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Properties

Label 144.7.q.c
Level $144$
Weight $7$
Character orbit 144.q
Analytic conductor $33.128$
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] = [144,7,Mod(65,144)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), 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: \(33.1277880413\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{13} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - 4 \beta_1 + 1) q^{3} + ( - \beta_{9} - \beta_{3} + \beta_{2} + \cdots + 48) q^{5}+ \cdots + ( - 7 \beta_{10} + 5 \beta_{9} + \cdots + 185) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - 4 \beta_1 + 1) q^{3} + ( - \beta_{9} - \beta_{3} + \beta_{2} + \cdots + 48) q^{5}+ \cdots + (591 \beta_{11} - 3336 \beta_{10} + \cdots - 535446) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 42 q^{3} + 432 q^{5} - 240 q^{7} + 2190 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 42 q^{3} + 432 q^{5} - 240 q^{7} + 2190 q^{9} - 378 q^{11} + 1680 q^{13} + 10872 q^{15} + 2820 q^{19} + 24876 q^{21} + 76248 q^{23} + 8094 q^{25} - 127008 q^{27} + 97092 q^{29} - 21480 q^{31} - 246258 q^{33} - 25536 q^{37} - 42204 q^{39} - 410562 q^{41} - 71430 q^{43} + 13716 q^{45} - 347652 q^{47} - 135954 q^{49} - 336402 q^{51} - 580392 q^{55} - 522282 q^{57} - 369738 q^{59} + 135744 q^{61} + 103800 q^{63} - 753840 q^{65} + 289938 q^{67} + 2059272 q^{69} - 977700 q^{73} + 2115342 q^{75} - 159192 q^{77} + 764796 q^{79} - 1428282 q^{81} - 396900 q^{83} + 1619568 q^{85} - 3072636 q^{87} - 355584 q^{91} - 2526576 q^{93} + 2089260 q^{95} - 38874 q^{97} - 4398804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 119 \nu^{11} - 59366 \nu^{9} - 10447223 \nu^{7} - 794976432 \nu^{5} - 25420007664 \nu^{3} + \cdots - 25705589760 ) / 51411179520 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1180295 \nu^{11} + 52300304 \nu^{10} - 477211526 \nu^{9} + 14435849504 \nu^{8} + \cdots + 14\!\cdots\!60 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2658601 \nu^{11} - 49792016 \nu^{10} - 625663450 \nu^{9} - 13943243552 \nu^{8} + \cdots + 77\!\cdots\!60 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79977 \nu^{11} - 234158 \nu^{10} + 22976562 \nu^{9} - 160687484 \nu^{8} + \cdots - 10\!\cdots\!40 ) / 1753335434880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26659 \nu^{11} + 7658854 \nu^{9} + 735955987 \nu^{7} + 28995600888 \nu^{5} + \cdots - 806248446336 \nu ) / 292222572480 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17176253 \nu^{11} + 113412560 \nu^{10} + 5974044146 \nu^{9} + 38860753760 \nu^{8} + \cdots + 60\!\cdots\!40 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17065017 \nu^{11} - 633297280 \nu^{10} + 4985960394 \nu^{9} - 212182020928 \nu^{8} + \cdots - 25\!\cdots\!00 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26881907 \nu^{11} + 333283088 \nu^{10} + 8275037726 \nu^{9} + 109710797216 \nu^{8} + \cdots + 14\!\cdots\!20 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51755593 \nu^{11} + 51054112 \nu^{10} + 16919791690 \nu^{9} + 32755005952 \nu^{8} + \cdots + 14\!\cdots\!20 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 62970535 \nu^{11} - 326409152 \nu^{10} + 20571500038 \nu^{9} - 105607994624 \nu^{8} + \cdots - 10\!\cdots\!60 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 93897127 \nu^{11} - 170078512 \nu^{10} + 32061713686 \nu^{9} - 56906799904 \nu^{8} + \cdots - 89\!