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Properties

Label 17.4.c.a
Level $17$
Weight $4$
Character orbit 17.c
Analytic conductor $1.003$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [17,4,Mod(4,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.c (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(1.00303247010\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 46x^{6} + 561x^{4} + 836x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{4} q^{3} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 5) q^{4}+ \cdots + (3 \beta_{5} - 3 \beta_{4} + \cdots - 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{4} q^{3} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 5) q^{4}+ \cdots + ( - 52 \beta_{7} + 189 \beta_{4} + \cdots - 248) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{4} + 14 q^{5} + 22 q^{6} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 36 q^{4} + 14 q^{5} + 22 q^{6} + 2 q^{7} + 78 q^{10} - 108 q^{11} - 174 q^{12} - 88 q^{13} + 108 q^{14} + 420 q^{16} - 10 q^{17} + 428 q^{18} - 306 q^{20} - 260 q^{21} + 30 q^{22} - 22 q^{23} - 862 q^{24} + 540 q^{27} - 764 q^{28} + 46 q^{29} - 120 q^{30} + 610 q^{31} + 816 q^{33} + 1002 q^{34} + 1172 q^{35} - 574 q^{37} - 768 q^{38} - 844 q^{39} - 342 q^{40} - 968 q^{41} + 550 q^{44} - 1154 q^{45} - 944 q^{46} - 368 q^{47} + 2494 q^{48} + 468 q^{50} + 296 q^{51} + 2564 q^{52} - 1592 q^{54} - 1996 q^{55} + 684 q^{56} - 300 q^{57} + 266 q^{58} + 1258 q^{61} - 2516 q^{62} + 122 q^{63} - 3044 q^{64} + 628 q^{65} + 764 q^{67} + 1914 q^{68} + 1812 q^{69} + 1266 q^{71} + 1404 q^{72} - 1732 q^{73} + 1538 q^{74} + 1292 q^{75} - 2836 q^{78} + 914 q^{79} + 498 q^{80} + 280 q^{81} - 280 q^{82} - 2952 q^{84} - 2498 q^{85} - 4244 q^{86} + 442 q^{88} - 2156 q^{89} + 2478 q^{90} - 1632 q^{91} - 1768 q^{92} + 1484 q^{95} + 3998 q^{96} + 1836 q^{97} + 6728 q^{98} - 2088 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 46x^{6} + 561x^{4} + 836x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} - 23\nu^{2} - 16 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 46\nu^{5} + 545\nu^{3} + 468\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 42\nu^{5} + 172\nu^{4} - 413\nu^{3} + 1864\nu^{2} + 396\nu + 1120 ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 42\nu^{5} + 172\nu^{4} + 413\nu^{3} + 1864\nu^{2} - 396\nu + 1120 ) / 160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 4\nu^{6} - 226\nu^{5} + 180\nu^{4} - 2673\nu^{3} + 2128\nu^{2} - 3156\nu + 2208 ) / 160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 4\nu^{6} + 226\nu^{5} + 180\nu^{4} + 2673\nu^{3} + 2128\nu^{2} + 3156\nu + 2208 ) / 160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 8\beta_{3} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -23\beta_{7} - 23\beta_{6} + 23\beta_{5} + 23\beta_{4} - 33\beta_{2} + 260 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -33\beta_{7} + 33\beta_{6} + 13\beta_{5} - 13\beta_{4} + 304\beta_{3} + 477\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 523\beta_{7} + 523\beta_{6} - 503\beta_{5} - 503\beta_{4} + 953\beta_{2} - 5868 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 973\beta_{7} - 973\beta_{6} - 53\beta_{5} + 53\beta_{4} - 9464\beta_{3} - 10965\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 4.93651i
− 0.648995i
1.11783i
4.46767i
− 4.46767i
− 1.11783i
0.648995i
4.93651i
− 3.93651i −0.299807 + 0.299807i −7.49613 1.37942 − 1.37942i 1.18019 + 1.18019i 17.9849 + 17.9849i − 1.98349i 26.8202i −5.43011 − 5.43011i
4.2 0.351005i 2.28193 − 2.28193i 7.87680 −9.32676 + 9.32676i 0.800971 + 0.800971i −23.5385 − 23.5385i 5.57284i 16.5856i −3.27374 − 3.27374i
4.3 2.11783i −5.92758 + 5.92758i 3.51478 10.1567 − 10.1567i −12.5536 − 12.5536i 3.21600 + 3.21600i 24.3864i − 43.2725i 21.5102 + 21.5102i
4.4 5.46767i 3.94546 − 3.94546i −21.8954 4.79064 − 4.79064i 21.5725 + 21.5725i 3.33761 + 3.33761i − 75.9757i − 4.13329i 26.1937 + 26.1937i
13.1 − 5.46767i 3.94546 + 3.94546i −21.8954 4.79064 + 4.79064i 21.5725 − 21.5725i 3.33761 − 3.33761i 75.9757i 4.13329i 26.1937 − 26.1937i
13.2 − 2.11783i −5.92758 − 5.92758i 3.51478 10.1567 + 10.1567i −12.5536 + 12.5536i 3.21600 − 3.21600i − 24.3864i 43.2725i 21.5102 − 21.5102i
13.3 − 0.351005i 2.28193 + 2.28193i 7.87680 −9.32676 − 9.32676i 0.800971 − 0.800971i −23.5385 + 23.5385i − 5.57284i − 16.5856i −3.27374 + 3.27374i
13.4 3.93651i −0.299807 − 0.299807i −7.49613 1.37942 + 1.37942i 1.18019 − 1.18019i 17.9849 − 17.9849i 1.98349i − 26.8202i −5.43011 + 5.43011i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.4.c.a ✓ 8
3.b odd 2 1 153.4.f.a 8
4.b odd 2 1 272.4.o.e 8
17.c even 4 1 inner 17.4.c.a ✓ 8
17.d even 8 2 289.4.a.f 8
17.d even 8 2 289.4.b.c 8
51.f odd 4 1 153.4.f.a 8
68.f odd 4 1 272.4.o.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.c.a ✓ 8 1.a even 1 1 trivial
17.4.c.a ✓ 8 17.c even 4 1 inner
153.4.f.a 8 3.b odd 2 1
153.4.f.a 8 51.f odd 4 1
272.4.o.e 8 4.b odd 2 1
272.4.o.e 8 68.f odd 4 1
289.4.a.f 8 17.d even 8 2
289.4.b.c 8 17.d even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 50 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} - 180 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( T^{8} - 14 T^{7} + \cdots + 6270016 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 330366976 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 40571627776 \) Copy content Toggle raw display
$13$ \( (T^{4} + 44 T^{3} + \cdots - 468640)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2286918209536 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23983351398400 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{4} + 184 T^{3} + \cdots + 1730640896)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{4} - 382 T^{3} + \cdots - 80371889536)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1078 T^{3} + \cdots - 22878545920)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
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