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Properties

Label 444.2.bb.b.25.1
Level $444$
Weight $2$
Character 444.25
Analytic conductor $3.545$
Analytic rank $0$
Dimension $24$
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] = [444,2,Mod(25,444)] 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("444.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 25.1
Character \(\chi\) \(=\) 444.25
Dual form 444.2.bb.b.373.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.766044 - 0.642788i) q^{3} +(-1.30455 - 3.58423i) q^{5} +(1.22792 - 0.446926i) q^{7} +(0.173648 + 0.984808i) q^{9} +(0.127639 + 0.221076i) q^{11} +(-0.901194 - 0.158905i) q^{13} +(-1.30455 + 3.58423i) q^{15} +(-7.91638 + 1.39587i) q^{17} +(1.61393 - 1.92341i) q^{19} +(-1.22792 - 0.446926i) q^{21} +(-4.97367 - 2.87155i) q^{23} +(-7.31465 + 6.13772i) q^{25} +(0.500000 - 0.866025i) q^{27} +(1.74476 - 1.00734i) q^{29} -5.60117i q^{31} +(0.0443284 - 0.251399i) q^{33} +(-3.20377 - 3.81811i) q^{35} +(2.30611 + 5.62866i) q^{37} +(0.588212 + 0.701004i) q^{39} +(1.19772 - 6.79262i) q^{41} +0.222022i q^{43} +(3.30325 - 1.90713i) q^{45} +(1.86356 - 3.22778i) q^{47} +(-4.05427 + 3.40194i) q^{49} +(6.96155 + 4.01925i) q^{51} +(6.08425 + 2.21449i) q^{53} +(0.625878 - 0.745893i) q^{55} +(-2.47268 + 0.436001i) q^{57} +(3.20193 - 8.79723i) q^{59} +(1.62798 + 0.287057i) q^{61} +(0.653362 + 1.13166i) q^{63} +(0.606104 + 3.43739i) q^{65} +(2.24357 - 0.816593i) q^{67} +(1.96426 + 5.39675i) q^{69} +(1.53636 + 1.28916i) q^{71} -8.71637 q^{73} +9.54859 q^{75} +(0.255534 + 0.214419i) q^{77} +(-3.50919 - 9.64143i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(2.44400 + 13.8606i) q^{83} +(15.3305 + 26.5532i) q^{85} +(-1.98406 - 0.349844i) q^{87} +(-0.732278 + 2.01192i) q^{89} +(-1.17761 + 0.207645i) q^{91} +(-3.60036 + 4.29075i) q^{93} +(-8.99939 - 3.27551i) q^{95} +(5.79292 + 3.34455i) q^{97} +(-0.195554 + 0.164089i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{7} + 6 q^{11} - 12 q^{13} - 15 q^{17} + 12 q^{19} + 3 q^{21} + 18 q^{25} + 12 q^{27} - 9 q^{29} + 18 q^{33} + 12 q^{35} + 12 q^{37} + 15 q^{39} + 15 q^{41} - 15 q^{47} + 15 q^{49} + 9 q^{51}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{18}\right)\)

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
\(3\) −0.766044 − 0.642788i −0.442276 − 0.371114i
\(4\) 0 0
\(5\) −1.30455 − 3.58423i −0.583414 − 1.60292i −0.782304 − 0.622896i \(-0.785956\pi\)
0.198890 − 0.980022i \(-0.436266\pi\)
\(6\) 0 0
\(7\) 1.22792 − 0.446926i 0.464110 − 0.168922i −0.0993726 − 0.995050i \(-0.531684\pi\)
0.563482 + 0.826128i \(0.309461\pi\)
\(8\) 0 0
\(9\) 0.173648 + 0.984808i 0.0578827 + 0.328269i
\(10\) 0 0
\(11\) 0.127639 + 0.221076i 0.0384845 + 0.0666571i 0.884626 − 0.466301i \(-0.154414\pi\)
−0.846142 + 0.532958i \(0.821080\pi\)
\(12\) 0 0
\(13\) −0.901194 − 0.158905i −0.249946 − 0.0440722i 0.0472712 − 0.998882i \(-0.484947\pi\)
−0.297217 + 0.954810i \(0.596059\pi\)
\(14\) 0 0
\(15\) −1.30455 + 3.58423i −0.336834 + 0.925445i
\(16\) 0 0
\(17\) −7.91638 + 1.39587i −1.92001 + 0.338549i −0.998731 − 0.0503648i \(-0.983962\pi\)
−0.921274 + 0.388914i \(0.872850\pi\)
\(18\) 0 0
\(19\) 1.61393 − 1.92341i 0.370261 − 0.441259i −0.548454 − 0.836180i \(-0.684784\pi\)
0.918715 + 0.394921i \(0.129228\pi\)
\(20\) 0 0
\(21\) −1.22792 − 0.446926i −0.267954 − 0.0975272i
\(22\) 0 0
\(23\) −4.97367 − 2.87155i −1.03708 − 0.598759i −0.118076 − 0.993005i \(-0.537673\pi\)
−0.919005 + 0.394245i \(0.871006\pi\)
\(24\) 0 0
\(25\) −7.31465 + 6.13772i −1.46293 + 1.22754i
\(26\) 0 0
\(27\) 0.500000 − 0.866025i 0.0962250 − 0.166667i
\(28\) 0 0
