Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(23.0952700531\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 97.3
Root \(0.500000 - 4.03013i\) of defining polynomial
Character \(\chi\) \(=\) 144.97
Dual form 144.6.i.b.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(8.58066 - 13.0143i) q^{3} +(-39.2420 - 67.9691i) q^{5} +(110.566 - 191.505i) q^{7} +(-95.7446 - 223.343i) q^{9} +(115.144 - 199.436i) q^{11} +(385.793 + 668.213i) q^{13} +(-1221.29 - 72.5123i) q^{15} -769.880 q^{17} +383.367 q^{19} +(-1543.58 - 3082.17i) q^{21} +(193.178 + 334.594i) q^{23} +(-1517.37 + 2628.15i) q^{25} +(-3728.20 - 670.377i) q^{27} +(394.883 - 683.958i) q^{29} +(1609.97 + 2788.54i) q^{31} +(-1607.51 - 3209.82i) q^{33} -17355.2 q^{35} +2465.33 q^{37} +(12006.7 + 712.877i) q^{39} +(4621.73 + 8005.07i) q^{41} +(5315.76 - 9207.16i) q^{43} +(-11423.2 + 15272.1i) q^{45} +(976.032 - 1690.54i) q^{47} +(-16046.0 - 27792.4i) q^{49} +(-6606.08 + 10019.5i) q^{51} -32589.2 q^{53} -18074.0 q^{55} +(3289.54 - 4989.25i) q^{57} +(-11916.1 - 20639.2i) q^{59} +(-18804.4 + 32570.2i) q^{61} +(-53357.3 - 6358.42i) q^{63} +(30278.5 - 52444.0i) q^{65} +(-11525.6 - 19962.9i) q^{67} +(6012.10 + 356.959i) q^{69} +66050.4 q^{71} +65130.0 q^{73} +(21183.6 + 42298.8i) q^{75} +(-25462.0 - 44101.5i) q^{77} +(-35433.7 + 61373.0i) q^{79} +(-40714.9 + 42767.7i) q^{81} +(27643.5 - 47880.0i) q^{83} +(30211.6 + 52328.0i) q^{85} +(-5512.88 - 11007.9i) q^{87} +10598.6 q^{89} +170622. q^{91} +(50105.6 + 2974.94i) q^{93} +(-15044.1 - 26057.1i) q^{95} +(41409.9 - 71724.1i) q^{97} +(-55567.0 - 6621.74i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 54 q^{5} + 132 q^{7} - 177 q^{9} + 315 q^{11} - 744 q^{13} - 2286 q^{15} + 2898 q^{17} - 2262 q^{19} - 11076 q^{21} + 3168 q^{23} - 2883 q^{25} - 18144 q^{27} - 5148 q^{29} + 8610 q^{31}+ \cdots - 282168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 8.58066 − 13.0143i 0.550449 − 0.834868i
\(4\) 0 0
\(5\) −39.2420 − 67.9691i −0.701982 − 1.21587i −0.967770 − 0.251837i \(-0.918965\pi\)
0.265788 − 0.964032i \(-0.414368\pi\)
\(6\) 0 0
\(7\) 110.566 − 191.505i 0.852854 − 1.47719i −0.0257678 − 0.999668i \(-0.508203\pi\)
0.878622 − 0.477518i \(-0.158464\pi\)
\(8\) 0 0
\(9\) −95.7446 − 223.343i −0.394011 − 0.919106i
\(10\) 0 0
\(11\) 115.144 − 199.436i 0.286920 − 0.496960i −0.686153 − 0.727457i \(-0.740702\pi\)
0.973073 + 0.230497i \(0.0740353\pi\)
\(12\) 0 0
\(13\) 385.793 + 668.213i 0.633134 + 1.09662i 0.986907 + 0.161289i \(0.0515652\pi\)
−0.353773 + 0.935331i \(0.615101\pi\)
\(14\) 0 0
\(15\) −1221.29 − 72.5123i −1.40150 − 0.0832115i
\(16\) 0 0
\(17\) −769.880 −0.646101 −0.323051 − 0.946382i \(-0.604708\pi\)
−0.323051 + 0.946382i \(0.604708\pi\)
\(18\) 0 0
\(19\) 383.367 0.243630 0.121815 − 0.992553i \(-0.461129\pi\)
0.121815 + 0.992553i \(0.461129\pi\)
\(20\) 0 0
\(21\) −1543.58 − 3082.17i −0.763803 − 1.52514i
\(22\) 0 0
\(23\) 193.178 + 334.594i 0.0761444 + 0.131886i 0.901583 − 0.432605i \(-0.142406\pi\)
−0.825439 + 0.564491i \(0.809072\pi\)
\(24\) 0 0
\(25\) −1517.37 + 2628.15i −0.485557 + 0.841009i
\(26\) 0 0
\(27\) −3728.20 − 670.377i −0.984215 − 0.176974i
\(28\) 0 0
\(29\) 394.883 − 683.958i 0.0871914 − 0.151020i −0.819131 − 0.573606i \(-0.805544\pi\)
0.906323 + 0.422586i \(0.138877\pi\)
\(30\) 0 0
\(31\) 1609.97 + 2788.54i 0.300893 + 0.521163i 0.976339 − 0.216247i \(-0.0693818\pi\)
−0.675445 + 0.737410i \(0.736048\pi\)
\(32\) 0 0
\(33\) −1607.51 − 3209.82i −0.256961 − 0.513092i
\(34\) 0 0
