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Properties

Label 6003.2.a.t.1.3
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $22$
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] = [6003,2,Mod(1,6003)] 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("6003.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6003, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,-3,0,17,0] 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: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 6003.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.24483 q^{2} +3.03928 q^{4} +2.70888 q^{5} -1.81542 q^{7} -2.33301 q^{8} -6.08099 q^{10} +1.28860 q^{11} +4.53826 q^{13} +4.07531 q^{14} -0.841340 q^{16} -4.48595 q^{17} +2.67238 q^{19} +8.23305 q^{20} -2.89270 q^{22} +1.00000 q^{23} +2.33804 q^{25} -10.1876 q^{26} -5.51756 q^{28} +1.00000 q^{29} -5.79860 q^{31} +6.55469 q^{32} +10.0702 q^{34} -4.91775 q^{35} -5.91943 q^{37} -5.99906 q^{38} -6.31985 q^{40} -7.82902 q^{41} -5.28362 q^{43} +3.91642 q^{44} -2.24483 q^{46} +2.52885 q^{47} -3.70427 q^{49} -5.24851 q^{50} +13.7930 q^{52} +7.14230 q^{53} +3.49067 q^{55} +4.23538 q^{56} -2.24483 q^{58} -7.87478 q^{59} +0.0192849 q^{61} +13.0169 q^{62} -13.0315 q^{64} +12.2936 q^{65} -15.5222 q^{67} -13.6340 q^{68} +11.0395 q^{70} +8.18804 q^{71} -16.8537 q^{73} +13.2881 q^{74} +8.12212 q^{76} -2.33935 q^{77} +2.32730 q^{79} -2.27909 q^{80} +17.5749 q^{82} +8.65460 q^{83} -12.1519 q^{85} +11.8608 q^{86} -3.00632 q^{88} +17.6714 q^{89} -8.23883 q^{91} +3.03928 q^{92} -5.67686 q^{94} +7.23917 q^{95} -7.50809 q^{97} +8.31546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24483 −1.58734 −0.793669 − 0.608350i \(-0.791832\pi\)
−0.793669 + 0.608350i \(0.791832\pi\)
\(3\) 0 0
\(4\) 3.03928 1.51964
\(5\) 2.70888 1.21145 0.605724 − 0.795675i \(-0.292883\pi\)
0.605724 + 0.795675i \(0.292883\pi\)
\(6\) 0 0
\(7\) −1.81542 −0.686163 −0.343081 − 0.939306i \(-0.611471\pi\)
−0.343081 + 0.939306i \(0.611471\pi\)
\(8\) −2.33301 −0.824843
\(9\) 0 0
\(10\) −6.08099 −1.92298
\(11\) 1.28860 0.388528 0.194264 − 0.980949i \(-0.437768\pi\)
0.194264 + 0.980949i \(0.437768\pi\)
\(12\) 0 0
\(13\) 4.53826 1.25869 0.629343 − 0.777127i \(-0.283324\pi\)
0.629343 + 0.777127i \(0.283324\pi\)
\(14\) 4.07531 1.08917
\(15\) 0 0
\(16\) −0.841340 −0.210335
\(17\) −4.48595 −1.08800 −0.544001 − 0.839085i \(-0.683091\pi\)
−0.544001 + 0.839085i \(0.683091\pi\)
\(18\) 0 0
\(19\) 2.67238 0.613087 0.306543 − 0.951857i \(-0.400828\pi\)
0.306543 + 0.951857i \(0.400828\pi\)
\(20\) 8.23305 1.84097
\(21\) 0 0
\(22\) −2.89270 −0.616725
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 2.33804 0.467607
\(26\) −10.1876 −1.99796
\(27\) 0 0
\(28\) −5.51756 −1.04272
