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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 289.1
Root \(0.500000 - 5.23712i\) of defining polynomial
Character \(\chi\) \(=\) 432.289
Dual form 432.6.i.b.145.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-33.0434 - 57.2329i) q^{5} +(57.0952 - 98.8918i) q^{7} +(192.812 - 333.961i) q^{11} +(-516.287 - 894.236i) q^{13} -959.020 q^{17} +464.576 q^{19} +(-1151.70 - 1994.81i) q^{23} +(-621.235 + 1076.01i) q^{25} +(3549.13 - 6147.28i) q^{29} +(3881.53 + 6723.00i) q^{31} -7546.48 q^{35} +9317.57 q^{37} +(-6661.64 - 11538.3i) q^{41} +(-1056.29 + 1829.55i) q^{43} +(-1247.40 + 2160.56i) q^{47} +(1883.78 + 3262.80i) q^{49} +10044.2 q^{53} -25484.7 q^{55} +(-2720.33 - 4711.75i) q^{59} +(-17094.4 + 29608.4i) q^{61} +(-34119.8 + 59097.2i) q^{65} +(26792.6 + 46406.2i) q^{67} -970.010 q^{71} -72400.3 q^{73} +(-22017.3 - 38135.1i) q^{77} +(16098.9 - 27884.0i) q^{79} +(18046.6 - 31257.6i) q^{83} +(31689.3 + 54887.5i) q^{85} -42622.2 q^{89} -117910. q^{91} +(-15351.2 - 26589.0i) q^{95} +(21926.9 - 37978.6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47}+ \cdots + 46245 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −33.0434 − 57.2329i −0.591099 − 1.02381i −0.994085 − 0.108607i \(-0.965361\pi\)
0.402986 − 0.915206i \(-0.367972\pi\)
\(6\) 0 0
\(7\) 57.0952 − 98.8918i 0.440407 − 0.762808i −0.557312 − 0.830303i \(-0.688167\pi\)
0.997720 + 0.0674952i \(0.0215007\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 192.812 − 333.961i 0.480455 − 0.832173i −0.519293 − 0.854596i \(-0.673805\pi\)
0.999749 + 0.0224231i \(0.00713811\pi\)
\(12\) 0 0
\(13\) −516.287 − 894.236i −0.847292 − 1.46755i −0.883616 − 0.468212i \(-0.844898\pi\)
0.0363242 − 0.999340i \(-0.488435\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −959.020 −0.804832 −0.402416 − 0.915457i \(-0.631829\pi\)
−0.402416 + 0.915457i \(0.631829\pi\)
\(18\) 0 0
\(19\) 464.576 0.295239 0.147619 − 0.989044i \(-0.452839\pi\)
0.147619 + 0.989044i \(0.452839\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1151.70 − 1994.81i −0.453963 − 0.786288i 0.544664 − 0.838654i \(-0.316657\pi\)
−0.998628 + 0.0523662i \(0.983324\pi\)
\(24\) 0 0
\(25\) −621.235 + 1076.01i −0.198795 + 0.344323i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3549.13 − 6147.28i 0.783659 − 1.35734i −0.146138 − 0.989264i \(-0.546684\pi\)
0.929797 − 0.368073i \(-0.119982\pi\)
\(30\) 0 0
\(31\) 3881.53 + 6723.00i 0.725435 + 1.25649i 0.958795 + 0.284100i \(0.0916947\pi\)
−0.233360 + 0.972391i \(0.574972\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7546.48 −1.04130
\(36\) 0 0
\(37\) 9317.57 1.11892 0.559459 − 0.828858i \(-0.311009\pi\)
0.559459 + 0.828858i \(0.311009\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6661.64 − 11538.3i −0.618901 − 1.07197i −0.989687 − 0.143250i \(-0.954245\pi\)
0.370785 − 0.928719i \(-0.379089\pi\)
