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Properties

Label 289.2.d.e
Level $289$
Weight $2$
Character orbit 289.d
Analytic conductor $2.308$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,2,Mod(110,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.110");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.d (of order \(8\), degree \(4\), 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: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.229607785695641627262976.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 799x^{8} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{2} + ( - \beta_{7} - \beta_{6}) q^{3} + ( - \beta_{10} - 2 \beta_{8}) q^{4} + \beta_{15} q^{5} - 3 \beta_{9} q^{6} + ( - 2 \beta_{2} - \beta_1) q^{7} - 3 \beta_{12} q^{8} + ( - \beta_{13} - \beta_{12}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{2} + ( - \beta_{7} - \beta_{6}) q^{3} + ( - \beta_{10} - 2 \beta_{8}) q^{4} + \beta_{15} q^{5} - 3 \beta_{9} q^{6} + ( - 2 \beta_{2} - \beta_1) q^{7} - 3 \beta_{12} q^{8} + ( - \beta_{13} - \beta_{12}) q^{9} + ( - 3 \beta_{2} + \beta_1) q^{10} + 3 \beta_{9} q^{11} + (\beta_{15} - 2 \beta_{14}) q^{12} + ( - \beta_{10} - 2 \beta_{8}) q^{13} + (3 \beta_{7} + \beta_{6}) q^{14} - 3 \beta_{3} q^{15} + (\beta_{4} + 2) q^{16} + 3 q^{18} + ( - 3 \beta_{5} - 2 \beta_{3}) q^{19} + ( - 3 \beta_{7} + 2 \beta_{6}) q^{20} + (2 \beta_{10} - 3 \beta_{8}) q^{21} - 3 \beta_{15} q^{22} + \beta_{11} q^{23} + (3 \beta_{2} + 3 \beta_1) q^{24} + ( - \beta_{13} - 2 \beta_{12}) q^{25} + (2 \beta_{13} - 3 \beta_{12}) q^{26} + (\beta_{2} - 2 \beta_1) q^{27} - \beta_{9} q^{28} + ( - 3 \beta_{15} + 3 \beta_{14}) q^{29} + (3 \beta_{10} + 3 \beta_{8}) q^{30} + ( - \beta_{7} - 2 \beta_{6}) q^{31} + (\beta_{5} + 4 \beta_{3}) q^{32} + ( - 3 \beta_{4} - 3) q^{33} + ( - \beta_{4} - 3) q^{35} + (\beta_{5} + 3 \beta_{3}) q^{36} + (4 \beta_{7} + 2 \beta_{6}) q^{37} + (2 \beta_{10} + 11 \beta_{8}) q^{38} + (\beta_{15} - 2 \beta_{14}) q^{39} - 3 \beta_{11} q^{40} - 6 \beta_{2} q^{41} + (3 \beta_{13} + 6 \beta_{12}) q^{42} + (2 \beta_{13} + 7 \beta_{12}) q^{43} + (3 \beta_{2} - 3 \beta_1) q^{44} - 3 \beta_{9} q^{45} + (\beta_{15} - 3 \beta_{14}) q^{46} + 3 \beta_{8} q^{47} + ( - 5 \beta_{7} - 2 \beta_{6}) q^{48} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{49} + (\beta_{4} + 3) q^{50} + (3 \beta_{4} - 4) q^{52} + (3 \beta_{5} - 6 \beta_{3}) q^{53} + (6 \beta_{7} - 3 \beta_{6}) q^{54} + ( - 3 \beta_{10} - 3 \beta_{8}) q^{55} + (3 \beta_{15} + 6 \beta_{14}) q^{56} + (\beta_{11} + 10 \beta_{9}) q^{57} + (9 \beta_{2} - 6 \beta_1) q^{58} + 6 \beta_{12} q^{59} + ( - 3 \beta_{13} + 3 \beta_{12}) q^{60} + (5 \beta_{2} + 6 \beta_1) q^{61} + (\beta_{11} - 6 \beta_{9}) q^{62} + (2 \beta_{15} + 5 \beta_{14}) q^{63} + ( - 6 \beta_{10} - 5 \beta_{8}) q^{64} + ( - 3 \beta_{7} + 2 \beta_{6}) q^{65} + 9 \beta_{3} q^{66} + (2 \beta_{4} - 8) q^{67} - 3 q^{69} + ( - 2 \beta_{5} + \beta_{3}) q^{70} + ( - 6 \beta_{7} - 4 \beta_{6}) q^{71} - 3 \beta_{10} q^{72} + ( - 2 \beta_{15} - 5 \beta_{14}) q^{73} + (2 \beta_{11} + 6 \beta_{9}) q^{74} + (5 \beta_{2} + 2 \beta_1) q^{75} + ( - 5 \beta_{13} + 8 \beta_{12}) q^{76} + ( - 3 \beta_{13} - 6 \beta_{12}) q^{77} + ( - 3 \beta_{2} + 3 \beta_1) q^{78} + ( - 4 \beta_{11} + 2 \beta_{9}) q^{79} + (\beta_{15} + 3 \beta_{14}) q^{80} + (2 \beta_{10} - 6 \beta_{8}) q^{81} + 6 \beta_{6} q^{82} + (5 \beta_{5} - 4 \beta_{3}) q^{83} + (\beta_{4} + 1) q^{84} + ( - 5 \beta_{4} - 6) q^{86} + (3 \beta_{5} + 9 \beta_{3}) q^{87} + 9 \beta_{7} q^{88} + ( - 4 \beta_{10} + 2 \beta_{8}) q^{89} + 3 \beta_{15} q^{90} - \beta_{9} q^{91} + ( - 3 \beta_{2} + 2 \beta_1) q^{92} + ( - \beta_{13} - 7 \beta_{12}) q^{93} - 3 \beta_{13} q^{94} + (9 \beta_{2} - 2 \beta_1) q^{95} + 3 \beta_{11} q^{96} + (3 \beta_{15} - 4 \beta_{14}) q^{97} + (3 \beta_{10} + 12 \beta_{8}) q^{98} + (3 \beta_{7} + 3 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{16} + 48 q^{18} - 24 q^{33} - 40 q^{35} + 40 q^{50} - 88 q^{52} - 144 q^{67} - 48 q^{69} + 8 q^{84} - 56 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 799x^{8} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 508\nu ) / 651 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 1159\nu^{2} ) / 1953 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 291 ) / 217 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{10} - 2683\nu^{2} ) / 1953 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 1159\nu^{3} ) / 1953 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{11} + 2683\nu^{3} ) / 5859 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{12} - 880\nu^{4} ) / 2511 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -19\nu^{13} - 14209\nu^{5} ) / 52731 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\nu^{12} + 14209\nu^{4} ) / 17577 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{13} - 880\nu^{5} ) / 2511 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -40\nu^{14} - 32689\nu^{6} ) / 158193 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -19\nu^{14} - 14209\nu^{6} ) / 52731 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -97\nu^{15} - 75316\nu^{7} ) / 474579 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -40\nu^{15} - 32689\nu^{7} ) / 158193 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + 4\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{10} - 19\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{11} + 21\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 40\beta_{13} - 57\beta_{12} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -97\beta_{15} + 120\beta_{14} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 217\beta_{4} - 291 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 651\beta_{2} - 508\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1159\beta_{5} - 2683\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3477\beta_{7} - 2683\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6160\beta_{10} + 14209\beta_{8} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14209\beta_{11} - 18480\beta_{9} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -32689\beta_{13} + 42627\beta_{12} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 75316\beta_{15} - 98067\beta_{14} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1
