Properties

Label 289.3.e.a
Level $289$
Weight $3$
Character orbit 289.e
Analytic conductor $7.875$
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] = [289,3,Mod(40,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([15]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 289.e (of order \(16\), degree \(8\), 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: \(7.87467964001\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{16}^{7} + \zeta_{16}^{6} - 1) q^{2} + (\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{2} - 2 \zeta_{16} - 2) q^{3} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} - 2 \zeta_{16}^{5} - \zeta_{16}^{4} + 1) q^{4} + (3 \zeta_{16}^{7} - 3 \zeta_{16}^{6} - \zeta_{16}^{5} - 2 \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} - 2 \zeta_{16} - 1) q^{5} + ( - 5 \zeta_{16}^{7} - 3 \zeta_{16}^{6} + 3 \zeta_{16} + 5) q^{6} + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \zeta_{16}^{5} - \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{2} - 2 \zeta_{16} - 2) q^{7} + (3 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - \zeta_{16}^{5} - 2 \zeta_{16}^{4} + 3 \zeta_{16}^{3} + \zeta_{16}^{2} - 1) q^{8} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 7 \zeta_{16}^{2} + 3 \zeta_{16} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{16}^{7} + \zeta_{16}^{6} - 1) q^{2} + (\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{2} - 2 \zeta_{16} - 2) q^{3} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} - 2 \zeta_{16}^{5} - \zeta_{16}^{4} + 1) q^{4} + (3 \zeta_{16}^{7} - 3 \zeta_{16}^{6} - \zeta_{16}^{5} - 2 \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} - 2 \zeta_{16} - 1) q^{5} + ( - 5 \zeta_{16}^{7} - 3 \zeta_{16}^{6} + 3 \zeta_{16} + 5) q^{6} + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \zeta_{16}^{5} - \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{2} - 2 \zeta_{16} - 2) q^{7} + (3 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - \zeta_{16}^{5} - 2 \zeta_{16}^{4} + 3 \zeta_{16}^{3} + \zeta_{16}^{2} - 1) q^{8} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 7 \zeta_{16}^{2} + 3 \zeta_{16} + 7) q^{9} + ( - 6 \zeta_{16}^{7} - \zeta_{16}^{6} + \zeta_{16}^{5} + 6 \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 2) q^{10} + (5 \zeta_{16}^{5} - 3 \zeta_{16}^{4} - 3 \zeta_{16}^{3} + 5 \zeta_{16}^{2}) q^{11} + (5 \zeta_{16}^{7} + 5 \zeta_{16}^{6} + 4 \zeta_{16}^{5} + 3 \zeta_{16}^{4} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots - 4) q^{12}+ \cdots + (32 \zeta_{16}^{7} - 17 \zeta_{16}^{6} + 15 \zeta_{16}^{5} + 15 \zeta_{16}^{4} - 17 \zeta_{16}^{3} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 16 q^{3} + 8 q^{4} - 8 q^{5} + 40 q^{6} - 16 q^{7} - 8 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 16 q^{3} + 8 q^{4} - 8 q^{5} + 40 q^{6} - 16 q^{7} - 8 q^{8} + 56 q^{9} + 16 q^{10} - 32 q^{12} + 16 q^{14} + 80 q^{15} - 136 q^{18} + 80 q^{19} - 48 q^{20} + 64 q^{21} - 40 q^{22} - 16 q^{23} + 88 q^{24} + 136 q^{25} - 40 q^{26} - 64 q^{27} - 40 q^{28} + 144 q^{29} - 168 q^{30} + 88 q^{31} - 16 