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Properties

Label 444.2.r.a.397.1
Level $444$
Weight $2$
Character 444.397
Analytic conductor $3.545$
Analytic rank $0$
Dimension $2$
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(85,444)] 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("444.85"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(6)) 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.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 397.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 444.397
Dual form 444.2.r.a.85.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-3.00000 + 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +6.00000 q^{11} +(1.50000 - 0.866025i) q^{13} +(3.00000 + 1.73205i) q^{15} +(6.00000 + 3.46410i) q^{17} +(3.00000 - 1.73205i) q^{19} +(-0.500000 + 0.866025i) q^{21} +3.46410i q^{23} +(3.50000 - 6.06218i) q^{25} +1.00000 q^{27} +6.92820i q^{29} -1.73205i q^{31} +(-3.00000 - 5.19615i) q^{33} +(3.00000 + 1.73205i) q^{35} +(-5.00000 - 3.46410i) q^{37} +(-1.50000 - 0.866025i) q^{39} +(-6.00000 - 10.3923i) q^{41} +12.1244i q^{43} -3.46410i q^{45} +12.0000 q^{47} +(3.00000 - 5.19615i) q^{49} -6.92820i q^{51} +(-3.00000 + 5.19615i) q^{53} +(-18.0000 + 10.3923i) q^{55} +(-3.00000 - 1.73205i) q^{57} +(-3.00000 - 1.73205i) q^{59} +(6.00000 - 3.46410i) q^{61} +1.00000 q^{63} +(-3.00000 + 5.19615i) q^{65} +(2.50000 + 4.33013i) q^{67} +(3.00000 - 1.73205i) q^{69} +7.00000 q^{73} -7.00000 q^{75} +(-3.00000 - 5.19615i) q^{77} +(1.50000 - 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-6.00000 + 10.3923i) q^{83} -24.0000 q^{85} +(6.00000 - 3.46410i) q^{87} +(9.00000 + 5.19615i) q^{89} +(-1.50000 - 0.866025i) q^{91} +(-1.50000 + 0.866025i) q^{93} +(-6.00000 + 10.3923i) q^{95} -1.73205i q^{97} +(-3.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 6 q^{5} - q^{7} - q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} + 12 q^{17} + 6 q^{19} - q^{21} + 7 q^{25} + 2 q^{27} - 6 q^{33} + 6 q^{35} - 10 q^{37} - 3 q^{39} - 12 q^{41} + 24 q^{47} + 6 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) −3.00000 + 1.73205i −1.34164 + 0.774597i −0.987048 − 0.160424i \(-0.948714\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i 0.755929 − 0.654654i \(-0.227186\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 1.50000 − 0.866025i 0.416025 − 0.240192i −0.277350 − 0.960769i \(-0.589456\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 3.00000 + 1.73205i 0.774597 + 0.447214i
\(16\) 0 0
\(17\) 6.00000 + 3.46410i 1.45521 + 0.840168i 0.998770 − 0.0495842i \(-0.0157896\pi\)
0.456444 + 0.889752i \(0.349123\pi\)
\(18\) 0 0
\(19\) 3.00000 − 1.73205i 0.688247 − 0.397360i −0.114708 − 0.993399i \(-0.536593\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −0.500000 + 0.866025i −0.109109 + 0.188982i
\(22\) 0 0
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 3.50000 − 6.06218i 0.700000 − 1.21244i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 − 0.155543i \(-0.950287\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) 0 0
\(33\) −3.00000 − 5.19615i −0.522233 − 0.904534i
\(34\) 0 0
\(35\) 3.00000 + 1.73205i 0.507093 + 0.292770i
\(36\) 0 0
\(37\) −5.00000 − 3.46410i −0.821995 − 0.569495i
\(38\) 0 0
\(39\) −1.50000 − 0.866025i −0.240192 − 0.138675i
\(40\) 0 0
\(41\) −6.00000 − 10.3923i −0.937043 − 1.62301i −0.770950 − 0.636895i \(-0.780218\pi\)
−0.166092 − 0.986110i \(-0.553115\pi\)
\(42\) 0 0
\(43\) 12.1244i 1.84895i 0.381246 + 0.924473i \(0.375495\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) − 3.46410i − 0.516398i
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) − 6.92820i − 0.970143i
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −18.0000 + 10.3923i −2.42712 + 1.40130i
\(56\) 0 0
\(57\) −3.00000 − 1.73205i −0.397360 − 0.229416i
\(58\) 0 0
\(59\) −3.00000 − 1.73205i −0.390567 − 0.225494i 0.291839 − 0.956467i \(-0.405733\pi\)
−0.682406 + 0.730974i \(0.739066\pi\)
\(60\) 0 0
\(61\) 6.00000 − 3.46410i 0.768221 − 0.443533i −0.0640184 − 0.997949i \(-0.520392\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) 2.50000 + 4.33013i 0.305424 + 0.529009i 0.977356 − 0.211604i \(-0.0678686\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 3.00000 − 1.73205i 0.361158 − 0.208514i
\(70\) 0 0
\(71\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 0 0
\(77\) −3.00000 − 5.19615i −0.341882 − 0.592157i
\(78\) 0 0
\(79\) 1.50000 − 0.866025i 0.168763 − 0.0974355i −0.413239 − 0.910622i \(-0.635603\pi\)
0.582003 + 0.813187i \(0.302269\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −6.00000 + 10.3923i −0.658586 + 1.14070i 0.322396 + 0.946605i \(0.395512\pi\)
−0.980982 + 0.194099i \(0.937822\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) 0 0
\(87\) 6.00000 − 3.46410i 0.643268 − 0.371391i
\(88\) 0 0
\(89\) 9.00000 + 5.19615i 0.953998 + 0.550791i 0.894321 − 0.447427i \(-0.147659\pi\)
0.0596775 + 0.998218i \(0.480993\pi\)
\(90\) 0 0
\(91\) −1.50000 − 0.866025i −0.157243 − 0.0907841i
\(92\) 0 0
\(93\) −1.50000 + 0.866025i −0.155543 + 0.0898027i
\(94\) 0 0
\(95\) −6.00000 + 10.3923i −0.615587 + 1.06623i
\(96\) 0 0
\(97\) − 1.73205i − 0.175863i −0.996127 − 0.0879316i \(-0.971974\pi\)
0.996127 − 0.0879316i \(-0.0280257\pi\)
\(98\) 0 0
\(99\) −3.00000 + 5.19615i −0.301511 + 0.522233i
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.r.a.397.1 yes 2
3.2 odd 2 1332.2.bi.f.397.1 2
4.3 odd 2 1776.2.bz.a.1729.1 2
37.11 even 6 inner 444.2.r.a.85.1 ✓ 2
111.11 odd 6 1332.2.bi.f.973.1 2
148.11 odd 6 1776.2.bz.a.529.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.r.a.85.1 ✓ 2 37.11 even 6 inner
444.2.r.a.397.1 yes 2 1.1 even 1 trivial
1332.2.bi.f.397.1 2 3.2 odd 2
1332.2.bi.f.973.1 2 111.11 odd 6
1776.2.bz.a.529.1 2 148.11 odd 6
1776.2.bz.a.1729.1 2 4.3 odd 2