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Properties

Label 144.8.i.c.49.4
Level $144$
Weight $8$
Character 144.49
Analytic conductor $44.983$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,8,Mod(49,144)] 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("144.49"); S:= CuspForms(chi, 8); 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, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.9834436697\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{15} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.4
Root \(0.500000 - 1.48508i\) of defining polynomial
Character \(\chi\) \(=\) 144.49
Dual form 144.8.i.c.97.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(22.8679 + 40.7929i) q^{3} +(47.9866 - 83.1153i) q^{5} +(189.000 + 327.358i) q^{7} +(-1141.12 + 1865.70i) q^{9} +(-3436.63 - 5952.41i) q^{11} +(4826.64 - 8359.99i) q^{13} +(4487.87 + 56.8389i) q^{15} +21431.3 q^{17} -5518.94 q^{19} +(-9031.83 + 15195.9i) q^{21} +(31486.4 - 54536.1i) q^{23} +(34457.1 + 59681.4i) q^{25} +(-102202. - 3884.83i) q^{27} +(-111113. - 192454. i) q^{29} +(57729.1 - 99989.7i) q^{31} +(164228. - 276309. i) q^{33} +36277.9 q^{35} +81737.7 q^{37} +(451403. + 5717.02i) q^{39} +(-298773. + 517491. i) q^{41} +(33874.2 + 58671.8i) q^{43} +(100310. + 184373. i) q^{45} +(151740. + 262822. i) q^{47} +(340329. - 589468. i) q^{49} +(490089. + 874243. i) q^{51} +846755. q^{53} -659649. q^{55} +(-126207. - 225134. i) q^{57} +(793119. - 1.37372e6i) q^{59} +(1.12706e6 + 1.95213e6i) q^{61} +(-826422. - 20936.6i) q^{63} +(-463228. - 802335. i) q^{65} +(1.51172e6 - 2.61838e6i) q^{67} +(2.94471e6 + 37294.8i) q^{69} +4.41675e6 q^{71} +2.21484e6 q^{73} +(-1.64661e6 + 2.77039e6i) q^{75} +(1.29905e6 - 2.25002e6i) q^{77} +(153821. + 266426. i) q^{79} +(-2.17868e6 - 4.25795e6i) q^{81} +(-1.57735e6 - 2.73204e6i) q^{83} +(1.02841e6 - 1.78127e6i) q^{85} +(5.30981e6 - 8.93363e6i) q^{87} +1.93441e6 q^{89} +3.64895e6 q^{91} +(5.39901e6 + 68378.5i) q^{93} +(-264836. + 458709. i) q^{95} +(-4.94528e6 - 8.56548e6i) q^{97} +(1.50270e7 + 380695. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{3} - 180 q^{5} + 84 q^{7} + 990 q^{9} + 8460 q^{11} - 1848 q^{13} + 1188 q^{15} + 30564 q^{17} - 24432 q^{19} - 187224 q^{21} + 51588 q^{23} + 4746 q^{25} - 322272 q^{27} - 414648 q^{29} - 8196 q^{31}+ \cdots + 49382676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 22.8679 + 40.7929i 0.488993 + 0.872288i
\(4\) 0 0
\(5\) 47.9866 − 83.1153i 0.171682 − 0.297362i −0.767326 − 0.641257i \(-0.778413\pi\)
