↑

Properties

Label 144.9.q.a.65.5
Level $144$
Weight $9$
Character 144.65
Analytic conductor $58.663$
Analytic rank $0$
Dimension $14$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,9,Mod(65,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.65"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,93] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.6625198488\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.5
Root \(-0.447645 - 0.775344i\) of defining polynomial
Character \(\chi\) \(=\) 144.65
Dual form 144.9.q.a.113.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(44.1317 - 67.9220i) q^{3} +(-604.549 + 349.037i) q^{5} +(1124.45 - 1947.61i) q^{7} +(-2665.79 - 5995.02i) q^{9} +(-18817.0 - 10864.0i) q^{11} +(-9209.84 - 15951.9i) q^{13} +(-2972.49 + 56465.7i) q^{15} +56538.1i q^{17} +212375. q^{19} +(-82661.6 - 162326. i) q^{21} +(-14399.3 + 8313.41i) q^{23} +(48340.5 - 83728.3i) q^{25} +(-524840. - 83504.2i) q^{27} +(-294955. - 170292. i) q^{29} +(82902.7 + 143592. i) q^{31} +(-1.56833e6 + 798642. i) q^{33} +1.56990e6i q^{35} +1.11833e6 q^{37} +(-1.48993e6 - 78433.6i) q^{39} +(-3.60414e6 + 2.08085e6i) q^{41} +(-1.28793e6 + 2.23075e6i) q^{43} +(3.70408e6 + 2.69382e6i) q^{45} +(-8.10049e6 - 4.67682e6i) q^{47} +(353608. + 612466. i) q^{49} +(3.84018e6 + 2.49512e6i) q^{51} +4.75879e6i q^{53} +1.51677e7 q^{55} +(9.37246e6 - 1.44249e7i) q^{57} +(8.39702e6 - 4.84802e6i) q^{59} +(-3.04570e6 + 5.27531e6i) q^{61} +(-1.46735e7 - 1.54919e6i) q^{63} +(1.11356e7 + 6.42914e6i) q^{65} +(1.02601e7 + 1.77710e7i) q^{67} +(-70799.4 + 1.34491e6i) q^{69} +2.08627e7i q^{71} +9.02632e6 q^{73} +(-3.55364e6 - 6.97845e6i) q^{75} +(-4.23177e7 + 2.44321e7i) q^{77} +(-1.67238e7 + 2.89665e7i) q^{79} +(-2.88338e7 + 3.19630e7i) q^{81} +(5.09741e7 + 2.94299e7i) q^{83} +(-1.97339e7 - 3.41800e7i) q^{85} +(-2.45835e7 + 1.25187e7i) q^{87} +8.65172e7i q^{89} -4.14242e7 q^{91} +(1.34117e7 + 706023. i) q^{93} +(-1.28391e8 + 7.41267e7i) q^{95} +(-4.68859e7 + 8.12087e7i) q^{97} +(-1.49677e7 + 1.41770e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 93 q^{3} + 438 q^{5} - 922 q^{7} + 17973 q^{9} + 28677 q^{11} + 1684 q^{13} + 75276 q^{15} + 269630 q^{19} + 354054 q^{21} + 1000452 q^{23} + 65177 q^{25} + 524826 q^{27} + 3797682 q^{29} + 164132 q^{31}+ \cdots + 511060752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 44.1317 − 67.9220i 0.544835 − 0.838543i
\(4\) 0 0
\(5\) −604.549 + 349.037i −0.967279 + 0.558459i −0.898405 − 0.439167i \(-0.855274\pi\)
−0.0688730 + 0.997625i \(0.521940\pi\)
