8 −15104.4 28259.9
1.5 26.6330 −85.9747 197.318 −1460.58 −2289.77 −446.232 −8380.94 −12291.3 −38899.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)

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 17.10.a.a 5
3.b odd 2 1 153.10.a.c 5
4.b odd 2 1 272.10.a.f 5
17.b even 2 1 289.10.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.a 5 1.a even 1 1 trivial
153.10.a.c 5 3.b odd 2 1
272.10.a.f 5 4.b odd 2 1
289.10.a.a 5 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 33T_{2}^{4} - 1162T_{2}^{3} - 24920T_{2}^{2} + 344192T_{2} + 457728 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(17))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 33 T^{4} - 1162 T^{3} + \cdots + 457728 \) Copy content Toggle raw display
$3$ \( T^{5} + 236 T^{4} + \cdots - 801447696 \) Copy content Toggle raw display
$5$ \( T^{5} - 1480 T^{4} + \cdots - 60\!\cdots\!20 \) Copy content Toggle raw display
$7$ \( T^{5} + 13202 T^{4} + \cdots - 26\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{5} + 68036 T^{4} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{5} + 158862 T^{4} + \cdots + 46\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( (T + 83521)^{5} \) Copy content Toggle raw display
$19$ \( T^{5} + 370992 T^{4} + \cdots - 23\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{5} - 1645870 T^{4} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{5} - 3668616 T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + 7262362 T^{4} + \cdots - 63\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{5} + 31420708 T^{4} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{5} + 7996938 T^{4} + \cdots + 84\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{5} + 56908268 T^{4} + \cdots - 35\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{5} + 16903336 T^{4} + \cdots - 53\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{5} + 83362982 T^{4} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{5} + 37946604 T^{4} + \cdots + 43\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{5} + 77685452 T^{4} + \cdots - 81\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{5} + 304503600 T^{4} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{5} + 476602922 T^{4} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{5} + 289980486 T^{4} + \cdots + 39\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{5} + 828240610 T^{4} + \cdots + 88\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{5} - 194681148 T^{4} + \cdots - 48\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{5} - 376848106 T^{4} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{5} - 692035246 T^{4} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
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