0.939008 + 0.343895i \(0.111746\pi\)
\(6\) 0 0
\(7\) 189.000 + 327.358i 0.208266 + 0.360728i 0.951169 − 0.308672i \(-0.0998846\pi\)
−0.742902 + 0.669400i \(0.766551\pi\)
\(8\) 0 0
\(9\) −1141.12 + 1865.70i −0.521773 + 0.853085i
\(10\) 0 0
\(11\) −3436.63 − 5952.41i −0.778499 − 1.34840i −0.932807 − 0.360377i \(-0.882648\pi\)
0.154308 − 0.988023i \(-0.450685\pi\)
\(12\) 0 0
\(13\) 4826.64 − 8359.99i 0.609317 − 1.05537i −0.382036 − 0.924147i \(-0.624777\pi\)
0.991353 − 0.131220i \(-0.0418896\pi\)
\(14\) 0 0
\(15\) 4487.87 + 56.8389i 0.343337 + 0.00434836i
\(16\) 0 0
\(17\) 21431.3 1.05798 0.528989 − 0.848629i \(-0.322571\pi\)
0.528989 + 0.848629i \(0.322571\pi\)
\(18\) 0 0
\(19\) −5518.94 −0.184594 −0.0922972 − 0.995732i \(-0.529421\pi\)
−0.0922972 + 0.995732i \(0.529421\pi\)
\(20\) 0 0
\(21\) −9031.83 + 15195.9i −0.212818 + 0.358062i
\(22\) 0 0
\(23\) 31486.4 − 54536.1i 0.539605 − 0.934623i −0.459320 − 0.888271i \(-0.651907\pi\)
0.998925 − 0.0463526i \(-0.0147598\pi\)
\(24\) 0 0
\(25\) 34457.1 + 59681.4i 0.441050 + 0.763922i
\(26\) 0 0
\(27\) −102202. − 3884.83i −0.999278 − 0.0379839i
\(28\) 0 0
\(29\) −111113. − 192454.i −0.846004 − 1.46532i −0.884747 − 0.466072i \(-0.845669\pi\)
0.0387428 − 0.999249i \(-0.487665\pi\)
\(30\) 0 0
\(31\) 57729.1 − 99989.7i 0.348040 − 0.602822i −0.637862 − 0.770151i \(-0.720181\pi\)
0.985901 + 0.167329i \(0.0535141\pi\)
\(32\) 0 0
\(33\) 164228. − 276309.i 0.795513 − 1.33843i
\(34\) 0 0
\(35\) 36277.9 0.143023
\(36\) 0 0
\(37\) 81737.7 0.265287 0.132644 − 0.991164i \(-0.457653\pi\)
0.132644 + 0.991164i \(0.457653\pi\)
\(38\) 0 0
\(39\) 451403. + 5717.02i 1.21854 + 0.0154328i
\(40\) 0 0
\(41\) −298773. + 517491.i −0.677015 + 1.17262i 0.298860 + 0.954297i \(0.403394\pi\)
−0.975875 + 0.218328i \(0.929940\pi\)
\(42\) 0 0
\(43\) 33874.2 + 58671.8i 0.0649725 + 0.112536i 0.896682 − 0.442676i \(-0.145971\pi\)
−0.831709 + 0.555211i \(0.812637\pi\)
\(44\) 0 0
\(45\) 100310. + 184373.i 0.164096 + 0.301615i
\(46\) 0 0
\(47\) 151740. + 262822.i 0.213186 + 0.369249i 0.952710 − 0.303881i \(-0.0982827\pi\)
−0.739524 + 0.673130i \(0.764949\pi\)
\(48\) 0 0
\(49\) 340329. − 589468.i 0.413250 − 0.715770i
\(50\) 0 0
\(51\) 490089. + 874243.i 0.517343 + 0.922861i
\(52\) 0 0
\(53\) 846755. 0.781254 0.390627 − 0.920549i \(-0.372258\pi\)
0.390627 + 0.920549i \(0.372258\pi\)
\(54\) 0 0
\(55\) −659649. −0.534618
\(56\) 0 0
\(57\) −126207. − 225134.i −0.0902653 − 0.161019i
\(58\) 0 0
\(59\) 793119. − 1.37372e6i 0.502755 − 0.870797i −0.497240 − 0.867613i \(-0.665653\pi\)
