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Properties

Label 144.4.i.f
Level $144$
Weight $4$
Character orbit 144.i
Analytic conductor $8.496$
Analytic rank $0$
Dimension $10$
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] = [144,4,Mod(49,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), 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: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 20x^{8} + 11x^{7} + 284x^{6} + 98x^{5} + 1567x^{4} + 780x^{3} + 6513x^{2} + 972x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{8} + \beta_1 - 1) q^{5} + ( - \beta_{9} - \beta_{4}) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{3} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{8} + \beta_1 - 1) q^{5} + ( - \beta_{9} - \beta_{4}) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{3} + \cdots + 4) q^{9}+ \cdots + (\beta_{9} + 50 \beta_{8} + 31 \beta_{7} + \cdots + 119) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{3} - 5 q^{5} + 3 q^{7} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{3} - 5 q^{5} + 3 q^{7} + 44 q^{9} - 25 q^{11} - 29 q^{13} + 59 q^{15} + 56 q^{17} - 128 q^{19} + 201 q^{21} + 89 q^{23} - 322 q^{25} - 376 q^{27} - 129 q^{29} + 241 q^{31} - 821 q^{33} + 486 q^{35} + 732 q^{37} + 143 q^{39} - 171 q^{41} + 803 q^{43} + 1439 q^{45} - 477 q^{47} - 1072 q^{49} + 1252 q^{51} + 748 q^{53} - 2938 q^{55} + 1382 q^{57} - 607 q^{59} - 1349 q^{61} - 2361 q^{63} - 527 q^{65} + 1549 q^{67} - 4823 q^{69} + 1624 q^{71} + 3856 q^{73} + 1120 q^{75} - 903 q^{77} + 1727 q^{79} + 3572 q^{81} - 1025 q^{83} - 2902 q^{85} + 2181 q^{87} + 4620 q^{89} - 5970 q^{91} + 3209 q^{93} - 2012 q^{95} - 2875 q^{97} - 1793 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 20x^{8} + 11x^{7} + 284x^{6} + 98x^{5} + 1567x^{4} + 780x^{3} + 6513x^{2} + 972x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 62716607 \nu^{9} + 60280747 \nu^{8} - 1256375540 \nu^{7} - 685723757 \nu^{6} + \cdots - 539986752 ) / 60861064212 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22624787 \nu^{9} + 181801400 \nu^{8} - 370019320 \nu^{7} + 5502995321 \nu^{6} + \cdots + 298449333963 ) / 15215266053 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28573957 \nu^{9} - 14660097 \nu^{8} + 765739366 \nu^{7} - 399290955 \nu^{6} + \cdots + 299133832806 ) / 10143510702 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 177727651 \nu^{9} - 660227855 \nu^{8} + 3432127720 \nu^{7} - 2569241147 \nu^{6} + \cdots - 336949188408 ) / 60861064212 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 137101645 \nu^{9} + 154863949 \nu^{8} + 1892511586 \nu^{7} + 7294548109 \nu^{6} + \cdots + 150967504278 ) / 30430532106 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 188911283 \nu^{9} - 829158949 \nu^{8} + 4253293862 \nu^{7} - 3900967045 \nu^{6} + \cdots + 378042902190 ) / 30430532106 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 446905099 \nu^{9} + 887156039 \nu^{8} - 13918168600 \nu^{7} + 19540194611 \nu^{6} + \cdots - 179773716600 ) / 60861064212 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1534654717 \nu^{9} + 1831664177 \nu^{8} - 29325422920 \nu^{7} - 19664815483 \nu^{6} + \cdots - 1196846607276 ) / 60861064212 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1582694957 \nu^{9} + 1131248413 \nu^{8} - 29809237556 \nu^{7} - 26414831879 \nu^{6} + \cdots + 317803639032 ) / 60861064212 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - 2\beta_{8} + \beta_{7} - \beta_{6} - 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - 4\beta_{8} - \beta_{6} - 4\beta_{5} - 4\beta_{4} + \beta_{3} + 91\beta _1 - 92 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{7} - 4\beta_{6} - 26\beta_{5} + 12\beta_{4} - 7\beta_{3} + 26\beta_{2} - 7\beta _1 - 91 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{9} + 38\beta_{8} - 19\beta_{7} + 12\beta_{6} + 21\beta_{5} + 62\beta_{4} + 38\beta_{2} - 526\beta _1 + 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23 \beta_{9} + 196 \beta_{8} - 45 \beta_{7} + 119 \beta_{6} + 334 \beta_{5} + 199 \beta_{4} + \cdots + 1120 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 467 \beta_{7} + 340 \beta_{6} + 1504 \beta_{5} - 1020 \beta_{4} - 103 \beta_{3} - 1384 \beta_{2} + \cdots + 14233 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 128 \beta_{9} - 3187 \beta_{8} + 1975 \beta_{7} - 1140 \beta_{6} - 2505 \beta_{5} - 5797 \beta_{4} + \cdots - 305 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1159 \beta_{9} - 24634 \beta_{8} + 6588 \beta_{7} - 15073 \beta_{6} - 46378 \beta_{5} + \cdots - 221162 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 37661 \beta_{7} - 29500 \beta_{6} - 112472 \beta_{5} + 88500 \beta_{4} - 511 \beta_{3} + \cdots - 791227 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.0750226 − 0.129943i
2.08481 + 3.61100i
−1.35977 − 2.35519i
−1.40499 − 2.43351i
1.25497 + 2.17367i
−0.0750226 + 0.129943i
2.08481 − 3.61100i
−1.35977 + 2.35519i
−1.40499 + 2.43351i
1.25497 − 2.17367i
0 −5.15100 + 0.683516i 0 −10.1111 + 17.5129i 0 −12.3712 − 21.4276i 0 26.0656 − 7.04158i 0
49.2 0 −4.59704 − 2.42224i 0 9.15919 − 15.8642i 0 7.48933 + 12.9719i 0 15.2655 + 22.2703i 0
49.3 0 0.714746 + 5.14676i 0 1.49098 − 2.58245i 0 5.85543 + 10.1419i 0 −25.9783 + 7.35726i 0
49.4 0 1.84482 − 4.85764i 0 −6.19843 + 10.7360i 0 15.3163 + 26.5286i 0 −20.1933 − 17.9229i 0
49.5 0 5.18847 − 0.282448i 0 3.15936 − 5.47217i 0 −14.7898 − 25.6167i 0 26.8404 − 2.93095i 0
97.1 0 −5.15100 − 0.683516i 0 −10.1111 − 17.5129i 0 −12.3712 + 21.4276i 0 26.0656 + 7.04158i 0
97.2 0 −4.59704 + 2.42224i 0 9.15919 + 15.8642i 0 7.48933 − 12.9719i 0 15.2655 − 22.2703i 0
97.3 0 0.714746 − 5.14676i 0 1.49098 + 2.58245i 0 5.85543 − 10.1419i 0 −25.9783 − 7.35726i 0
97.4 0 1.84482 + 4.85764i 0 −6.19843 − 10.7360i 0 15.3163 − 26.5286i 0 −20.1933 + 17.9229i 0
97.5 0 5.18847 + 0.282448i 0 3.15936 + 5.47217i 0 −14.7898 + 25.6167i 0 26.8404 + 2.93095i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.i.f 10
3.b odd 2 1 432.4.i.f 10
4.b odd 2 1 72.4.i.b ✓ 10
9.c even 3 1 inner 144.4.i.f 10
9.c even 3 1 1296.4.a.bd 5
9.d odd 6 1 432.4.i.f 10
9.d odd 6 1 1296.4.a.bc 5
12.b even 2 1 216.4.i.b 10
36.f odd 6 1 72.4.i.b ✓ 10
36.f odd 6 1 648.4.a.l 5
36.h even 6 1 216.4.i.b 10
36.h even 6 1 648.4.a.k 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.b ✓ 10 4.b odd 2 1
72.4.i.b ✓ 10 36.f odd 6 1
144.4.i.f 10 1.a even 1 1 trivial
144.4.i.f 10 9.c even 3 1 inner
216.4.i.b 10 12.b even 2 1
216.4.i.b 10 36.h even 6 1
432.4.i.f 10 3.b odd 2 1
432.4.i.f 10 9.d odd 6 1
648.4.a.k 5 36.h even 6 1
648.4.a.l 5 36.f odd 6 1
1296.4.a.bc 5 9.d odd 6 1
1296.4.a.bd 5 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 5 T_{5}^{9} + 486 T_{5}^{8} - 111 T_{5}^{7} + 181830 T_{5}^{6} + 57429 T_{5}^{5} + \cdots + 7487094784 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 4 T^{9} + \cdots + 14348907 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 7487094784 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 15465374220816 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 6004376846689 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 137434637455504 \) Copy content Toggle raw display
$17$ \( (T^{5} - 28 T^{4} + \cdots - 465026264)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 64 T^{4} + \cdots + 219796928)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{5} - 366 T^{4} + \cdots - 525481299072)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 55\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{5} - 374 T^{4} + \cdots - 758189779072)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} - 812 T^{4} + \cdots - 24420797440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 28702127905256)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 887800297007520)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 76\!\cdots\!09 \) Copy content Toggle raw display
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