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Properties

Label 17.14.b.a
Level $17$
Weight $14$
Character orbit 17.b
Analytic conductor $18.229$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [17,14,Mod(16,17)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17.16"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 17.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.2292579218\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 18192152 x^{16} + 135370932931108 x^{14} + \cdots + 81\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{6}\cdot 17^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{3} + 2710) q^{4} + ( - \beta_{12} - 3 \beta_1) q^{5} + ( - \beta_{10} + 3 \beta_1) q^{6} + ( - \beta_{12} + \beta_{11} + 19 \beta_1) q^{7} + ( - \beta_{4} - 8 \beta_{3} + \cdots - 644) q^{8}+ \cdots + (57180 \beta_{17} + \cdots - 3316889813 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 48786 q^{4} - 16386 q^{8} - 7686490 q^{9} + 16136148 q^{13} + 124323840 q^{15} + 65611842 q^{16} + 25023898 q^{17} - 123776710 q^{18} - 516719736 q^{19} - 665521472 q^{21} - 2475440166 q^{25}+ \cdots - 48142146348894 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 18192152 x^{16} + 135370932931108 x^{14} + \cdots + 81\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!47 \nu^{16} + \cdots + 49\!\cdots\!92 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!37 \nu^{16} + \cdots - 34\!\cdots\!32 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!97 \nu^{16} + \cdots + 16\!\cdots\!08 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59\!\cdots\!87 \nu^{16} + \cdots + 14\!\cdots\!92 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37\!\cdots\!29 \nu^{16} + \cdots + 38\!\cdots\!56 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39\!\cdots\!71 \nu^{16} + \cdots - 47\!\cdots\!44 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!87 \nu^{16} + \cdots + 17\!\cdots\!32 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61\!\cdots\!13 \nu^{16} + \cdots - 96\!\cdots\!68 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!47 \nu^{17} + \cdots - 47\!\cdots\!92 \nu ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!87 \nu^{17} + \cdots + 91\!\cdots\!68 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!29 \nu^{17} + \cdots - 13\!\cdots\!56 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\!\cdots\!99 \nu^{17} + \cdots + 61\!\cdots\!64 \nu ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 55\!\cdots\!37 \nu^{17} + \cdots - 21\!\cdots\!32 \nu ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 86\!\cdots\!49 \nu^{17} + \cdots - 23\!\cdots\!64 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 25\!\cdots\!37 \nu^{17} + \cdots - 48\!\cdots\!32 \nu ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 68\!\cdots\!83 \nu^{17} + \cdots - 39\!\cdots\!68 \nu ) / 31\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 33\beta_{3} - 630\beta_{2} - 2021431 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 12 \beta_{17} + 88 \beta_{16} - 108 \beta_{15} - 225 \beta_{14} + 12 \beta_{13} + 234 \beta_{12} + \cdots - 3309545 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2340 \beta_{9} - 61434 \beta_{8} - 20136 \beta_{7} - 15585 \beta_{6} - 4436500 \beta_{5} + \cdots + 6690782165977 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 66553500 \beta_{17} - 588778504 \beta_{16} + 741556188 \beta_{15} + 1195415370 \beta_{14} + \cdots + 12633714645050 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9510682320 \beta_{9} + 425315727396 \beta_{8} + 164738836992 \beta_{7} + 144579893406 \beta_{6} + \cdots - 25\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 325013208544920 \beta_{17} + \cdots - 51\!\cdots\!88 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33\!\cdots\!00 \beta_{9} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 15\!\cdots\!80 \beta_{17} + \cdots + 22\!\cdots\!52 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 67\!\cdots\!60 \beta_{9} + \cdots - 45\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 73\!\cdots\!88 \beta_{17} + \cdots - 99\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 58\!\cdots\!40 \beta_{9} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 34\!\cdots\!08 \beta_{17} + \cdots + 45\!\cdots\!88 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 39\!\cdots\!80 \beta_{9} + \cdots - 91\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 16\!\cdots\!44 \beta_{17} + \cdots - 21\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 23\!\cdots\!60 \beta_{9} + \cdots + 42\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 79\!\cdots\!76 \beta_{17} + \cdots + 99\!\cdots\!04 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
− 1610.78i
1610.78i
− 591.744i
591.744i
− 1711.77i
1711.77i
− 700.781i
700.781i
− 2206.85i
2206.85i
− 132.632i
132.632i
− 1985.62i
1985.62i
− 1456.81i
1456.81i
− 934.529i
934.529i
−160.159 − 1610.78i 17458.8 13282.0i 257980.i − 80458.9i −1.48416e6 −1.00029e6 − 2.12722e6i
16.2 −160.159 1610.78i 17458.8 − 13282.0i − 257980.i 80458.9i −1.48416e6 −1.00029e6 2.12722e6i
16.3 −125.237 − 591.744i 7492.39 − 41487.1i 74108.5i − 240258.i 87617.3 1.24416e6 5.19573e6i
16.4 −125.237 591.744i 7492.39 41487.1i − 74108.5i 240258.i 87617.3 1.24416e6 − 5.19573e6i
16.5 −66.6386 − 1711.77i −3751.29 − 6896.67i 114070.i 473258.i 795885. −1.33583e6 459584.i
16.6 −66.6386 1711.77i −3751.29 6896.67i − 114070.i − 473258.i 795885. −1.33583e6 − 459584.i
16.7 −54.7247 − 700.781i −5197.21 60101.0i 38350.0i − 240607.i 732720. 1.10323e6 − 3.28901e6i
16.8 −54.7247 700.781i −5197.21 − 60101.0i − 38350.0i 240607.i 732720. 1.10323e6 3.28901e6i
16.9 −3.92602 − 2206.85i −8176.59 − 19074.9i 8664.15i − 561094.i 64263.4 −3.27588e6 74888.4i
16.10 −3.92602 2206.85i −8176.59 19074.9i − 8664.15i 561094.i 64263.4 −3.27588e6 − 74888.4i
16.11 39.8458 − 132.632i −6604.32 − 13351.8i − 5284.84i − 43497.8i −589570. 1.57673e6 − 532014.i
16.12 39.8458 132.632i −6604.32 13351.8i 5284.84i 43497.8i −589570. 1.57673e6 532014.i
16.13 96.0241 − 1985.62i 1028.63 63719.4i − 190668.i 387010.i −687856. −2.34838e6 6.11860e6i
16.14 96.0241 1985.62i 1028.63 − 63719.4i 190668.i − 387010.i −687856. −2.34838e6 − 6.11860e6i
16.15 114.114 − 1456.81i 4830.06 − 42689.7i − 166243.i 143521.i −383645. −527971. − 4.87151e6i
16.16 114.114 1456.81i 4830.06 42689.7i 166243.i − 143521.i −383645. −527971. 4.87151e6i
16.17 159.701 − 934.529i 17312.5 15558.4i − 149246.i − 466546.i 1.45655e6 720978. 2.48470e6i
16.18 159.701 934.529i 17312.5 − 15558.4i 149246.i 466546.i 1.45655e6 720978. − 2.48470e6i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 16.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.14.b.a ✓ 18
3.b odd 2 1 153.14.d.b 18
17.b even 2 1 inner 17.14.b.a ✓ 18
51.c odd 2 1 153.14.d.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.14.b.a ✓ 18 1.a even 1 1 trivial
17.14.b.a ✓ 18 17.b even 2 1 inner
153.14.d.b 18 3.b odd 2 1
153.14.d.b 18 51.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{9} + \cdots + 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 81\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots - 46\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 91\!\cdots\!77 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 55\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 49\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots - 94\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 36\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 96\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 49\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
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