0.999995 − 0.00318395i \(-0.00101348\pi\)
\(60\) 0 0
\(61\) 1.12706e6 + 1.95213e6i 0.635760 + 1.10117i 0.986353 + 0.164641i \(0.0526467\pi\)
−0.350593 + 0.936528i \(0.614020\pi\)
\(62\) 0 0
\(63\) −826422. − 20936.6i −0.416399 − 0.0105491i
\(64\) 0 0
\(65\) −463228. − 802335.i −0.209218 − 0.362376i
\(66\) 0 0
\(67\) 1.51172e6 − 2.61838e6i 0.614060 − 1.06358i −0.376489 − 0.926421i \(-0.622869\pi\)
0.990549 − 0.137162i \(-0.0437980\pi\)
\(68\) 0 0
\(69\) 2.94471e6 + 37294.8i 1.07912 + 0.0136671i
\(70\) 0 0
\(71\) 4.41675e6 1.46453 0.732266 − 0.681018i \(-0.238463\pi\)
0.732266 + 0.681018i \(0.238463\pi\)
\(72\) 0 0
\(73\) 2.21484e6 0.666366 0.333183 − 0.942862i \(-0.391877\pi\)
0.333183 + 0.942862i \(0.391877\pi\)
\(74\) 0 0
\(75\) −1.64661e6 + 2.77039e6i −0.450689 + 0.758275i
\(76\) 0 0
\(77\) 1.29905e6 − 2.25002e6i 0.324271 − 0.561653i
\(78\) 0 0
\(79\) 153821. + 266426.i 0.0351011 + 0.0607969i 0.883042 − 0.469293i \(-0.155491\pi\)
−0.847941 + 0.530090i \(0.822158\pi\)
\(80\) 0 0
\(81\) −2.17868e6 − 4.25795e6i −0.455507 − 0.890232i
\(82\) 0 0
\(83\) −1.57735e6 − 2.73204e6i −0.302798 − 0.524462i 0.673970 − 0.738758i \(-0.264588\pi\)
−0.976769 + 0.214296i \(0.931254\pi\)
\(84\) 0 0
\(85\) 1.02841e6 − 1.78127e6i 0.181636 − 0.314603i
\(86\) 0 0
\(87\) 5.30981e6 − 8.93363e6i 0.864493 − 1.45449i
\(88\) 0 0
\(89\) 1.93441e6 0.290859 0.145430 − 0.989369i \(-0.453544\pi\)
0.145430 + 0.989369i \(0.453544\pi\)
\(90\) 0 0
\(91\) 3.64895e6 0.507601
\(92\) 0 0
\(93\) 5.39901e6 + 68378.5i 0.696023 + 0.00881514i
\(94\) 0 0
\(95\) −264836. + 458709.i −0.0316916 + 0.0548914i
\(96\) 0 0
\(97\) −4.94528e6 − 8.56548e6i −0.550161 − 0.952907i −0.998262 − 0.0589243i \(-0.981233\pi\)
0.448101 − 0.893983i \(-0.352100\pi\)
\(98\) 0 0
\(99\) 1.50270e7 + 380695.i 1.55650 + 0.0394325i
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 144.8.i.c.49.4 12
3.2 odd 2 432.8.i.c.145.3 12
4.3 odd 2 9.8.c.a.4.4 ✓ 12
9.2 odd 6 432.8.i.c.289.3 12
9.7 even 3 inner 144.8.i.c.97.4 12
12.11 even 2 27.8.c.a.10.3 12
36.7 odd 6 9.8.c.a.7.4 yes 12
36.11 even 6 27.8.c.a.19.3 12
36.23 even 6 81.8.a.c.1.4 6
36.31 odd 6 81.8.a.e.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.4 ✓ 12 4.3 odd 2
9.8.c.a.7.4 yes 12 36.7 odd 6
27.8.c.a.10.3 12 12.11 even 2
27.8.c.a.19.3 12 36.11 even 6
81.8.a.c.1.4 6 36.23 even 6
81.8.a.e.1.3 6 36.31 odd 6
144.8.i.c.49.4 12 1.1 even 1 trivial
144.8.i.c.97.4 12 9.7 even 3 inner
432.8.i.c.145.3 12 3.2 odd 2
432.8.i.c.289.3 12 9.2 odd 6