q^{32} + 80 q^{35} + 72 q^{36} - 24 q^{37} - 8 q^{38} + 96 q^{40} + 144 q^{41} - 88 q^{42} - 48 q^{43} - 112 q^{44} - 8 q^{46} + 80 q^{47} + 40 q^{48} - 72 q^{49} + 240 q^{52} - 88 q^{53} + 256 q^{54} + 8 q^{55} + 64 q^{56} - 200 q^{57} - 144 q^{58} + 40 q^{59} + 96 q^{60} + 88 q^{61} - 24 q^{62} - 216 q^{63} + 120 q^{64} - 160 q^{65} + 120 q^{66} - 208 q^{69} - 96 q^{70} - 48 q^{71} - 24 q^{72} - 224 q^{73} + 104 q^{74} - 272 q^{75} - 192 q^{76} - 120 q^{77} + 232 q^{78} + 136 q^{79} + 80 q^{80} + 224 q^{81} - 144 q^{82} - 504 q^{83} + 288 q^{86} - 216 q^{87} - 112 q^{88} - 288 q^{89} + 344 q^{90} + 16 q^{91} - 48 q^{92} - 248 q^{93} - 72 q^{94} + 144 q^{95} + 80 q^{96} - 208 q^{97} - 16 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
0.923880 0.382683i
−0.923880 0.382683i
−0.382683 0.923880i
0.382683 0.923880i
−0.382683 + 0.923880i
0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 + 0.382683i
−2.63099 1.08979i −5.09606 + 1.01367i 2.90602 + 2.90602i −2.02560 + 3.03153i 14.5124 + 2.88669i −1.83409 2.74491i −0.119585 0.288703i 16.6274 6.88730i 8.63307 5.76843i
65.1 −0.783227 + 0.324423i −0.318152 + 1.59946i −2.32023 + 2.32023i 5.68246 3.79690i −0.269716 1.35595i −0.751697 0.502268i 2.36223 5.70292i 5.85788 + 2.42641i −3.21885 + 4.81736i
75.1 0.0897902 0.216773i 0.779037 1.16591i 2.78950 + 2.78950i −0.0672509 0.338093i −0.182788 0.273561i −1.40054 + 7.04101i 1.72225 0.713379i 2.69170 + 6.49834i −0.0793278 0.0157793i
131.1 −0.675577 1.63099i −3.36482 + 2.24830i 0.624715 0.624715i −7.58960 1.50967i 5.94015 + 3.96908i −4.01367 + 0.798369i −7.96489 3.29916i 2.82302 6.81537i 2.66511 + 13.3984i
158.1 0.0897902 + 0.216773i 0.779037 + 1.16591i 2.78950 2.78950i −0.0672509 + 0.338093i −0.182788 + 0.273561i −1.40054 7.04101i 1.72225 + 0.713379i 2.69170 6.49834i −0.0793278 + 0.0157793i
214.1 −0.675577 + 1.63099i −3.36482 2.24830i 0.624715 + 0.624715i −7.58960 + 1.50967i 5.94015 3.96908i −4.01367 0.798369i −7.96489 + 3.29916i 2.82302 + 6.81537i 2.66511 13.3984i
224.1 −2.63099 + 1.08979i −5.09606 1.01367i 2.90602 2.90602i −2.02560 3.03153i 14.5124 2.88669i −1.83409 + 2.74491i −0.119585 + 0.288703i 16.6274 + 6.88730i 8.63307 + 5.76843i
249.1 −0.783227 0.324423i −0.318152 1.59946i −2.32023 2.32023i 5.68246 + 3.79690i −0.269716 + 1.35595i −0.751697 + 0.502268i 2.36223 + 5.70292i 5.85788 2.42641i −3.21885 4.81736i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.3.e.a 8
17.b even 2 1 289.3.e.e 8
17.c even 4 1 289.3.e.j 8
17.c even 4 1 289.3.e.n 8
17.d even 8 1 17.3.e.b 8
17.d even 8 1 289.3.e.f 8
17.d even 8 1 289.3.e.g 8
17.d even 8 1 289.3.e.h 8
17.e odd 16 1 17.3.e.b 8
17.e odd 16 1 inner 289.3.e.a 8
17.e odd 16 1 289.3.e.e 8
17.e odd 16 1 289.3.e.f 8
17.e odd 16 1 289.3.e.g 8
17.e odd 16 1 289.3.e.h 8
17.e odd 16 1 289.3.e.j 8
17.e odd 16 1 289.3.e.n 8
51.g odd 8 1 153.3.p.a 8
51.i even 16 1 153.3.p.a 8
68.g odd 8 1 272.3.bh.b 8
68.i even 16 1 272.3.bh.b 8
85.k odd 8 1 425.3.t.d 8
85.m even 8 1 425.3.u.a 8
85.n odd 8 1 425.3.t.b 8
85.o even 16 1 425.3.t.b 8
85.p odd 16 1 425.3.u.a 8
