Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,6,Mod(145,432)] 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("432.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(69.2858101592\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 145.3
Root \(0.500000 + 4.03013i\) of defining polynomial
Character \(\chi\) \(=\) 432.145
Dual form 432.6.i.b.289.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(39.2420 - 67.9691i) q^{5} +(110.566 + 191.505i) q^{7} +(-115.144 - 199.436i) q^{11} +(385.793 - 668.213i) q^{13} +769.880 q^{17} +383.367 q^{19} +(-193.178 + 334.594i) q^{23} +(-1517.37 - 2628.15i) q^{25} +(-394.883 - 683.958i) q^{29} +(1609.97 - 2788.54i) q^{31} +17355.2 q^{35} +2465.33 q^{37} +(-4621.73 + 8005.07i) q^{41} +(5315.76 + 9207.16i) q^{43} +(-976.032 - 1690.54i) q^{47} +(-16046.0 + 27792.4i) q^{49} +32589.2 q^{53} -18074.0 q^{55} +(11916.1 - 20639.2i) q^{59} +(-18804.4 - 32570.2i) q^{61} +(-30278.5 - 52444.0i) q^{65} +(-11525.6 + 19962.9i) q^{67} -66050.4 q^{71} +65130.0 q^{73} +(25462.0 - 44101.5i) q^{77} +(-35433.7 - 61373.0i) q^{79} +(-27643.5 - 47880.0i) q^{83} +(30211.6 - 52328.0i) q^{85} -10598.6 q^{89} +170622. q^{91} +(15044.1 - 26057.1i) q^{95} +(41409.9 + 71724.1i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47}+ \cdots + 46245 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 39.2420 − 67.9691i 0.701982 − 1.21587i −0.265788 − 0.964032i \(-0.585632\pi\)
0.967770 − 0.251837i \(-0.0810346\pi\)
\(6\) 0 0
\(7\) 110.566 + 191.505i 0.852854 + 1.47719i 0.878622 + 0.477518i \(0.158464\pi\)
−0.0257678 + 0.999668i \(0.508203\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −115.144 − 199.436i −0.286920 − 0.496960i 0.686153 − 0.727457i \(-0.259298\pi\)
−0.973073 + 0.230497i \(0.925965\pi\)
\(12\) 0 0
\(13\) 385.793 − 668.213i 0.633134 − 1.09662i −0.353773 − 0.935331i \(-0.615101\pi\)
0.986907 − 0.161289i \(-0.0515652\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 769.880 0.646101 0.323051 − 0.946382i \(-0.395292\pi\)
0.323051 + 0.946382i \(0.395292\pi\)
\(18\) 0 0
\(19\) 383.367 0.243630 0.121815 − 0.992553i \(-0.461129\pi\)
0.121815 + 0.992553i \(0.461129\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −193.178 + 334.594i −0.0761444 + 0.131886i −0.901583 − 0.432605i \(-0.857594\pi\)
0.825439 + 0.564491i \(0.190928\pi\)
\(24\) 0 0
\(25\) −1517.37 − 2628.15i −0.485557 − 0.841009i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −394.883 − 683.958i −0.0871914 − 0.151020i 0.819131 − 0.573606i \(-0.194456\pi\)
−0.906323 + 0.422586i \(0.861123\pi\)
\(30\) 0 0
\(31\) 1609.97 − 2788.54i 0.300893 − 0.521163i −0.675445 − 0.737410i \(-0.736048\pi\)
0.976339 + 0.216247i \(0.0693818\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −4621.73 + 8005.07i −0.429383 + 0.743713i −0.996819 − 0.0797047i \(-0.974602\pi\)
0.567436 + 0.823418i \(0.307936\pi\)
\(42\) 0 0