\(42\) 0 0
\(43\) −1056.29 + 1829.55i −0.0871190 + 0.150895i −0.906292 − 0.422652i \(-0.861099\pi\)
0.819173 + 0.573546i \(0.194433\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1247.40 + 2160.56i −0.0823686 + 0.142667i −0.904267 − 0.426968i \(-0.859582\pi\)
0.821898 + 0.569634i \(0.192915\pi\)
\(48\) 0 0
\(49\) 1883.78 + 3262.80i 0.112083 + 0.194133i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10044.2 0.491163 0.245582 − 0.969376i \(-0.421021\pi\)
0.245582 + 0.969376i \(0.421021\pi\)
\(54\) 0 0
\(55\) −25484.7 −1.13599
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2720.33 − 4711.75i −0.101740 − 0.176219i 0.810662 − 0.585515i \(-0.199108\pi\)
−0.912402 + 0.409296i \(0.865774\pi\)
\(60\) 0 0
\(61\) −17094.4 + 29608.4i −0.588206 + 1.01880i 0.406261 + 0.913757i \(0.366832\pi\)
−0.994467 + 0.105046i \(0.966501\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −34119.8 + 59097.2i −1.00167 + 1.73494i
\(66\) 0 0
\(67\) 26792.6 + 46406.2i 0.729169 + 1.26296i 0.957235 + 0.289311i \(0.0934263\pi\)
−0.228066 + 0.973646i \(0.573240\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −970.010 −0.0228365 −0.0114183 − 0.999935i \(-0.503635\pi\)
−0.0114183 + 0.999935i \(0.503635\pi\)
\(72\) 0 0
\(73\) −72400.3 −1.59013 −0.795066 − 0.606523i \(-0.792564\pi\)
−0.795066 + 0.606523i \(0.792564\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −22017.3 − 38135.1i −0.423192 − 0.732990i
\(78\) 0 0
\(79\) 16098.9 − 27884.0i 0.290220 − 0.502676i −0.683642 − 0.729818i \(-0.739605\pi\)
0.973862 + 0.227142i \(0.0729382\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 18046.6 − 31257.6i 0.287541 − 0.498036i −0.685681 − 0.727902i \(-0.740496\pi\)
0.973222 + 0.229866i \(0.0738289\pi\)
\(84\) 0 0
\(85\) 31689.3 + 54887.5i 0.475735 + 0.823998i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −42622.2 −0.570375 −0.285188 − 0.958472i \(-0.592056\pi\)
−0.285188 + 0.958472i \(0.592056\pi\)
\(90\) 0 0
\(91\) −117910. −1.49261
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15351.2 − 26589.0i −0.174515 − 0.302269i
\(96\) 0 0
\(97\) 21926.9 − 37978.6i 0.236618 − 0.409835i −0.723123 − 0.690719i \(-0.757294\pi\)
0.959742 + 0.280884i \(0.0906275\pi\)
\(98\) 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.289.1 6
3.2 odd 2 144.6.i.b.97.1 6
4.3 odd 2 54.6.c.b.19.1 6
9.4 even 3 inner 432.6.i.b.145.1 6
9.5 odd 6 144.6.i.b.49.1 6
12.11 even 2 18.6.c.b.7.3 ✓ 6
36.7 odd 6 162.6.a.i.1.3 3
36.11 even 6 162.6.a.j.1.1 3
36.23 even 6 18.6.c.b.13.3 yes 6
36.31 odd 6 54.6.c.b.37.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.3 ✓ 6 12.11 even 2
18.6.c.b.13.3 yes 6 36.23 even 6
54.6.c.b.19.1 6 4.3 odd 2
54.6.c.b.37.1 6 36.31 odd 6
144.6.i.b.49.1 6 9.5 odd 6
144.6.i.b.97.1 6 3.2 odd 2
162.6.a.i.1.3 3 36.7 odd 6
162.6.a.j.1.1 3 36.11 even 6
432.6.i.b.145.1 6 9.4 even 3 inner
432.6.i.b.289.1 6 1.1 even 1 trivial