\cdots\!40 ) / 84160100874240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{11} + 2\beta_{10} - 2\beta_{8} + 6\beta_{6} - \beta_{5} - 6\beta_{3} - 22\beta _1 - 6 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 3 \beta_{10} - \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + \cdots - 1127 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 115 \beta_{11} - 99 \beta_{10} - 4 \beta_{9} + 79 \beta_{8} - 4 \beta_{7} - 249 \beta_{6} - 75 \beta_{5} + \cdots + 1621 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 237 \beta_{11} - 583 \beta_{10} + 80 \beta_{9} + 157 \beta_{8} + 772 \beta_{7} + 2191 \beta_{6} + \cdots + 102599 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14607 \beta_{11} + 10535 \beta_{10} + 1804 \beta_{9} - 7803 \beta_{8} + 1804 \beta_{7} + \cdots - 295549 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 40901 \beta_{11} + 90283 \beta_{10} - 20204 \beta_{9} - 20697 \beta_{8} - 118968 \beta_{7} + \cdots - 11285535 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1949507 \beta_{11} - 1254095 \beta_{10} - 351192 \beta_{9} + 866627 \beta_{8} - 351192 \beta_{7} + \cdots + 45646137 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6337301 \beta_{11} - 13085271 \beta_{10} + 3689736 \beta_{9} + 2647565 \beta_{8} + 17185676 \beta_{7} + \cdots + 1377954079 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 266230991 \beta_{11} + 161067231 \beta_{10} + 55958900 \beta_{9} - 104331251 \beta_{8} + \cdots - 6678928493 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 934620621 \beta_{11} + 1851791483 \beta_{10} - 593723620 \beta_{9} - 340897001 \beta_{8} + \cdots - 178598701207 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 36751740435 \beta_{11} - 21554669479 \beta_{10} - 8281784624 \beta_{9} + 13288290939 \beta_{8} + \cdots + 955050931721 ) / 18 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
11.8022i
3.87527i
− 7.20150i
− 8.88570i
− 4.28281i
8.15670i
− 11.8022i
− 3.87527i
7.20150i
8.88570i
4.28281i
− 8.15670i
0 −25.6924 + 8.30052i 0 −9.39126 + 5.42205i 0 −322.041 + 557.792i 0 591.203 − 426.521i 0
65.2 0 −25.6485 − 8.43532i 0 1.59771 − 0.922438i 0 −6.34411 + 10.9883i 0 586.691 + 432.707i 0
65.3 0 11.1983 + 24.5683i 0 −39.5602 + 22.8401i 0 245.097 − 424.521i 0 −478.198 + 550.243i 0
65.4 0 14.9408 + 22.4894i 0 202.253 − 116.771i 0 −95.5752 + 165.541i 0 −282.543 + 672.020i 0
65.5 0 22.5447 − 14.8572i 0 156.951 − 90.6160i 0 −104.306 + 180.663i 0 287.526 − 669.903i 0
65.6 0 23.6571 − 13.0130i 0 −95.8504 + 55.3393i 0 163.169 − 282.617i 0 390.321 − 615.703i 0
113.1 0 −25.6924 − 8.30052i 0 −9.39126 − 5.42205i 0 −322.041 − 557.792i 0 591.203 + 426.521i 0
113.2 0 −25.6485 + 8.43532i 0 1.59771 + 0.922438i 0 −6.34411 − 10.9883i 0 586.691 − 432.707i 0
113.3 0 11.1983 − 24.5683i 0 −39.5602 − 22.8401i 0 245.097 + 424.521i 0 −478.198 − 550.243i 0
113.4 0 14.9408 − 22.4894i 0 202.253 + 116.771i 0 −95.5752 − 165.541i 0 −282.543 − 672.020i 0
113.5 0 22.5447 + 14.8572i 0 156.951 + 90.6160i 0 −104.306 − 180.663i 0 287.526 + 669.903i 0
113.6 0 23.6571 + 13.0130i 0 −95.8504 − 55.3393i 0 163.169 + 282.617i 0 390.321 + 615.703i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.7.q.c 12
3.b odd 2 1 432.7.q.b 12
4.b odd 2 1 18.7.d.a ✓ 12
9.c even 3 1 432.7.q.b 12
9.d odd 6 1 inner 144.7.q.c 12
12.b even 2 1 54.7.d.a 12
36.f odd 6 1 54.7.d.a 12
36.f odd 6 1 162.7.b.c 12
36.h even 6 1 18.7.d.a ✓ 12
36.h even 6 1 162.7.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a ✓ 12 4.b odd 2 1
18.7.d.a ✓ 12 36.h even 6 1
54.7.d.a 12 12.b even 2 1
54.7.d.a 12 36.f odd 6 1
144.7.q.c 12 1.a even 1 1 trivial
144.7.q.c 12 9.d odd 6 1 inner
162.7.b.c 12 36.f odd 6 1
162.7.b.c 12 36.h even 6 1
432.7.q.b 12 3.b odd 2 1
432.7.q.b 12 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 432 T_{5}^{11} + 42390 T_{5}^{10} + 8561376 T_{5}^{9} - 951920181 T_{5}^{8} + \cdots + 18\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 54\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 49\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
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