\(43\) 5315.76 + 9207.16i 0.438424 + 0.759372i 0.997568 − 0.0696983i \(-0.0222037\pi\)
−0.559145 + 0.829070i \(0.688870\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −976.032 − 1690.54i −0.0644495 − 0.111630i 0.832000 − 0.554775i \(-0.187196\pi\)
−0.896450 + 0.443146i \(0.853862\pi\)
\(48\) 0 0
\(49\) −16046.0 + 27792.4i −0.954720 + 1.65362i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 32589.2 1.59362 0.796809 − 0.604231i \(-0.206520\pi\)
0.796809 + 0.604231i \(0.206520\pi\)
\(54\) 0 0
\(55\) −18074.0 −0.805651
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11916.1 − 20639.2i 0.445659 − 0.771903i −0.552439 − 0.833553i \(-0.686303\pi\)
0.998098 + 0.0616498i \(0.0196362\pi\)
\(60\) 0 0
\(61\) −18804.4 − 32570.2i −0.647046 − 1.12072i −0.983825 − 0.179133i \(-0.942671\pi\)
0.336779 − 0.941584i \(-0.390663\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −30278.5 − 52444.0i −0.888897 − 1.53962i
\(66\) 0 0
\(67\) −11525.6 + 19962.9i −0.313672 + 0.543295i −0.979154 − 0.203118i \(-0.934892\pi\)
0.665483 + 0.746413i \(0.268226\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −66050.4 −1.55500 −0.777498 − 0.628885i \(-0.783511\pi\)
−0.777498 + 0.628885i \(0.783511\pi\)
\(72\) 0 0
\(73\) 65130.0 1.43045 0.715227 − 0.698893i \(-0.246324\pi\)
0.715227 + 0.698893i \(0.246324\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 25462.0 − 44101.5i 0.489402 − 0.847669i
\(78\) 0 0
\(79\) −35433.7 − 61373.0i −0.638776 − 1.10639i −0.985702 − 0.168500i \(-0.946108\pi\)
0.346925 − 0.937893i \(-0.387226\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −27643.5 − 47880.0i −0.440451 − 0.762884i 0.557272 − 0.830330i \(-0.311848\pi\)
−0.997723 + 0.0674462i \(0.978515\pi\)
\(84\) 0 0
\(85\) 30211.6 − 52328.0i 0.453551 − 0.785574i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10598.6 −0.141831 −0.0709156 − 0.997482i \(-0.522592\pi\)
−0.0709156 + 0.997482i \(0.522592\pi\)
\(90\) 0 0
\(91\) 170622. 2.15988
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15044.1 − 26057.1i 0.171024 − 0.296222i
\(96\) 0 0
\(97\) 41409.9 + 71724.1i 0.446864 + 0.773991i 0.998180 − 0.0603057i \(-0.0192076\pi\)
−0.551316 + 0.834296i \(0.685874\pi\)
\(98\) 0 0
\(99\) 0 0
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 432.6.i.b.145.3 6
3.2 odd 2 144.6.i.b.49.3 6
4.3 odd 2 54.6.c.b.37.3 6
9.2 odd 6 144.6.i.b.97.3 6
9.7 even 3 inner 432.6.i.b.289.3 6
12.11 even 2 18.6.c.b.13.1 yes 6
36.7 odd 6 54.6.c.b.19.3 6
36.11 even 6 18.6.c.b.7.1 ✓ 6
36.23 even 6 162.6.a.j.1.3 3
36.31 odd 6 162.6.a.i.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.1 ✓ 6 36.11 even 6
18.6.c.b.13.1 yes 6 12.11 even 2
54.6.c.b.19.3 6 36.7 odd 6
54.6.c.b.37.3 6 4.3 odd 2
144.6.i.b.49.3 6 3.2 odd 2
144.6.i.b.97.3 6 9.2 odd 6
162.6.a.i.1.1 3 36.31 odd 6
162.6.a.j.1.3 3 36.23 even 6
432.6.i.b.145.3 6 1.1 even 1 trivial
432.6.i.b.289.3 6 9.7 even 3 inner