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Properties

Label 3344.2.a.q.1.2
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [3344,2,Mod(1,3344)] 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("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.523976 q^{3} +2.72545 q^{5} +4.67750 q^{7} -2.72545 q^{9} +1.00000 q^{11} -2.67750 q^{13} +1.42807 q^{15} +0.201472 q^{17} +1.00000 q^{19} +2.45090 q^{21} +1.79853 q^{23} +2.42807 q^{25} -3.00000 q^{27} -4.92692 q^{29} +2.57193 q^{31} +0.523976 q^{33} +12.7483 q^{35} +4.40294 q^{37} -1.40294 q^{39} +4.97487 q^{41} +11.7734 q^{43} -7.42807 q^{45} +12.4989 q^{47} +14.8790 q^{49} +0.105567 q^{51} -4.10557 q^{53} +2.72545 q^{55} +0.523976 q^{57} +5.24943 q^{59} +1.59706 q^{61} -12.7483 q^{63} -7.29738 q^{65} -9.08044 q^{67} +0.942386 q^{69} -12.8214 q^{71} +2.20147 q^{73} +1.27225 q^{75} +4.67750 q^{77} -11.8538 q^{79} +6.60442 q^{81} +13.6775 q^{83} +0.549103 q^{85} -2.58159 q^{87} -5.45090 q^{89} -12.5240 q^{91} +1.34763 q^{93} +2.72545 q^{95} +16.8059 q^{97} -2.72545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.523976 0.302518 0.151259 − 0.988494i \(-0.451667\pi\)
0.151259 + 0.988494i \(0.451667\pi\)
\(4\) 0 0
\(5\) 2.72545 1.21886 0.609429 − 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(6\) 0 0
\(7\) 4.67750 1.76793 0.883964 − 0.467556i \(-0.154865\pi\)
0.883964 + 0.467556i \(0.154865\pi\)
\(8\) 0 0
\(9\) −2.72545 −0.908483
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.67750 −0.742604 −0.371302 − 0.928512i \(-0.621089\pi\)
−0.371302 + 0.928512i \(0.621089\pi\)
\(14\) 0 0
\(15\) 1.42807 0.368726
\(16\) 0 0
\(17\) 0.201472 0.0488642 0.0244321 − 0.999701i \(-0.492222\pi\)
0.0244321 + 0.999701i \(0.492222\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.45090 0.534830
\(22\) 0 0
\(23\) 1.79853 0.375019 0.187509 − 0.982263i \(-0.439958\pi\)
0.187509 + 0.982263i \(0.439958\pi\)
\(24\) 0 0
\(25\) 2.42807 0.485614
\(26\) 0 0
\(27\) −3.00000 −0.577350
\(28\) 0 0
\(29\) −4.92692 −0.914906 −0.457453 − 0.889234i \(-0.651238\pi\)
−0.457453 + 0.889234i \(0.651238\pi\)
\(30\) 0 0
\(31\) 2.57193 0.461932 0.230966 − 0.972962i \(-0.425811\pi\)
0.230966 + 0.972962i \(0.425811\pi\)
\(32\) 0 0
\(33\) 0.523976 0.0912126
\(34\) 0 0
\(35\) 12.7483 2.15485
\(36\) 0 0
\(37\) 4.40294 0.723840 0.361920 − 0.932209i \(-0.382121\pi\)
0.361920 + 0.932209i \(0.382121\pi\)
\(38\) 0 0
\(39\) −1.40294 −0.224651
\(40\) 0 0
\(41\) 4.97487 0.776945 0.388472 − 0.921460i \(-0.373003\pi\)
0.388472 + 0.921460i \(0.373003\pi\)
\(42\) 0 0
\(43\) 11.7734 1.79543 0.897713 − 0.440580i \(-0.145227\pi\)
0.897713 + 0.440580i \(0.145227\pi\)
\(44\) 0 0
\(45\) −7.42807 −1.10731
\(46\) 0 0
\(47\) 12.4989 1.82314 0.911572 − 0.411140i \(-0.134869\pi\)
0.911572 + 0.411140i \(0.134869\pi\)
\(48\) 0 0
\(49\) 14.8790 2.12557
\(50\) 0 0
\(51\) 0.105567 0.0147823
\(52\) 0 0
\(53\) −4.10557 −0.563943 −0.281971 − 0.959423i \(-0.590988\pi\)
−0.281971 + 0.959423i \(0.590988\pi\)
\(54\) 0 0
\(55\) 2.72545 0.367499
\(56\) 0 0
\(57\) 0.523976 0.0694024
\(58\) 0 0
\(59\) 5.24943 0.683417 0.341708 − 0.939806i \(-0.388994\pi\)
0.341708 + 0.939806i \(0.388994\pi\)
\(60\) 0 0
\(61\) 1.59706 0.204482 0.102241 − 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(62\) 0 0
\(63\) −12.7483 −1.60613
\(64\) 0 0
\(65\) −7.29738 −0.905128
\(66\) 0 0
\(67\) −9.08044 −1.10935 −0.554676 − 0.832066i \(-0.687158\pi\)
−0.554676 + 0.832066i \(0.687158\pi\)
\(68\) 0 0
\(69\) 0.942386 0.113450
\(70\) 0 0
\(71\) −12.8214 −1.52161 −0.760807 − 0.648978i \(-0.775197\pi\)
−0.760807 + 0.648978i \(0.775197\pi\)
\(72\) 0 0
\(73\) 2.20147 0.257663 0.128831 − 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(74\) 0 0
\(75\) 1.27225 0.146907
\(76\) 0 0
\(77\) 4.67750 0.533050
\(78\) 0 0
\(79\) −11.8538 −1.33366 −0.666831 − 0.745209i \(-0.732350\pi\)
−0.666831 + 0.745209i \(0.732350\pi\)
\(80\) 0 0
\(81\) 6.60442 0.733824
\(82\) 0 0
\(83\) 13.6775 1.50130 0.750650 − 0.660700i \(-0.229740\pi\)
0.750650 + 0.660700i \(0.229740\pi\)
\(84\) 0 0
\(85\) 0.549103 0.0595585
\(86\) 0 0
\(87\) −2.58159 −0.276776
\(88\) 0 0
\(89\) −5.45090 −0.577794 −0.288897 − 0.957360i \(-0.593289\pi\)
−0.288897 + 0.957360i \(0.593289\pi\)
\(90\) 0 0
\(91\) −12.5240 −1.31287
\(92\) 0 0
\(93\) 1.34763 0.139743
\(94\) 0 0
\(95\) 2.72545 0.279625
\(96\) 0 0
\(97\) 16.8059 1.70638 0.853190 − 0.521601i \(-0.174665\pi\)
0.853190 + 0.521601i \(0.174665\pi\)
\(98\) 0 0
\(99\) −2.72545 −0.273918
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 3344.2.a.q.1.2 3
4.3 odd 2 418.2.a.g.1.2 ✓ 3
12.11 even 2 3762.2.a.bg.1.1 3
44.43 even 2 4598.2.a.bo.1.2 3
76.75 even 2 7942.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.2 ✓ 3 4.3 odd 2
3344.2.a.q.1.2 3 1.1 even 1 trivial
3762.2.a.bg.1.1 3 12.11 even 2
4598.2.a.bo.1.2 3 44.43 even 2
7942.2.a.bi.1.2 3 76.75 even 2