Properties

Label 17.18.a.b
Level $17$
Weight $18$
Character orbit 17.a
Self dual yes
Analytic conductor $31.148$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [17,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 18 \)
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: \(31.1477548486\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 1187158 x^{10} - 42381124 x^{9} + 516627945000 x^{8} + 35391783973088 x^{7} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{34}\cdot 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 44) q^{2} + (\beta_{2} + 2 \beta_1 + 1790) q^{3} + (\beta_{3} + \beta_{2} - 33 \beta_1 + 68719) q^{4} + ( - \beta_{5} + \beta_{3} - 8 \beta_{2} - 316 \beta_1 + 147570) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 48 \beta_{2} - 3102 \beta_1 - 274087) q^{6} + (\beta_{7} - 6 \beta_{5} + 23 \beta_{3} + 283 \beta_{2} + 4508 \beta_1 + 1864497) q^{7} + (2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - 18 \beta_{5} + \cdots + 3802062) q^{8}+ \cdots + ( - 9 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + \cdots + 62231549) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 44) q^{2} + (\beta_{2} + 2 \beta_1 + 1790) q^{3} + (\beta_{3} + \beta_{2} - 33 \beta_1 + 68719) q^{4} + ( - \beta_{5} + \beta_{3} - 8 \beta_{2} - 316 \beta_1 + 147570) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 48 \beta_{2} - 3102 \beta_1 - 274087) q^{6} + (\beta_{7} - 6 \beta_{5} + 23 \beta_{3} + 283 \beta_{2} + 4508 \beta_1 + 1864497) q^{7} + (2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - 18 \beta_{5} + \cdots + 3802062) q^{8}+ \cdots + (1709547604 \beta_{11} + 919273308 \beta_{10} + \cdots - 93\!\cdots\!36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 527 q^{2} + 21484 q^{3} + 824597 q^{4} + 1770512 q^{5} - 3292232 q^{6} + 22379060 q^{7} + 45571509 q^{8} + 746877740 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 527 q^{2} + 21484 q^{3} + 824597 q^{4} + 1770512 q^{5} - 3292232 q^{6} + 22379060 q^{7} + 45571509 q^{8} + 746877740 q^{9} + 824176946 q^{10} + 1508850220 q^{11} + 4380824560 q^{12} + 2071135824 q^{13} - 9581529548 q^{14} - 15603628992 q^{15} + 20409782305 q^{16} + 83709089292 q^{17} - 195058406893 q^{18} - 1108652840 q^{19} + 304149438198 q^{20} + 711976780056 q^{21} + 350258857160 q^{22} + 816663191324 q^{23} - 118287259488 q^{24} + 841645606900 q^{25} + 1135562035330 q^{26} + 2668523518384 q^{27} + 7745458522756 q^{28} - 4667447777712 q^{29} + 15034204946608 q^{30} + 23367156664948 q^{31} + 16779331087925 q^{32} + 25952886361216 q^{33} + 3676224171407 q^{34} + 52808829544064 q^{35} + 178683325289537 q^{36} + 45103867774096 q^{37} + 141942362802260 q^{38} + 187488571444048 q^{39} + 231489638087062 q^{40} + 106534751749128 q^{41} + 82630767165312 q^{42} + 350248430413232 q^{43} + 468477290587728 q^{44} + 238676787528480 q^{45} + 587080431637364 q^{46} + 419195348129576 q^{47} + 10\!\cdots\!72 q^{48}+ \cdots - 11\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 1187158 x^{10} - 42381124 x^{9} + 516627945000 x^{8} + 35391783973088 x^{7} + \cdots + 61\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{11} + \cdots + 88\!\cdots\!20 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!93 \nu^{11} + \cdots - 23\!\cdots\!20 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!51 \nu^{11} + \cdots - 20\!\cdots\!40 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\!\cdots\!13 \nu^{11} + \cdots - 26\!\cdots\!20 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!33 \nu^{11} + \cdots + 31\!\cdots\!80 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\!\cdots\!03 \nu^{11} + \cdots - 65\!\cdots\!20 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!41 \nu^{11} + \cdots + 12\!\cdots\!60 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 78\!\cdots\!19 \nu^{11} + \cdots - 50\!\cdots\!60 ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25\!\cdots\!29 \nu^{11} + \cdots - 22\!\cdots\!40 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!87 \nu^{11} + \cdots - 52\!