\(29\) 1.74476 − 1.00734i 0.323993 − 0.187058i −0.329178 − 0.944268i \(-0.606772\pi\)
0.653171 + 0.757210i \(0.273438\pi\)
\(30\) 0 0
\(31\) − 5.60117i − 1.00600i −0.864286 − 0.503000i \(-0.832230\pi\)
0.864286 − 0.503000i \(-0.167770\pi\)
\(32\) 0 0
\(33\) 0.0443284 − 0.251399i 0.00771658 − 0.0437629i
\(34\) 0 0
\(35\) −3.20377 − 3.81811i −0.541536 − 0.645378i
\(36\) 0 0
\(37\) 2.30611 + 5.62866i 0.379122 + 0.925347i
\(38\) 0 0
\(39\) 0.588212 + 0.701004i 0.0941894 + 0.112251i
\(40\) 0 0
\(41\) 1.19772 − 6.79262i 0.187053 − 1.06083i −0.736237 − 0.676724i \(-0.763399\pi\)
0.923290 − 0.384105i \(-0.125490\pi\)
\(42\) 0 0
\(43\) 0.222022i 0.0338581i 0.999857 + 0.0169291i \(0.00538894\pi\)
−0.999857 + 0.0169291i \(0.994611\pi\)
\(44\) 0 0
\(45\) 3.30325 − 1.90713i 0.492419 − 0.284298i
\(46\) 0 0
\(47\) 1.86356 − 3.22778i 0.271828 − 0.470820i −0.697502 − 0.716583i \(-0.745705\pi\)
0.969330 + 0.245763i \(0.0790385\pi\)
\(48\) 0 0
\(49\) −4.05427 + 3.40194i −0.579181 + 0.485991i
\(50\) 0 0
\(51\) 6.96155 + 4.01925i 0.974812 + 0.562808i
\(52\) 0 0
\(53\) 6.08425 + 2.21449i 0.835736 + 0.304183i 0.724211 − 0.689579i \(-0.242204\pi\)
0.111525 + 0.993762i \(0.464426\pi\)
\(54\) 0 0
\(55\) 0.625878 − 0.745893i 0.0843934 − 0.100576i
\(56\) 0 0
\(57\) −2.47268 + 0.436001i −0.327515 + 0.0577497i
\(58\) 0 0
\(59\) 3.20193 − 8.79723i 0.416856 − 1.14530i −0.536618 − 0.843825i \(-0.680298\pi\)
0.953474 − 0.301476i \(-0.0974795\pi\)
\(60\) 0 0
\(61\) 1.62798 + 0.287057i 0.208442 + 0.0367539i 0.276894 − 0.960901i \(-0.410695\pi\)
−0.0684523 + 0.997654i \(0.521806\pi\)
\(62\) 0 0
\(63\) 0.653362 + 1.13166i 0.0823159 + 0.142575i
\(64\) 0 0
\(65\) 0.606104 + 3.43739i 0.0751780 + 0.426356i
\(66\) 0 0
\(67\) 2.24357 − 0.816593i 0.274096 − 0.0997627i −0.201315 − 0.979526i \(-0.564522\pi\)
0.475411 + 0.879764i \(0.342299\pi\)
\(68\) 0 0
\(69\) 1.96426 + 5.39675i 0.236469 + 0.649692i
\(70\) 0 0
\(71\) 1.53636 + 1.28916i 0.182332 + 0.152995i 0.729385 − 0.684103i \(-0.239806\pi\)
−0.547054 + 0.837098i \(0.684251\pi\)
\(72\) 0 0
\(73\) −8.71637 −1.02017 −0.510087 − 0.860123i \(-0.670387\pi\)
−0.510087 + 0.860123i \(0.670387\pi\)
\(74\) 0 0
\(75\) 9.54859 1.10258
\(76\) 0 0
\(77\) 0.255534 + 0.214419i 0.0291209 + 0.0244353i
\(78\) 0 0
\(79\) −3.50919 − 9.64143i −0.394815 − 1.08475i −0.964776 − 0.263074i \(-0.915264\pi\)
0.569961 − 0.821672i \(-0.306958\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.104410 + 0.0380022i
\(82\) 0 0
\(83\) 2.44400 + 13.8606i 0.268264 + 1.52140i 0.759576 + 0.650419i \(0.225407\pi\)
−0.491312 + 0.870984i \(0.663482\pi\)
\(84\) 0 0
\(85\) 15.3305 + 26.5532i 1.66282 + 2.88010i
\(86\) 0 0
\(87\) −1.98406 − 0.349844i −0.212714 − 0.0375072i
\(88\) 0 0
\(89\) −0.732278 + 2.01192i −0.0776213 + 0.213263i −0.972434 − 0.233180i \(-0.925087\pi\)
0.894812 + 0.446442i \(0.147309\pi\)
\(90\) 0 0
\(91\) −1.17761 + 0.207645i −0.123447 + 0.0217671i
\(92\) 0 0
\(93\) −3.60036 + 4.29075i −0.373340 + 0.444930i
\(94\) 0 0
\(95\) −8.99939 − 3.27551i −0.923318 − 0.336060i
\(96\) 0 0
\(97\) 5.79292 + 3.34455i 0.588182 + 0.339587i 0.764378 − 0.644768i \(-0.223046\pi\)
−0.176196 + 0.984355i \(0.556379\pi\)
\(98\) 0 0
\(99\) −0.195554 + 0.164089i −0.0196539 + 0.0164916i
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 444.2.bb.b.25.1 ✓ 24
3.2 odd 2 1332.2.ct.e.469.4 24
37.3 even 18 inner 444.2.bb.b.373.1 yes 24
111.77 odd 18 1332.2.ct.e.1261.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.bb.b.25.1 ✓ 24 1.1 even 1 trivial
444.2.bb.b.373.1 yes 24 37.3 even 18 inner
1332.2.ct.e.469.4 24 3.2 odd 2
1332.2.ct.e.1261.4 24 111.77 odd 18