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Properties

Label 4650.2.a.bz.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
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] = [4650,2,Mod(1,4650)] 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("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-2,2,0,2,1,-2,2,0,5,-2,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} +1.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} -9.06226 q^{37} +3.53113 q^{38} +6.00000 q^{39} -9.06226 q^{41} -3.53113 q^{42} +0.468871 q^{43} -1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} -1.00000 q^{48} +5.46887 q^{49} -4.00000 q^{51} -6.00000 q^{52} -5.53113 q^{53} +1.00000 q^{54} +3.53113 q^{56} +3.53113 q^{57} +7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} -3.53113 q^{63} +1.00000 q^{64} -1.53113 q^{66} +11.0623 q^{67} +4.00000 q^{68} -1.53113 q^{69} -4.46887 q^{71} -1.00000 q^{72} +0.468871 q^{73} +9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -6.00000 q^{78} -0.468871 q^{79} +1.00000 q^{81} +9.06226 q^{82} +8.00000 q^{83} +3.53113 q^{84} -0.468871 q^{86} +1.53113 q^{88} +1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} -1.00000 q^{93} +11.0623 q^{94} +1.00000 q^{96} +16.1245 q^{97} -5.46887 q^{98} -1.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −3.53113 −1.33464 −0.667321 − 0.744771i \(-0.732559\pi\)
−0.667321 + 0.744771i \(0.732559\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.53113 −0.461653 −0.230826 − 0.972995i \(-0.574143\pi\)
−0.230826 + 0.972995i \(0.574143\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 3.53113 0.943734
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.53113 −0.810097 −0.405048 − 0.914295i \(-0.632745\pi\)
−0.405048 + 0.914295i \(0.632745\pi\)
\(20\) 0 0
\(21\) 3.53113 0.770555
\(22\) 1.53113 0.326438
\(23\) 1.53113 0.319262 0.159631 − 0.987177i \(-0.448969\pi\)
0.159631 + 0.987177i \(0.448969\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) −3.53113 −0.667321
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 1.53113 0.266535
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −9.06226 −1.48983 −0.744913 − 0.667162i \(-0.767509\pi\)
−0.744913 + 0.667162i \(0.767509\pi\)
\(38\) 3.53113 0.572825
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −9.06226 −1.41529 −0.707643 − 0.706570i \(-0.750242\pi\)
−0.707643 + 0.706570i \(0.750242\pi\)
\(42\) −3.53113 −0.544865
\(43\) 0.468871 0.0715022 0.0357511 − 0.999361i \(-0.488618\pi\)
0.0357511 + 0.999361i \(0.488618\pi\)
\(44\) −1.53113 −0.230826
\(45\) 0 0
\(46\) −1.53113 −0.225753
\(47\) −11.0623 −1.61360 −0.806798 − 0.590827i \(-0.798801\pi\)
−0.806798 + 0.590827i \(0.798801\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.46887 0.781267
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −6.00000 −0.832050
\(53\) −5.53113 −0.759759 −0.379879 − 0.925036i \(-0.624035\pi\)
−0.379879 + 0.925036i \(0.624035\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.53113 0.471867
\(57\) 3.53113 0.467709
\(58\) 0 0
\(59\) 7.06226 0.919428 0.459714 − 0.888067i \(-0.347952\pi\)
0.459714 + 0.888067i \(0.347952\pi\)
\(60\) 0 0
\(61\) −11.0623 −1.41638 −0.708188 − 0.706023i \(-0.750487\pi\)
−0.708188 + 0.706023i \(0.750487\pi\)
\(62\) −1.00000 −0.127000
\(63\) −3.53113 −0.444880
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.53113 −0.188469
\(67\) 11.0623 1.35147 0.675735 − 0.737145i \(-0.263826\pi\)
0.675735 + 0.737145i \(0.263826\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.53113 −0.184326
\(70\) 0 0
\(71\) −4.46887 −0.530357 −0.265179 − 0.964199i \(-0.585431\pi\)
−0.265179 + 0.964199i \(0.585431\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0.468871 0.0548772 0.0274386 − 0.999623i \(-0.491265\pi\)
0.0274386 + 0.999623i \(0.491265\pi\)
\(74\) 9.06226 1.05347
\(75\) 0 0
\(76\) −3.53113 −0.405048
\(77\) 5.40661 0.616141
\(78\) −6.00000 −0.679366
\(79\) −0.468871 −0.0527521 −0.0263761 − 0.999652i \(-0.508397\pi\)
−0.0263761 + 0.999652i \(0.508397\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.06226 1.00076
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 3.53113 0.385278
\(85\) 0 0
\(86\) −0.468871 −0.0505597
\(87\) 0 0
\(88\) 1.53113 0.163219
\(89\) 1.53113 0.162299 0.0811497 − 0.996702i \(-0.474141\pi\)
0.0811497 + 0.996702i \(0.474141\pi\)
\(90\) 0 0
\(91\) 21.1868 2.22098
\(92\) 1.53113 0.159631
\(93\) −1.00000 −0.103695
\(94\) 11.0623 1.14098
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 16.1245 1.63720 0.818598 − 0.574367i \(-0.194752\pi\)
0.818598 + 0.574367i \(0.194752\pi\)
\(98\) −5.46887 −0.552439
\(99\) −1.53113 −0.153884
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 4650.2.a.bz.1.1 2
5.2 odd 4 4650.2.d.bg.3349.1 4
5.3 odd 4 4650.2.d.bg.3349.4 4
5.4 even 2 930.2.a.q.1.2 ✓ 2
15.14 odd 2 2790.2.a.bf.1.2 2
20.19 odd 2 7440.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.q.1.2 ✓ 2 5.4 even 2
2790.2.a.bf.1.2 2 15.14 odd 2
4650.2.a.bz.1.1 2 1.1 even 1 trivial
4650.2.d.bg.3349.1 4 5.2 odd 4
4650.2.d.bg.3349.4 4 5.3 odd 4
7440.2.a.bd.1.1 2 20.19 odd 2