1.20361 − 0.498551i
−1.20361 + 0.498551i
2.12749 − 0.881234i
−2.12749 + 0.881234i
1.20361 + 0.498551i
−1.20361 − 0.498551i
2.12749 + 0.881234i
−2.12749 − 0.881234i
−0.881234 + 2.12749i
0.881234 − 2.12749i
−0.498551 + 1.20361i
0.498551 − 1.20361i
−0.881234 − 2.12749i
0.881234 + 2.12749i
−0.498551 − 1.20361i
0.498551 + 1.20361i
−0.921201 + 0.921201i −0.881234 + 2.12749i 0.302776i 1.20361 + 0.498551i −1.14805 − 2.77164i −3.05137 + 1.26392i −2.12132 − 2.12132i −1.62831 − 1.62831i −1.56803 + 0.649500i
110.2 −0.921201 + 0.921201i 0.881234 − 2.12749i 0.302776i −1.20361 − 0.498551i 1.14805 + 2.77164i 3.05137 − 1.26392i −2.12132 − 2.12132i −1.62831 − 1.62831i 1.56803 − 0.649500i
110.3 1.62831 − 1.62831i −0.498551 + 1.20361i − 3.30278i 2.12749 + 0.881234i 1.14805 + 2.77164i −0.279728 + 0.115867i −2.12132 − 2.12132i 0.921201 + 0.921201i 4.89913 − 2.02928i
110.4 1.62831 − 1.62831i 0.498551 − 1.20361i − 3.30278i −2.12749 − 0.881234i −1.14805 − 2.77164i 0.279728 − 0.115867i −2.12132 − 2.12132i 0.921201 + 0.921201i −4.89913 + 2.02928i
134.1 −0.921201 − 0.921201i −0.881234 − 2.12749i − 0.302776i 1.20361 − 0.498551i −1.14805 + 2.77164i −3.05137 − 1.26392i −2.12132 + 2.12132i −1.62831 + 1.62831i −1.56803 − 0.649500i
134.2 −0.921201 − 0.921201i 0.881234 + 2.12749i − 0.302776i −1.20361 + 0.498551i 1.14805 − 2.77164i 3.05137 + 1.26392i −2.12132 + 2.12132i −1.62831 + 1.62831i 1.56803 + 0.649500i
134.3 1.62831 + 1.62831i −0.498551 − 1.20361i 3.30278i 2.12749 − 0.881234i 1.14805 − 2.77164i −0.279728 − 0.115867i −2.12132 + 2.12132i 0.921201 − 0.921201i 4.89913 + 2.02928i
134.4 1.62831 + 1.62831i 0.498551 + 1.20361i 3.30278i −2.12749 + 0.881234i −1.14805 + 2.77164i 0.279728 + 0.115867i −2.12132 + 2.12132i 0.921201 − 0.921201i −4.89913 − 2.02928i
155.1 −1.62831 − 1.62831i −1.20361 + 0.498551i 3.30278i −0.881234 − 2.12749i 2.77164 + 1.14805i 0.115867 − 0.279728i 2.12132 − 2.12132i −0.921201 + 0.921201i −2.02928 + 4.89913i
155.2 −1.62831 − 1.62831i 1.20361 − 0.498551i 3.30278i 0.881234 + 2.12749i −2.77164 − 1.14805i −0.115867 + 0.279728i 2.12132 − 2.12132i −0.921201 + 0.921201i 2.02928 − 4.89913i
155.3 0.921201 + 0.921201i −2.12749 + 0.881234i − 0.302776i −0.498551 − 1.20361i −2.77164 − 1.14805i 1.26392 − 3.05137i 2.12132 − 2.12132i 1.62831 − 1.62831i 0.649500 − 1.56803i
155.4 0.921201 + 0.921201i 2.12749 − 0.881234i − 0.302776i 0.498551 + 1.20361i 2.77164 + 1.14805i −1.26392 + 3.05137i 2.12132 − 2.12132i 1.62831 − 1.62831i −0.649500 + 1.56803i
179.1 −1.62831 + 1.62831i −1.20361 − 0.498551i − 3.30278i −0.881234 + 2.12749i 2.77164 − 1.14805i 0.115867 + 0.279728i 2.12132 + 2.12132i −0.921201 − 0.921201i −2.02928 − 4.89913i
179.2 −1.62831 + 1.62831i 1.20361 + 0.498551i − 3.30278i 0.881234 − 2.12749i −2.77164 + 1.14805i −0.115867 − 0.279728i 2.12132 + 2.12132i −0.921201 − 0.921201i 2.02928 + 4.89913i
179.3 0.921201 − 0.921201i −2.12749 − 0.881234i 0.302776i −0.498551 + 1.20361i −2.77164 + 1.14805i 1.26392 + 3.05137i 2.12132 + 2.12132i 1.62831 + 1.62831i 0.649500 + 1.56803i