\(35\) −17355.2 −2.39475
\(36\) 0 0
\(37\) 2465.33 0.296054 0.148027 − 0.988983i \(-0.452708\pi\)
0.148027 + 0.988983i \(0.452708\pi\)
\(38\) 0 0
\(39\) 12006.7 + 712.877i 1.26404 + 0.0750505i
\(40\) 0 0
\(41\) 4621.73 + 8005.07i 0.429383 + 0.743713i 0.996819 − 0.0797047i \(-0.0253977\pi\)
−0.567436 + 0.823418i \(0.692064\pi\)
\(42\) 0 0
\(43\) 5315.76 − 9207.16i 0.438424 − 0.759372i −0.559145 − 0.829070i \(-0.688870\pi\)
0.997568 + 0.0696983i \(0.0222037\pi\)
\(44\) 0 0
\(45\) −11423.2 + 15272.1i −0.840923 + 1.12426i
\(46\) 0 0
\(47\) 976.032 − 1690.54i 0.0644495 − 0.111630i −0.832000 − 0.554775i \(-0.812804\pi\)
0.896450 + 0.443146i \(0.146138\pi\)
\(48\) 0 0
\(49\) −16046.0 − 27792.4i −0.954720 − 1.65362i
\(50\) 0 0
\(51\) −6606.08 + 10019.5i −0.355646 + 0.539410i
\(52\) 0 0
\(53\) −32589.2 −1.59362 −0.796809 − 0.604231i \(-0.793480\pi\)
−0.796809 + 0.604231i \(0.793480\pi\)
\(54\) 0 0
\(55\) −18074.0 −0.805651
\(56\) 0 0
\(57\) 3289.54 − 4989.25i 0.134106 − 0.203399i
\(58\) 0 0
\(59\) −11916.1 − 20639.2i −0.445659 − 0.771903i 0.552439 − 0.833553i \(-0.313697\pi\)
−0.998098 + 0.0616498i \(0.980364\pi\)
\(60\) 0 0
\(61\) −18804.4 + 32570.2i −0.647046 + 1.12072i 0.336779 + 0.941584i \(0.390663\pi\)
−0.983825 + 0.179133i \(0.942671\pi\)
\(62\) 0 0
\(63\) −53357.3 − 6358.42i −1.69372 − 0.201836i
\(64\) 0 0
\(65\) 30278.5 − 52444.0i 0.888897 − 1.53962i
\(66\) 0 0
\(67\) −11525.6 − 19962.9i −0.313672 − 0.543295i 0.665483 − 0.746413i \(-0.268226\pi\)
−0.979154 + 0.203118i \(0.934892\pi\)
\(68\) 0 0
\(69\) 6012.10 + 356.959i 0.152021 + 0.00902600i
\(70\) 0 0
\(71\) 66050.4 1.55500 0.777498 − 0.628885i \(-0.216489\pi\)
0.777498 + 0.628885i \(0.216489\pi\)
\(72\) 0 0
\(73\) 65130.0 1.43045 0.715227 − 0.698893i \(-0.246324\pi\)
0.715227 + 0.698893i \(0.246324\pi\)
\(74\) 0 0
\(75\) 21183.6 + 42298.8i 0.434858 + 0.868309i
\(76\) 0 0
\(77\) −25462.0 − 44101.5i −0.489402 − 0.847669i
\(78\) 0 0
\(79\) −35433.7 + 61373.0i −0.638776 + 1.10639i 0.346925 + 0.937893i \(0.387226\pi\)
−0.985702 + 0.168500i \(0.946108\pi\)
\(80\) 0 0
\(81\) −40714.9 + 42767.7i −0.689511 + 0.724275i
\(82\) 0 0
\(83\) 27643.5 − 47880.0i 0.440451 − 0.762884i −0.557272 − 0.830330i \(-0.688152\pi\)
0.997723 + 0.0674462i \(0.0214851\pi\)
\(84\) 0 0
\(85\) 30211.6 + 52328.0i 0.453551 + 0.785574i
\(86\) 0 0
\(87\) −5512.88 − 11007.9i −0.0780873 − 0.155922i
\(88\) 0 0
\(89\) 10598.6 0.141831 0.0709156 − 0.997482i \(-0.477408\pi\)
0.0709156 + 0.997482i \(0.477408\pi\)
\(90\) 0 0
\(91\) 170622. 2.15988
\(92\) 0 0
\(93\) 50105.6 + 2974.94i 0.600729 + 0.0356673i
\(94\) 0 0
\(95\) −15044.1 − 26057.1i −0.171024 − 0.296222i
\(96\) 0 0
\(97\) 41409.9 − 71724.1i 0.446864 − 0.773991i −0.551316 − 0.834296i \(-0.685874\pi\)
0.998180 + 0.0603057i \(0.0192076\pi\)
\(98\) 0 0
\(99\) −55567.0 − 6621.74i −0.569809 − 0.0679023i
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 144.6.i.b.97.3 6
3.2 odd 2 432.6.i.b.289.3 6
4.3 odd 2 18.6.c.b.7.1 ✓ 6
9.4 even 3 inner 144.6.i.b.49.3 6
9.5 odd 6 432.6.i.b.145.3 6
12.11 even 2 54.6.c.b.19.3 6
36.7 odd 6 162.6.a.j.1.3 3
36.11 even 6 162.6.a.i.1.1 3
36.23 even 6 54.6.c.b.37.3 6
36.31 odd 6 18.6.c.b.13.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.1 ✓ 6 4.3 odd 2
18.6.c.b.13.1 yes 6 36.31 odd 6
54.6.c.b.19.3 6 12.11 even 2
54.6.c.b.37.3 6 36.23 even 6
144.6.i.b.49.3 6 9.4 even 3 inner
144.6.i.b.97.3 6 1.1 even 1 trivial
162.6.a.i.1.1 3 36.11 even 6
162.6.a.j.1.3 3 36.7 odd 6
432.6.i.b.145.3 6 9.5 odd 6
432.6.i.b.289.3 6 3.2 odd 2