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Properties

Label 4600.2.a.bd.1.1
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
Dimension $5$
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] = [4600,2,Mod(1,4600)] 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("4600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,-4,0,3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.521397.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} - 3x^{2} + 18x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.61696\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.61696 q^{3} -3.83744 q^{7} +3.84849 q^{9} -0.508005 q^{11} -1.01106 q^{13} -1.44705 q^{17} -0.508005 q^{19} +10.0424 q^{21} +1.00000 q^{23} -2.22047 q^{27} +7.51040 q^{29} -0.439038 q^{31} +1.32943 q^{33} +7.02642 q^{37} +2.64590 q^{39} +5.47041 q^{41} -6.72592 q^{43} +2.64098 q^{47} +7.72592 q^{49} +3.78688 q^{51} -4.77648 q^{53} +1.32943 q^{57} +3.85345 q^{59} -9.05844 q^{61} -14.7683 q^{63} -3.45696 q^{67} -2.61696 q^{69} -2.73649 q^{71} +9.21300 q^{73} +1.94944 q^{77} +10.5504 q^{79} -5.73458 q^{81} +1.40211 q^{83} -19.6544 q^{87} +6.77086 q^{89} +3.87986 q^{91} +1.14895 q^{93} +0.313420 q^{97} -1.95506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{7} + 3 q^{9} - 4 q^{13} - 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} - 18 q^{31} - 6 q^{33} - 10 q^{37} + 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} - 6 q^{51} - 10 q^{53} - 6 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) −2.61696 −1.51090 −0.755452 − 0.655204i \(-0.772583\pi\)
−0.755452 + 0.655204i \(0.772583\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.83744 −1.45041 −0.725207 − 0.688531i \(-0.758256\pi\)
−0.725207 + 0.688531i \(0.758256\pi\)
\(8\) 0 0
\(9\) 3.84849 1.28283
\(10\) 0 0
\(11\) −0.508005 −0.153169 −0.0765847 − 0.997063i \(-0.524402\pi\)
−0.0765847 + 0.997063i \(0.524402\pi\)
\(12\) 0 0
\(13\) −1.01106 −0.280417 −0.140208 − 0.990122i \(-0.544777\pi\)
−0.140208 + 0.990122i \(0.544777\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.44705 −0.350961 −0.175481 − 0.984483i \(-0.556148\pi\)
−0.175481 + 0.984483i \(0.556148\pi\)
\(18\) 0 0
\(19\) −0.508005 −0.116544 −0.0582722 − 0.998301i \(-0.518559\pi\)
−0.0582722 + 0.998301i \(0.518559\pi\)
\(20\) 0 0
\(21\) 10.0424 2.19144
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.22047 −0.427330
\(28\) 0 0
\(29\) 7.51040 1.39465 0.697323 − 0.716757i \(-0.254374\pi\)
0.697323 + 0.716757i \(0.254374\pi\)
\(30\) 0 0
\(31\) −0.439038 −0.0788536 −0.0394268 − 0.999222i \(-0.512553\pi\)
−0.0394268 + 0.999222i \(0.512553\pi\)
\(32\) 0 0
\(33\) 1.32943 0.231424
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.02642 1.15514 0.577568 − 0.816343i \(-0.304002\pi\)
0.577568 + 0.816343i \(0.304002\pi\)
\(38\) 0 0
\(39\) 2.64590 0.423683
\(40\) 0 0
\(41\) 5.47041 0.854335 0.427167 − 0.904173i \(-0.359512\pi\)
0.427167 + 0.904173i \(0.359512\pi\)
\(42\) 0 0
\(43\) −6.72592 −1.02569 −0.512847 − 0.858480i \(-0.671409\pi\)
−0.512847 + 0.858480i \(0.671409\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.64098 0.385227 0.192614 − 0.981275i \(-0.438304\pi\)
0.192614 + 0.981275i \(0.438304\pi\)
\(48\) 0 0
\(49\) 7.72592 1.10370
\(50\) 0 0
\(51\) 3.78688 0.530269
\(52\) 0 0
\(53\) −4.77648 −0.656100 −0.328050 − 0.944660i \(-0.606391\pi\)
−0.328050 + 0.944660i \(0.606391\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.32943 0.176087
\(58\) 0 0
\(59\) 3.85345 0.501676 0.250838 − 0.968029i \(-0.419294\pi\)
0.250838 + 0.968029i \(0.419294\pi\)
\(60\) 0 0
\(61\) −9.05844 −1.15981 −0.579907 − 0.814683i \(-0.696911\pi\)
−0.579907 + 0.814683i \(0.696911\pi\)
\(62\) 0 0
\(63\) −14.7683 −1.86064
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.45696 −0.422335 −0.211167 − 0.977450i \(-0.567727\pi\)
−0.211167 + 0.977450i \(0.567727\pi\)
\(68\) 0 0
\(69\) −2.61696 −0.315045
\(70\) 0 0
\(71\) −2.73649 −0.324762 −0.162381 − 0.986728i \(-0.551917\pi\)
−0.162381 + 0.986728i \(0.551917\pi\)
\(72\) 0 0
\(73\) 9.21300 1.07830 0.539150 − 0.842210i \(-0.318746\pi\)
0.539150 + 0.842210i \(0.318746\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.94944 0.222159
\(78\) 0 0
\(79\) 10.5504 1.18702 0.593508 − 0.804828i \(-0.297742\pi\)
0.593508 + 0.804828i \(0.297742\pi\)
\(80\) 0 0
\(81\) −5.73458 −0.637176
\(82\) 0 0
\(83\) 1.40211 0.153901 0.0769505 − 0.997035i \(-0.475482\pi\)
0.0769505 + 0.997035i \(0.475482\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −19.6544 −2.10718
\(88\) 0 0
\(89\) 6.77086 0.717710 0.358855 − 0.933393i \(-0.383167\pi\)
0.358855 + 0.933393i \(0.383167\pi\)
\(90\) 0 0
\(91\) 3.87986 0.406720
\(92\) 0 0
\(93\) 1.14895 0.119140
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.313420 0.0318230 0.0159115 − 0.999873i \(-0.494935\pi\)
0.0159115 + 0.999873i \(0.494935\pi\)
\(98\) 0 0
\(99\) −1.95506 −0.196490
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 4600.2.a.bd.1.1 ✓ 5
4.3 odd 2 9200.2.a.cv.1.5 5
5.2 odd 4 4600.2.e.w.4049.10 10
5.3 odd 4 4600.2.e.w.4049.1 10
5.4 even 2 4600.2.a.bf.1.5 yes 5
20.19 odd 2 9200.2.a.ct.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.1 ✓ 5 1.1 even 1 trivial
4600.2.a.bf.1.5 yes 5 5.4 even 2
4600.2.e.w.4049.1 10 5.3 odd 4
4600.2.e.w.4049.10 10 5.2 odd 4
9200.2.a.ct.1.1 5 20.19 odd 2
9200.2.a.cv.1.5 5 4.3 odd 2