\cdots\!80 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 55\beta _1 + 197855 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + 18 \beta_{5} + 2 \beta_{4} + 35 \beta_{3} + 153 \beta_{2} + 316888 \beta _1 + 10865646 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1012 \beta_{11} + 159 \beta_{10} - 1345 \beta_{9} - 285 \beta_{8} - 468 \beta_{7} + 1369 \beta_{6} + 4604 \beta_{5} + 444 \beta_{4} + 405035 \beta_{3} + 889335 \beta_{2} + 33391568 \beta _1 + 62692964896 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 841044 \beta_{11} - 629971 \beta_{10} - 574467 \beta_{9} - 648343 \beta_{8} + 1093188 \beta_{7} + 874779 \beta_{6} + 7634948 \beta_{5} + 1505540 \beta_{4} + \cdots + 6579798020940 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 555254204 \beta_{11} + 116709683 \beta_{10} - 774110381 \beta_{9} - 262931017 \beta_{8} - 212882572 \beta_{7} + 961945445 \beta_{6} + 3958738484 \beta_{5} + \cdots + 22\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 284056367836 \beta_{11} - 300733753547 \beta_{10} - 295540766795 \beta_{9} - 299946644111 \beta_{8} + 503043359276 \beta_{7} + 501407932019 \beta_{6} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 248284637042076 \beta_{11} + 59689510377043 \beta_{10} - 348145387717677 \beta_{9} - 159126389766089 \beta_{8} - 72630948452716 \beta_{7} + \cdots + 84\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 86\!\cdots\!36 \beta_{11} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10\!\cdots\!24 \beta_{11} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22\!\cdots\!84 \beta_{11} + \cdots + 84\!\cdots\!04 \) 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
679.099
593.609
484.598
438.925
84.5783
7.28517
−7.43695
−267.738
−351.322
−472.034
−564.709
−623.855
−635.099 21620.7 272279. −557056. −1.37313e7 2.41616e7 −8.96804e7 3.38316e8 3.53786e8
1.2 −549.609 −20215.1 170998. 363670. 1.11104e7 −612411. −2.19436e7 2.79512e8 −1.99876e8
1.3 −440.598 12134.5 63054.5 −636939. −5.34642e6 −2.19133e7 2.99684e7 1.81051e7 2.80634e8
1.4 −394.925 −2715.83 24893.8 1.09984e6 1.07255e6 2.44505e7 4.19324e7 −1.21764e8 −4.34356e8
1.5 −40.5783 −7620.30 −129425. 755663. 309219. −713602. 1.05705e7 −7.10711e7 −3.06635e7
1.6 36.7148 18306.2 −129724. 943214. 672108. −8.88671e6 −9.57508e6 2.05976e8 3.46300e7
1.7 51.4370 −334.377 −128426. −1.22185e6 −17199.3 −1.54284e7 −1.33478e7 −1.29028e8 −6.28483e7
1.8 311.738 1657.27 −33891.5 −805866. 516633. 2.47100e7 −5.14254e7 −1.26394e8 −2.51219e8
1.9 395.322 −17262.0 25207.2 −631340. −6.82405e6 −1.72779e7 −4.18507e7 1.68837e8 −2.49582e8
1.10 516.034 15990.3 135219. 1.06954e6 8.25154e6 2.48734e7 2.13988e6 1.26550e8 5.51917e8
1.11 608.709 −12610.8 239455. 1.65111e6 −7.67629e6 −2.47907e6 6.59734e7 2.98913e7 1.00504e9
1.12 667.855 12533.5 314959. −259472. 8.37056e6 −8.50519e6 1.22810e8 2.79484e7 −1.73290e8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 17.18.a.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.18.a.b 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 527 T_{2}^{11} - 1059866 T_{2}^{10} + 546096644 T_{2}^{9} + 398261067096 T_{2}^{8} - 202285178523872 T_{2}^{7} + \cdots - 12\!\cdots\!64 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(17))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 527 T^{11} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{12} - 21484 T^{11} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{12} - 1770512 T^{11} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} - 22379060 T^{11} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{12} - 1508850220 T^{11} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} - 2071135824 T^{11} + \cdots - 22\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( (T - 6975757441)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + 1108652840 T^{11} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} - 816663191324 T^{11} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + 4667447777712 T^{11} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 23367156664948 T^{11} + \cdots + 32\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{12} - 45103867774096 T^{11} + \cdots + 58\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{12} - 106534751749128 T^{11} + \cdots - 91\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{12} - 350248430413232 T^{11} + \cdots + 23\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{12} - 419195348129576 T^{11} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} - 56947812536104 T^{11} + \cdots - 28\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 55\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 33\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 52\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 85\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 75\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
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