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Properties

Label 4600.2.e.v.4049.9
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
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] = [4600,2,Mod(4049,4600)] 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("4600.4049"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-8,0,-8,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.642242103510016.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.9
Root \(-1.45894i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.v.4049.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.21042i q^{3} +2.22487i q^{7} -1.88594 q^{9} -1.57300 q^{11} +3.96189i q^{13} -0.294907i q^{17} +7.76572 q^{19} -4.91788 q^{21} -1.00000i q^{23} +2.46253i q^{27} +9.29233 q^{29} +9.18913 q^{31} -3.47698i q^{33} +10.5425i q^{37} -8.75744 q^{39} -2.34251 q^{41} +6.67460i q^{43} +1.38007i q^{47} +2.04998 q^{49} +0.651868 q^{51} -11.0395i q^{53} +17.1655i q^{57} +5.09378 q^{59} -8.91788 q^{61} -4.19597i q^{63} +1.12002i q^{67} +2.21042 q^{69} +7.60168 q^{71} +12.8549i q^{73} -3.49971i q^{77} -11.0211 q^{79} -11.1010 q^{81} -13.5257i q^{83} +20.5399i q^{87} -14.3475 q^{89} -8.81468 q^{91} +20.3118i q^{93} +0.199218i q^{97} +2.96658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91}+ \cdots - 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.21042i 1.27618i 0.769960 + 0.638092i \(0.220276\pi\)
−0.769960 + 0.638092i \(0.779724\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.22487i 0.840920i 0.907311 + 0.420460i \(0.138131\pi\)
−0.907311 + 0.420460i \(0.861869\pi\)
\(8\) 0 0
\(9\) −1.88594 −0.628648
\(10\) 0 0
\(11\) −1.57300 −0.474276 −0.237138 − 0.971476i \(-0.576209\pi\)
−0.237138 + 0.971476i \(0.576209\pi\)
\(12\) 0 0
\(13\) 3.96189i 1.09883i 0.835549 + 0.549416i \(0.185150\pi\)
−0.835549 + 0.549416i \(0.814850\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.294907i − 0.0715256i −0.999360 − 0.0357628i \(-0.988614\pi\)
0.999360 − 0.0357628i \(-0.0113861\pi\)
\(18\) 0 0
\(19\) 7.76572 1.78158 0.890789 − 0.454418i \(-0.150153\pi\)
0.890789 + 0.454418i \(0.150153\pi\)
\(20\) 0 0
\(21\) −4.91788 −1.07317
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.46253i 0.473914i
\(28\) 0 0
\(29\) 9.29233 1.72554 0.862771 − 0.505595i \(-0.168727\pi\)
0.862771 + 0.505595i \(0.168727\pi\)
\(30\) 0 0
\(31\) 9.18913 1.65042 0.825208 − 0.564829i \(-0.191058\pi\)
0.825208 + 0.564829i \(0.191058\pi\)
\(32\) 0 0
\(33\) − 3.47698i − 0.605264i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.5425i 1.73318i 0.499024 + 0.866588i \(0.333692\pi\)
−0.499024 + 0.866588i \(0.666308\pi\)
\(38\) 0 0
\(39\) −8.75744 −1.40231
\(40\) 0 0
\(41\) −2.34251 −0.365839 −0.182920 − 0.983128i \(-0.558555\pi\)
−0.182920 + 0.983128i \(0.558555\pi\)
\(42\) 0 0
\(43\) 6.67460i 1.01787i 0.860806 + 0.508933i \(0.169960\pi\)
−0.860806 + 0.508933i \(0.830040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.38007i 0.201304i 0.994922 + 0.100652i \(0.0320928\pi\)
−0.994922 + 0.100652i \(0.967907\pi\)
\(48\) 0 0
\(49\) 2.04998 0.292854
\(50\) 0 0
\(51\) 0.651868 0.0912798
\(52\) 0 0
\(53\) − 11.0395i − 1.51640i −0.652023 − 0.758199i \(-0.726080\pi\)
0.652023 − 0.758199i \(-0.273920\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 17.1655i 2.27362i
\(58\) 0 0
\(59\) 5.09378 0.663154 0.331577 − 0.943428i \(-0.392419\pi\)
0.331577 + 0.943428i \(0.392419\pi\)
\(60\) 0 0
\(61\) −8.91788 −1.14182 −0.570909 − 0.821014i \(-0.693409\pi\)
−0.570909 + 0.821014i \(0.693409\pi\)
\(62\) 0 0
\(63\) − 4.19597i − 0.528642i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.12002i 0.136832i 0.997657 + 0.0684160i \(0.0217945\pi\)
−0.997657 + 0.0684160i \(0.978205\pi\)
\(68\) 0 0
\(69\) 2.21042 0.266103
\(70\) 0 0
\(71\) 7.60168 0.902154 0.451077 − 0.892485i \(-0.351040\pi\)
0.451077 + 0.892485i \(0.351040\pi\)
\(72\) 0 0
\(73\) 12.8549i 1.50455i 0.658848 + 0.752276i \(0.271044\pi\)
−0.658848 + 0.752276i \(0.728956\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.49971i − 0.398828i
\(78\) 0 0
\(79\) −11.0211 −1.23997 −0.619984 − 0.784614i \(-0.712861\pi\)
−0.619984 + 0.784614i \(0.712861\pi\)
\(80\) 0 0
\(81\) −11.1010 −1.23345
\(82\) 0 0
\(83\) − 13.5257i − 1.48464i −0.670048 − 0.742318i \(-0.733726\pi\)
0.670048 − 0.742318i \(-0.266274\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 20.5399i 2.20211i
\(88\) 0 0
\(89\) −14.3475 −1.52084 −0.760418 − 0.649434i \(-0.775006\pi\)
−0.760418 + 0.649434i \(0.775006\pi\)
\(90\) 0 0
\(91\) −8.81468 −0.924030
\(92\) 0 0
\(93\) 20.3118i 2.10624i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.199218i 0.0202275i 0.999949 + 0.0101137i \(0.00321936\pi\)
−0.999949 + 0.0101137i \(0.996781\pi\)
\(98\) 0 0
\(99\) 2.96658 0.298153
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.e.v.4049.9 10
5.2 odd 4 4600.2.a.bc.1.5 ✓ 5
5.3 odd 4 4600.2.a.bg.1.1 yes 5
5.4 even 2 inner 4600.2.e.v.4049.2 10
20.3 even 4 9200.2.a.cs.1.5 5
20.7 even 4 9200.2.a.cw.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.5 ✓ 5 5.2 odd 4
4600.2.a.bg.1.1 yes 5 5.3 odd 4
4600.2.e.v.4049.2 10 5.4 even 2 inner
4600.2.e.v.4049.9 10 1.1 even 1 trivial
9200.2.a.cs.1.5 5 20.3 even 4
9200.2.a.cw.1.1 5 20.7 even 4