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −5.79860 −1.04146 −0.520730 − 0.853722i \(-0.674340\pi\)
−0.520730 + 0.853722i \(0.674340\pi\)
\(32\) 6.55469 1.15872
\(33\) 0 0
\(34\) 10.0702 1.72703
\(35\) −4.91775 −0.831251
\(36\) 0 0
\(37\) −5.91943 −0.973147 −0.486574 − 0.873640i \(-0.661754\pi\)
−0.486574 + 0.873640i \(0.661754\pi\)
\(38\) −5.99906 −0.973175
\(39\) 0 0
\(40\) −6.31985 −0.999255
\(41\) −7.82902 −1.22269 −0.611344 − 0.791365i \(-0.709371\pi\)
−0.611344 + 0.791365i \(0.709371\pi\)
\(42\) 0 0
\(43\) −5.28362 −0.805744 −0.402872 − 0.915256i \(-0.631988\pi\)
−0.402872 + 0.915256i \(0.631988\pi\)
\(44\) 3.91642 0.590423
\(45\) 0 0
\(46\) −2.24483 −0.330983
\(47\) 2.52885 0.368871 0.184436 − 0.982845i \(-0.440954\pi\)
0.184436 + 0.982845i \(0.440954\pi\)
\(48\) 0 0
\(49\) −3.70427 −0.529181
\(50\) −5.24851 −0.742251
\(51\) 0 0
\(52\) 13.7930 1.91275
\(53\) 7.14230 0.981070 0.490535 − 0.871421i \(-0.336801\pi\)
0.490535 + 0.871421i \(0.336801\pi\)
\(54\) 0 0
\(55\) 3.49067 0.470682
\(56\) 4.23538 0.565977
\(57\) 0 0
\(58\) −2.24483 −0.294761
\(59\) −7.87478 −1.02521 −0.512605 − 0.858625i \(-0.671319\pi\)
−0.512605 + 0.858625i \(0.671319\pi\)
\(60\) 0 0
\(61\) 0.0192849 0.00246918 0.00123459 − 0.999999i \(-0.499607\pi\)
0.00123459 + 0.999999i \(0.499607\pi\)
\(62\) 13.0169 1.65315
\(63\) 0 0
\(64\) −13.0315 −1.62894
\(65\) 12.2936 1.52483
\(66\) 0 0
\(67\) −15.5222 −1.89634 −0.948170 − 0.317764i \(-0.897068\pi\)
−0.948170 + 0.317764i \(0.897068\pi\)
\(68\) −13.6340 −1.65337
\(69\) 0 0
\(70\) 11.0395 1.31948
\(71\) 8.18804 0.971742 0.485871 − 0.874031i \(-0.338502\pi\)
0.485871 + 0.874031i \(0.338502\pi\)
\(72\) 0 0
\(73\) −16.8537 −1.97258 −0.986288 − 0.165035i \(-0.947226\pi\)
−0.986288 + 0.165035i \(0.947226\pi\)
\(74\) 13.2881 1.54471
\(75\) 0 0
\(76\) 8.12212 0.931671
\(77\) −2.33935 −0.266593
\(78\) 0 0
\(79\) 2.32730 0.261842 0.130921 − 0.991393i \(-0.458207\pi\)
0.130921 + 0.991393i \(0.458207\pi\)
\(80\) −2.27909 −0.254810
\(81\) 0 0
\(82\) 17.5749 1.94082
\(83\) 8.65460 0.949966 0.474983 − 0.879995i \(-0.342454\pi\)
0.474983 + 0.879995i \(0.342454\pi\)
\(84\) 0 0
\(85\) −12.1519 −1.31806
\(86\) 11.8608 1.27899
\(87\) 0 0
\(88\) −3.00632 −0.320475
\(89\) 17.6714 1.87316 0.936581 − 0.350451i \(-0.113972\pi\)
0.936581 + 0.350451i \(0.113972\pi\)
\(90\) 0 0
\(91\) −8.23883 −0.863664
\(92\) 3.03928 0.316867
\(93\) 0 0
\(94\) −5.67686 −0.585523
\(95\) 7.23917 0.742723
\(96\) 0 0
\(97\) −7.50809 −0.762331 −0.381166 − 0.924507i \(-0.624477\pi\)
−0.381166 + 0.924507i \(0.624477\pi\)
\(98\) 8.31546 0.839989
\(99\) 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 6003.2.a.t.1.3 ✓ 22
3.2 odd 2 6003.2.a.u.1.20 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.t.1.3 ✓ 22 1.1 even 1 trivial
6003.2.a.u.1.20 yes 22 3.2 odd 2