\(6\) 0 0
\(7\) 1124.45 − 1947.61i 0.468327 − 0.811167i −0.531017 − 0.847361i \(-0.678190\pi\)
0.999345 + 0.0361942i \(0.0115235\pi\)
\(8\) 0 0
\(9\) −2665.79 − 5995.02i −0.406309 − 0.913736i
\(10\) 0 0
\(11\) −18817.0 − 10864.0i −1.28523 − 0.742026i −0.307428 − 0.951571i \(-0.599468\pi\)
−0.977799 + 0.209545i \(0.932802\pi\)
\(12\) 0 0
\(13\) −9209.84 − 15951.9i −0.322462 − 0.558521i 0.658533 − 0.752552i \(-0.271177\pi\)
−0.980995 + 0.194031i \(0.937844\pi\)
\(14\) 0 0
\(15\) −2972.49 + 56465.7i −0.0587159 + 1.11537i
\(16\) 0 0
\(17\) 56538.1i 0.676932i 0.940979 + 0.338466i \(0.109908\pi\)
−0.940979 + 0.338466i \(0.890092\pi\)
\(18\) 0 0
\(19\) 212375. 1.62963 0.814815 − 0.579721i \(-0.196838\pi\)
0.814815 + 0.579721i \(0.196838\pi\)
\(20\) 0 0
\(21\) −82661.6 − 162326.i −0.425037 − 0.834665i
\(22\) 0 0
\(23\) −14399.3 + 8313.41i −0.0514551 + 0.0297076i −0.525507 − 0.850789i \(-0.676124\pi\)
0.474052 + 0.880497i \(0.342791\pi\)
\(24\) 0 0
\(25\) 48340.5 − 83728.3i 0.123752 − 0.214344i
\(26\) 0 0
\(27\) −524840. − 83504.2i −0.987578 − 0.157128i
\(28\) 0 0
\(29\) −294955. − 170292.i −0.417027 − 0.240771i 0.276778 − 0.960934i \(-0.410733\pi\)
−0.693804 + 0.720163i \(0.744067\pi\)
\(30\) 0 0
\(31\) 82902.7 + 143592.i 0.0897681 + 0.155483i 0.907413 − 0.420240i \(-0.138054\pi\)
−0.817645 + 0.575723i \(0.804721\pi\)
\(32\) 0 0
\(33\) −1.56833e6 + 798642.i −1.32246 + 0.673436i
\(34\) 0 0
\(35\) 1.56990e6i 1.04617i
\(36\) 0 0
\(37\) 1.11833e6 0.596709 0.298355 − 0.954455i \(-0.403562\pi\)
0.298355 + 0.954455i \(0.403562\pi\)
\(38\) 0 0
\(39\) −1.48993e6 − 78433.6i −0.644033 − 0.0339034i
\(40\) 0 0
\(41\) −3.60414e6 + 2.08085e6i −1.27546 + 0.736387i −0.976010 − 0.217725i \(-0.930136\pi\)
−0.299449 + 0.954112i \(0.596803\pi\)
\(42\) 0 0
\(43\) −1.28793e6 + 2.23075e6i −0.376719 + 0.652496i −0.990583 − 0.136916i \(-0.956281\pi\)
0.613864 + 0.789412i \(0.289614\pi\)
\(44\) 0 0
\(45\) 3.70408e6 + 2.69382e6i 0.903297 + 0.656930i
\(46\) 0 0
\(47\) −8.10049e6 − 4.67682e6i −1.66005 − 0.958428i −0.972690 − 0.232107i \(-0.925438\pi\)
−0.687356 − 0.726321i \(-0.741229\pi\)
\(48\) 0 0
\(49\) 353608. + 612466.i 0.0613391 + 0.106242i
\(50\) 0 0
\(51\) 3.84018e6 + 2.49512e6i 0.567637 + 0.368817i
\(52\) 0 0
\(53\) 4.75879e6i 0.603106i 0.953449 + 0.301553i \(0.0975050\pi\)
−0.953449 + 0.301553i \(0.902495\pi\)
\(54\) 0 0
\(55\) 1.51677e7 1.65756
\(56\) 0 0
\(57\) 9.37246e6 − 1.44249e7i 0.887880 − 1.36652i
\(58\) 0 0
\(59\) 8.39702e6 − 4.84802e6i 0.692974 − 0.400089i −0.111751 − 0.993736i \(-0.535646\pi\)
