Properties

Label 289.4.b.a
Level $289$
Weight $4$
Character orbit 289.b
Analytic conductor $17.052$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(288,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.288");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not 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: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{2} - 4 \beta q^{3} + q^{4} + 3 \beta q^{5} - 12 \beta q^{6} + 14 \beta q^{7} - 21 q^{8} - 37 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} - 4 \beta q^{3} + q^{4} + 3 \beta q^{5} - 12 \beta q^{6} + 14 \beta q^{7} - 21 q^{8} - 37 q^{9} + 9 \beta q^{10} + 12 \beta q^{11} - 4 \beta q^{12} - 58 q^{13} + 42 \beta q^{14} + 48 q^{15} - 71 q^{16} - 111 q^{18} - 116 q^{19} + 3 \beta q^{20} + 224 q^{21} + 36 \beta q^{22} + 30 \beta q^{23} + 84 \beta q^{24} + 89 q^{25} - 174 q^{26} + 40 \beta q^{27} + 14 \beta q^{28} + 15 \beta q^{29} + 144 q^{30} - 86 \beta q^{31} - 45 q^{32} + 192 q^{33} - 168 q^{35} - 37 q^{36} - 29 \beta q^{37} - 348 q^{38} + 232 \beta q^{39} - 63 \beta q^{40} + 171 \beta q^{41} + 672 q^{42} + 148 q^{43} + 12 \beta q^{44} - 111 \beta q^{45} + 90 \beta q^{46} + 288 q^{47} + 284 \beta q^{48} - 441 q^{49} + 267 q^{50} - 58 q^{52} - 318 q^{53} + 120 \beta q^{54} - 144 q^{55} - 294 \beta q^{56} + 464 \beta q^{57} + 45 \beta q^{58} - 252 q^{59} + 48 q^{60} - 55 \beta q^{61} - 258 \beta q^{62} - 518 \beta q^{63} + 433 q^{64} - 174 \beta q^{65} + 576 q^{66} - 484 q^{67} + 480 q^{69} - 504 q^{70} - 354 \beta q^{71} + 777 q^{72} + 181 \beta q^{73} - 87 \beta q^{74} - 356 \beta q^{75} - 116 q^{76} - 672 q^{77} + 696 \beta q^{78} + 242 \beta q^{79} - 213 \beta q^{80} - 359 q^{81} + 513 \beta q^{82} - 756 q^{83} + 224 q^{84} + 444 q^{86} + 240 q^{87} - 252 \beta q^{88} - 774 q^{89} - 333 \beta q^{90} - 812 \beta q^{91} + 30 \beta q^{92} - 1376 q^{93} + 864 q^{94} - 348 \beta q^{95} + 180 \beta q^{96} - 191 \beta q^{97} - 1323 q^{98} - 444 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 2 q^{4} - 42 q^{8} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 2 q^{4} - 42 q^{8} - 74 q^{9} - 116 q^{13} + 96 q^{15} - 142 q^{16} - 222 q^{18} - 232 q^{19} + 448 q^{21} + 178 q^{25} - 348 q^{26} + 288 q^{30} - 90 q^{32} + 384 q^{33} - 336 q^{35} - 74 q^{36} - 696 q^{38} + 1344 q^{42} + 296 q^{43} + 576 q^{47} - 882 q^{49} + 534 q^{50} - 116 q^{52} - 636 q^{53} - 288 q^{55} - 504 q^{59} + 96 q^{60} + 866 q^{64} + 1152 q^{66} - 968 q^{67} + 960 q^{69} - 1008 q^{70} + 1554 q^{72} - 232 q^{76} - 1344 q^{77} - 718 q^{81} - 1512 q^{83} + 448 q^{84} + 888 q^{86} + 480 q^{87} - 1548 q^{89} - 2752 q^{93} + 1728 q^{94} - 2646 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
288.1
1.00000i
1.00000i
3.00000 8.00000i 1.00000 6.00000i 24.0000i 28.0000i −21.0000 −37.0000 18.0000i
288.2 3.00000 8.00000i 1.00000 6.00000i 24.0000i 28.0000i −21.0000 −37.0000 18.0000i
\(n\): e.g. 2-40 or 990-1000
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 289.4.b.a 2
17.b even 2 1 inner 289.4.b.a 2
17.c even 4 1 17.4.a.a 1
17.c even 4 1 289.4.a.a 1
51.f odd 4 1 153.4.a.d 1
68.f odd 4 1 272.4.a.d 1
85.f odd 4 1 425.4.b.c 2
85.i odd 4 1 425.4.b.c 2
85.j even 4 1 425.4.a.d 1
119.f odd 4 1 833.4.a.a 1
136.i even 4 1 1088.4.a.l 1
136.j odd 4 1 1088.4.a.a 1
187.f odd 4 1 2057.4.a.d 1
204.l even 4 1 2448.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 17.c even 4 1
153.4.a.d 1 51.f odd 4 1
272.4.a.d 1 68.f odd 4 1
289.4.a.a 1 17.c even 4 1
289.4.b.a 2 1.a even 1 1 trivial
289.4.b.a 2 17.b even 2 1 inner
425.4.a.d 1 85.j even 4 1
425.4.b.c 2 85.f odd 4 1
425.4.b.c 2 85.i odd 4 1
833.4.a.a 1 119.f odd 4 1
1088.4.a.a 1 136.j odd 4 1
1088.4.a.l 1 136.i even 4 1
2057.4.a.d 1 187.f odd 4 1
2448.4.a.f 1 204.l even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{2} + 784 \) Copy content Toggle raw display
$11$ \( T^{2} + 576 \) Copy content Toggle raw display
$13$ \( (T + 58)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3600 \) Copy content Toggle raw display
$29$ \( T^{2} + 900 \) Copy content Toggle raw display
$31$ \( T^{2} + 29584 \) Copy content Toggle raw display
$37$ \( T^{2} + 3364 \) Copy content Toggle raw display
$41$ \( T^{2} + 116964 \) Copy content Toggle raw display
$43$ \( (T - 148)^{2} \) Copy content Toggle raw display
$47$ \( (T - 288)^{2} \) Copy content Toggle raw display
$53$ \( (T + 318)^{2} \) Copy content Toggle raw display
$59$ \( (T + 252)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 12100 \) Copy content Toggle raw display
$67$ \( (T + 484)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 501264 \) Copy content Toggle raw display
$73$ \( T^{2} + 131044 \) Copy content Toggle raw display
$79$ \( T^{2} + 234256 \) Copy content Toggle raw display
$83$ \( (T + 756)^{2} \) Copy content Toggle raw display
$89$ \( (T + 774)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 145924 \) Copy content Toggle raw display
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