85.r even 16 1 425.3.t.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.b 8 17.d even 8 1
17.3.e.b 8 17.e odd 16 1
153.3.p.a 8 51.g odd 8 1
153.3.p.a 8 51.i even 16 1
272.3.bh.b 8 68.g odd 8 1
272.3.bh.b 8 68.i even 16 1
289.3.e.a 8 1.a even 1 1 trivial
289.3.e.a 8 17.e odd 16 1 inner
289.3.e.e 8 17.b even 2 1
289.3.e.e 8 17.e odd 16 1
289.3.e.f 8 17.d even 8 1
289.3.e.f 8 17.e odd 16 1
289.3.e.g 8 17.d even 8 1
289.3.e.g 8 17.e odd 16 1
289.3.e.h 8 17.d even 8 1
289.3.e.h 8 17.e odd 16 1
289.3.e.j 8 17.c even 4 1
289.3.e.j 8 17.e odd 16 1
289.3.e.n 8 17.c even 4 1
289.3.e.n 8 17.e odd 16 1
425.3.t.b 8 85.n odd 8 1
425.3.t.b 8 85.o even 16 1
425.3.t.d 8 85.k odd 8 1
425.3.t.d 8 85.r even 16 1
425.3.u.a 8 85.m even 8 1
425.3.u.a 8 85.p odd 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{8} + 8T_{2}^{7} + 28T_{2}^{6} + 56T_{2}^{5} + 72T_{2}^{4} + 48T_{2}^{3} + 12T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 16T_{3}^{7} + 100T_{3}^{6} + 304T_{3}^{5} + 484T_{3}^{4} + 624T_{3}^{3} + 1184T_{3}^{2} + 544T_{3} + 2312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 8 T^{7} + 28 T^{6} + 56 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 16 T^{7} + 100 T^{6} + \cdots + 2312 \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} - 36 T^{6} + \cdots + 4418 \) Copy content Toggle raw display
$7$ \( T^{8} + 16 T^{7} + 164 T^{6} + \cdots + 7688 \) Copy content Toggle raw display
$11$ \( T^{8} - 84 T^{6} - 240 T^{5} + \cdots + 2221832 \) Copy content Toggle raw display
$13$ \( T^{8} - 784 T^{5} + \cdots + 16273156 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 80 T^{7} + \cdots + 34090452496 \) Copy content Toggle raw display
$23$ \( T^{8} + 16 T^{7} + \cdots + 564614408 \) Copy content Toggle raw display
$29$ \( T^{8} - 144 T^{7} + \cdots + 55846825218 \) Copy content Toggle raw display
$31$ \( T^{8} - 88 T^{7} + \cdots + 182700453128 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + \cdots + 368317129538 \) Copy content Toggle raw display
$41$ \( T^{8} - 144 T^{7} + \cdots + 126445152962 \) Copy content Toggle raw display
$43$ \( T^{8} + 48 T^{7} + \cdots + 6626611216 \) Copy content Toggle raw display
$47$ \( T^{8} - 80 T^{7} + \cdots + 298321984 \) Copy content Toggle raw display
$53$ \( T^{8} + 88 T^{7} + \cdots + 41458660996 \) Copy content Toggle raw display
$59$ \( T^{8} - 40 T^{7} + \cdots + 2347596304 \) Copy content Toggle raw display
$61$ \( T^{8} - 88 T^{7} + \cdots + 22880163635522 \) Copy content Toggle raw display
$67$ \( T^{8} + 5648 T^{6} + \cdots + 883915868224 \) Copy content Toggle raw display
$71$ \( T^{8} + 48 T^{7} + \cdots + 53847249369608 \) Copy content Toggle raw display
$73$ \( T^{8} + 224 T^{7} + \cdots + 18825456624578 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 376288804594952 \) Copy content Toggle raw display
$83$ \( T^{8} + 504 T^{7} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 128947789048324 \) Copy content Toggle raw display
$97$ \( T^{8} + 208 T^{7} + \cdots + 18\!\cdots\!42 \) Copy content Toggle raw display
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