0.804725 + 0.593647i \(0.202313\pi\)
\(60\) 0 0
\(61\) −3.04570e6 + 5.27531e6i −0.219972 + 0.381003i −0.954799 − 0.297252i \(-0.903930\pi\)
0.734827 + 0.678255i \(0.237263\pi\)
\(62\) 0 0
\(63\) −1.46735e7 − 1.54919e6i −0.931478 − 0.0983430i
\(64\) 0 0
\(65\) 1.11356e7 + 6.42914e6i 0.623821 + 0.360164i
\(66\) 0 0
\(67\) 1.02601e7 + 1.77710e7i 0.509158 + 0.881888i 0.999944 + 0.0106075i \(0.00337652\pi\)
−0.490786 + 0.871280i \(0.663290\pi\)
\(68\) 0 0
\(69\) −70799.4 + 1.34491e6i −0.00312344 + 0.0593331i
\(70\) 0 0
\(71\) 2.08627e7i 0.820987i 0.911863 + 0.410494i \(0.134644\pi\)
−0.911863 + 0.410494i \(0.865356\pi\)
\(72\) 0 0
\(73\) 9.02632e6 0.317848 0.158924 − 0.987291i \(-0.449198\pi\)
0.158924 + 0.987291i \(0.449198\pi\)
\(74\) 0 0
\(75\) −3.55364e6 − 6.97845e6i −0.112313 − 0.220554i
\(76\) 0 0
\(77\) −4.23177e7 + 2.44321e7i −1.20381 + 0.695022i
\(78\) 0 0
\(79\) −1.67238e7 + 2.89665e7i −0.429366 + 0.743683i −0.996817 − 0.0797237i \(-0.974596\pi\)
0.567451 + 0.823407i \(0.307930\pi\)
\(80\) 0 0
\(81\) −2.88338e7 + 3.19630e7i −0.669826 + 0.742518i
\(82\) 0 0
\(83\) 5.09741e7 + 2.94299e7i 1.07408 + 0.620121i 0.929294 − 0.369342i \(-0.120417\pi\)
0.144788 + 0.989463i \(0.453750\pi\)
\(84\) 0 0
\(85\) −1.97339e7 − 3.41800e7i −0.378039 − 0.654782i
\(86\) 0 0
\(87\) −2.45835e7 + 1.25187e7i −0.429107 + 0.218515i
\(88\) 0 0
\(89\) 8.65172e7i 1.37893i 0.724319 + 0.689465i \(0.242154\pi\)
−0.724319 + 0.689465i \(0.757846\pi\)
\(90\) 0 0
\(91\) −4.14242e7 −0.604071
\(92\) 0 0
\(93\) 1.34117e7 + 706023.i 0.179288 + 0.00943815i
\(94\) 0 0
\(95\) −1.28391e8 + 7.41267e7i −1.57631 + 0.910081i
\(96\) 0 0
\(97\) −4.68859e7 + 8.12087e7i −0.529609 + 0.917309i 0.469795 + 0.882776i \(0.344328\pi\)
−0.999404 + 0.0345336i \(0.989005\pi\)
\(98\) 0 0
\(99\) −1.49677e7 + 1.41770e8i −0.155816 + 1.47585i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 144.9.q.a.65.5 14
3.2 odd 2 432.9.q.a.305.6 14
4.3 odd 2 9.9.d.a.2.4 ✓ 14
9.4 even 3 432.9.q.a.17.6 14
9.5 odd 6 inner 144.9.q.a.113.5 14
12.11 even 2 27.9.d.a.8.4 14
36.7 odd 6 81.9.b.a.80.7 14
36.11 even 6 81.9.b.a.80.8 14
36.23 even 6 9.9.d.a.5.4 yes 14
36.31 odd 6 27.9.d.a.17.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.4 ✓ 14 4.3 odd 2
9.9.d.a.5.4 yes 14 36.23 even 6
27.9.d.a.8.4 14 12.11 even 2
27.9.d.a.17.4 14 36.31 odd 6
81.9.b.a.80.7 14 36.7 odd 6
81.9.b.a.80.8 14 36.11 even 6
144.9.q.a.65.5 14 1.1 even 1 trivial
144.9.q.a.113.5 14 9.5 odd 6 inner
432.9.q.a.17.6 14 9.4 even 3
432.9.q.a.305.6 14 3.2 odd 2