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Properties

Label 289.4.b.e
Level $289$
Weight $4$
Character orbit 289.b
Analytic conductor $17.052$
Analytic rank $0$
Dimension $12$
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] = [289,4,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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 34 x^{10} + 124 x^{9} + 671 x^{8} - 1984 x^{7} - 5452 x^{6} + 8264 x^{5} + \cdots + 300356 \) 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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + 1) q^{2} + \beta_{11} q^{3} + ( - \beta_{6} + \beta_{3} + 2 \beta_{2} + 2) q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + 1) q^{2} + \beta_{11} q^{3} + ( - \beta_{6} + \beta_{3} + 2 \beta_{2} + 2) q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + ( - 47 \beta_{11} - 72 \beta_{10} + \cdots + 93 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{2} + 16 q^{4} + 96 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{2} + 16 q^{4} + 96 q^{8} + 36 q^{9} - 8 q^{13} + 192 q^{15} - 184 q^{16} + 352 q^{19} - 256 q^{21} + 492 q^{25} + 784 q^{26} + 744 q^{30} - 24 q^{32} - 1400 q^{33} - 632 q^{35} + 856 q^{36} - 624 q^{38} + 1664 q^{42} + 1200 q^{43} - 1512 q^{47} + 1052 q^{49} - 2856 q^{50} + 792 q^{52} + 2504 q^{53} - 1424 q^{55} + 3408 q^{59} + 2808 q^{60} + 272 q^{64} - 272 q^{66} - 1080 q^{67} - 344 q^{69} - 2600 q^{70} + 248 q^{72} - 896 q^{76} - 848 q^{77} - 2404 q^{81} + 2960 q^{83} + 4768 q^{84} - 1200 q^{86} + 160 q^{87} - 2144 q^{89} - 3800 q^{93} - 5984 q^{94} + 3464 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 34 x^{10} + 124 x^{9} + 671 x^{8} - 1984 x^{7} - 5452 x^{6} + 8264 x^{5} + \cdots + 300356 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 38\!\cdots\!39 \nu^{11} + \cdots - 14\!\cdots\!32 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 145162841529 \nu^{11} - 47041952383 \nu^{10} - 7410219357526 \nu^{9} + \cdots - 86\!\cdots\!82 ) / 87\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40\!\cdots\!08 \nu^{11} + \cdots - 79\!\cdots\!52 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 60\!\cdots\!85 \nu^{11} + \cdots - 15\!\cdots\!80 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!00 \nu^{11} + \cdots - 98\!\cdots\!32 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!00 \nu^{11} + \cdots + 98\!\cdots\!32 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 83\!\cdots\!12 \nu^{11} + \cdots - 14\!\cdots\!68 ) / 83\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!10 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!98 \nu^{11} + \cdots - 21\!\cdots\!52 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 57\!\cdots\!37 \nu^{11} + \cdots - 13\!\cdots\!28 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 62\!\cdots\!99 \nu^{11} + \cdots - 10\!\cdots\!16 ) / 16\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{11} + 3\beta_{10} - 2\beta_{9} - \beta_{8} - \beta_{7} + 7\beta_{6} + 27\beta_{5} + 3\beta_{4} + \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{10} - 8 \beta_{9} - 5 \beta_{8} - 6 \beta_{7} + 9 \beta_{6} + 24 \beta_{5} + 48 \beta_{4} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 100 \beta_{11} + 45 \beta_{10} - 49 \beta_{9} - 5 \beta_{8} - 100 \beta_{7} - 81 \beta_{6} + \cdots + 107 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 88 \beta_{11} - 26 \beta_{10} + 62 \beta_{9} - 164 \beta_{8} - 227 \beta_{7} - 167 \beta_{6} + \cdots - 1988 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 490 \beta_{11} - 49 \beta_{10} + 1218 \beta_{9} - 30 \beta_{8} - 2937 \beta_{7} - 4003 \beta_{6} + \cdots + 498 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1808 \beta_{11} - 120 \beta_{10} + 8688 \beta_{9} - 2961 \beta_{8} - 5452 \beta_{7} - 7847 \beta_{6} + \cdots - 46589 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21300 \beta_{11} - 20331 \beta_{10} + 79677 \beta_{9} - 899 \beta_{8} - 42058 \beta_{7} - 66435 \beta_{6} + \cdots - 31887 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 8640 \beta_{11} - 11918 \beta_{10} + 311046 \beta_{9} - 17994 \beta_{8} - 65145 \beta_{7} + \cdots - 407976 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 678766 \beta_{11} - 539363 \beta_{10} + 2038640 \beta_{9} - 9890 \beta_{8} + 40375 \beta_{7} + \cdots - 817256 \) Copy content Toggle raw display