179.4 0.921201 − 0.921201i 2.12749 + 0.881234i 0.302776i 0.498551 − 1.20361i 2.77164 − 1.14805i −1.26392 − 3.05137i 2.12132 + 2.12132i 1.62831 + 1.62831i −0.649500 − 1.56803i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 110.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.2.d.e 16
17.b even 2 1 inner 289.2.d.e 16
17.c even 4 2 inner 289.2.d.e 16
17.d even 8 4 inner 289.2.d.e 16
17.e odd 16 1 289.2.a.b ✓ 2
17.e odd 16 1 289.2.a.c yes 2
17.e odd 16 2 289.2.b.c 4
17.e odd 16 4 289.2.c.b 8
51.i even 16 1 2601.2.a.r 2
51.i even 16 1 2601.2.a.s 2
68.i even 16 1 4624.2.a.j 2
68.i even 16 1 4624.2.a.v 2
85.p odd 16 1 7225.2.a.m 2
85.p odd 16 1 7225.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.2.a.b ✓ 2 17.e odd 16 1
289.2.a.c yes 2 17.e odd 16 1
289.2.b.c 4 17.e odd 16 2
289.2.c.b 8 17.e odd 16 4
289.2.d.e 16 1.a even 1 1 trivial
289.2.d.e 16 17.b even 2 1 inner
289.2.d.e 16 17.c even 4 2 inner
289.2.d.e 16 17.d even 8 4 inner
2601.2.a.r 2 51.i even 16 1
2601.2.a.s 2 51.i even 16 1
4624.2.a.j 2 68.i even 16 1
4624.2.a.v 2 68.i even 16 1
7225.2.a.m 2 85.p odd 16 1
7225.2.a.n 2 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 \(S_{2}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{8} + 31T_{2}^{4} + 81 \) Copy content Toggle raw display
\( T_{3}^{16} + 799T_{3}^{8} + 6561 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 31 T^{4} + 81)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 799 T^{8} + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 799 T^{8} + 6561 \) Copy content Toggle raw display
$7$ \( T^{16} + 14159T^{8} + 1 \) Copy content Toggle raw display
$11$ \( (T^{8} + 6561)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 11 T^{2} + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1799 T^{4} + 707281)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 799 T^{8} + 6561 \) Copy content Toggle raw display
$29$ \( T^{16} + 92897199 T^{8} + 43046721 \) Copy content Toggle raw display
$31$ \( (T^{8} + 28561)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 3624704 T^{8} + 65536 \) Copy content Toggle raw display
$41$ \( (T^{8} + 1679616)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 8546 T^{4} + 279841)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{8} \) Copy content Toggle raw display
$53$ \( (T^{8} + 27783 T^{4} + 531441)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 1296)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( (T^{2} + 18 T + 68)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 2821109907456 \) Copy content Toggle raw display
$73$ \( T^{16} + 11195554 T^{8} + 6561 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 2821109907456 \) Copy content Toggle raw display
$83$ \( (T^{8} + 57967 T^{4} + 2313441)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 136 T^{2} + 1296)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + 200477279 T^{8} + 1 \) Copy content Toggle raw display
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