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

Character values

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

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

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
4.15292 + 1.84776i
4.15292 − 1.84776i
3.68604 + 0.765367i
3.68604 − 0.765367i
1.22788 − 0.765367i
1.22788 + 0.765367i
−0.705468 − 1.84776i
−0.705468 + 1.84776i
−2.49971 + 0.765367i
−2.49971 − 0.765367i
−3.86166 + 1.84776i
−3.86166 − 1.84776i
−3.15292 − 1.99138i 1.94089 5.00761i 6.27866i 2.78516i 19.1039 23.0344 − 15.7886i
288.2 −3.15292 1.99138i 1.94089 − 5.00761i − 6.27866i − 2.78516i 19.1039 23.0344 15.7886i
288.3 −2.68604 − 4.33137i −0.785167 2.08666i 11.6343i − 24.9985i 23.5973 8.23920 − 5.60485i
288.4 −2.68604 4.33137i −0.785167 − 2.08666i − 11.6343i 24.9985i 23.5973 8.23920 5.60485i
288.5 −0.227878 − 8.23916i −7.94807 − 2.75144i 1.87752i − 21.5220i 3.63421 −40.8838 0.626993i
288.6 −0.227878 8.23916i −7.94807 2.75144i − 1.87752i 21.5220i 3.63421 −40.8838 − 0.626993i
288.7 1.70547 − 4.44379i −5.09138 6.80983i − 7.57875i 13.8760i −22.3269 7.25269 11.6140i
288.8 1.70547 4.44379i −5.09138 − 6.80983i 7.57875i − 13.8760i −22.3269 7.25269 − 11.6140i
288.9 3.49971 − 2.82539i 4.24796 − 8.71933i − 9.88806i 6.85501i −13.1311 19.0171 − 30.5151i
288.10 3.49971 2.82539i 4.24796 8.71933i 9.88806i − 6.85501i −13.1311 19.0171 30.5151i
288.11 4.86166 − 5.06554i 15.6358 18.5634i − 24.6269i 14.0210i 37.1225 1.34032 90.2489i
288.12 4.86166 5.06554i 15.6358 − 18.5634i 24.6269i − 14.0210i 37.1225 1.34032 − 90.2489i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 288.12
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.e 12
17.b even 2 1 inner 289.4.b.e 12
17.c even 4 2 289.4.a.g 12
17.e odd 16 2 17.4.d.a ✓ 12
51.i even 16 2 153.4.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.d.a ✓ 12 17.e odd 16 2
153.4.l.a 12 51.i even 16 2
289.4.a.g 12 17.c even 4 2
289.4.b.e 12 1.a even 1 1 trivial
289.4.b.e 12 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 4T_{2}^{5} - 20T_{2}^{4} + 64T_{2}^{3} + 111T_{2}^{2} - 224T_{2} - 56 \) acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 4 T^{5} - 20 T^{4} + \cdots - 56)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 144 T^{10} + \cdots + 20428832 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1004236928 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 3993906708992 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( (T^{6} + 4 T^{5} + \cdots - 587761088)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - 176 T^{5} + \cdots + 37116511456)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 78\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 73\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 36\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 25553688086896)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 426029287281152)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 309016062416752)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 847356213991824)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 61\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 32\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 99\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 93\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 124020435364336)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
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