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Properties

Label 4600.2.e.w.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,-6,0,0,0,0,0,0,0,0,0,0] 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: 10.0.278379347567616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.9
Root \(2.29544i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.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)
 
\(f(q)\) \(=\) \(q+2.29544i q^{3} -1.61758i q^{7} -2.26905 q^{9} -2.79255 q^{11} -4.88663i q^{13} +3.54388i q^{17} +2.79255 q^{19} +3.71306 q^{21} +1.00000i q^{23} +1.67786i q^{27} +3.36282 q^{29} -4.84564 q^{31} -6.41013i q^{33} +3.87204i q^{37} +11.2170 q^{39} -0.327923 q^{41} +5.38343i q^{43} +0.0121087i q^{47} +4.38343 q^{49} -8.13476 q^{51} +0.866254i q^{53} +6.41013i q^{57} -2.96752 q^{59} -7.29816 q^{61} +3.67037i q^{63} +6.94082i q^{67} -2.29544 q^{69} -2.16864 q^{71} +6.02888i q^{73} +4.51718i q^{77} -6.50561 q^{79} -10.6586 q^{81} +7.88030i q^{83} +7.71915i q^{87} +9.71986 q^{89} -7.90452 q^{91} -11.1229i q^{93} +11.9952i q^{97} +6.33643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 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
\(3\) 2.29544i 1.32527i 0.748941 + 0.662637i \(0.230563\pi\)
−0.748941 + 0.662637i \(0.769437\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.61758i − 0.611388i −0.952130 − 0.305694i \(-0.901111\pi\)
0.952130 − 0.305694i \(-0.0988885\pi\)
\(8\) 0 0
\(9\) −2.26905 −0.756349
\(10\) 0 0
\(11\) −2.79255 −0.841985 −0.420993 − 0.907064i \(-0.638318\pi\)
−0.420993 + 0.907064i \(0.638318\pi\)
\(12\) 0 0
\(13\) − 4.88663i − 1.35531i −0.735381 − 0.677653i \(-0.762997\pi\)
0.735381 − 0.677653i \(-0.237003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.54388i 0.859516i 0.902944 + 0.429758i \(0.141401\pi\)
−0.902944 + 0.429758i \(0.858599\pi\)
\(18\) 0 0
\(19\) 2.79255 0.640655 0.320327 − 0.947307i \(-0.396207\pi\)
0.320327 + 0.947307i \(0.396207\pi\)
\(20\) 0 0
\(21\) 3.71306 0.810257
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.67786i 0.322904i
\(28\) 0 0
\(29\) 3.36282 0.624460 0.312230 − 0.950007i \(-0.398924\pi\)
0.312230 + 0.950007i \(0.398924\pi\)
\(30\) 0 0
\(31\) −4.84564 −0.870303 −0.435152 − 0.900357i \(-0.643305\pi\)
−0.435152 + 0.900357i \(0.643305\pi\)
\(32\) 0 0
\(33\) − 6.41013i − 1.11586i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.87204i 0.636559i 0.947997 + 0.318279i \(0.103105\pi\)
−0.947997 + 0.318279i \(0.896895\pi\)
\(38\) 0 0
\(39\) 11.2170 1.79615
\(40\) 0 0
\(41\) −0.327923 −0.0512130 −0.0256065 − 0.999672i \(-0.508152\pi\)
−0.0256065 + 0.999672i \(0.508152\pi\)
\(42\) 0 0
\(43\) 5.38343i 0.820965i 0.911868 + 0.410483i \(0.134640\pi\)
−0.911868 + 0.410483i \(0.865360\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.0121087i 0.00176623i 1.00000 0.000883117i \(0.000281105\pi\)
−1.00000 0.000883117i \(0.999719\pi\)
\(48\) 0 0
\(49\) 4.38343 0.626204
\(50\) 0 0
\(51\) −8.13476 −1.13909
\(52\) 0 0
\(53\) 0.866254i 0.118989i 0.998229 + 0.0594946i \(0.0189489\pi\)
−0.998229 + 0.0594946i \(0.981051\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.41013i 0.849043i
\(58\) 0 0
\(59\) −2.96752 −0.386338 −0.193169 − 0.981166i \(-0.561877\pi\)
−0.193169 + 0.981166i \(0.561877\pi\)
\(60\) 0 0
\(61\) −7.29816 −0.934434 −0.467217 − 0.884143i \(-0.654743\pi\)
−0.467217 + 0.884143i \(0.654743\pi\)
\(62\) 0 0
\(63\) 3.67037i 0.462423i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.94082i 0.847956i 0.905673 + 0.423978i \(0.139367\pi\)
−0.905673 + 0.423978i \(0.860633\pi\)
\(68\) 0 0
\(69\) −2.29544 −0.276339
\(70\) 0 0
\(71\) −2.16864 −0.257370 −0.128685 − 0.991685i \(-0.541076\pi\)
−0.128685 + 0.991685i \(0.541076\pi\)
\(72\) 0 0
\(73\) 6.02888i 0.705627i 0.935694 + 0.352813i \(0.114775\pi\)
−0.935694 + 0.352813i \(0.885225\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.51718i 0.514780i
\(78\) 0 0
\(79\) −6.50561 −0.731938 −0.365969 − 0.930627i \(-0.619262\pi\)
−0.365969 + 0.930627i \(0.619262\pi\)
\(80\) 0 0
\(81\) −10.6586 −1.18429
\(82\) 0 0
\(83\) 7.88030i 0.864976i 0.901640 + 0.432488i \(0.142364\pi\)
−0.901640 + 0.432488i \(0.857636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.71915i 0.827580i
\(88\) 0 0
\(89\) 9.71986 1.03030 0.515151 − 0.857099i \(-0.327736\pi\)
0.515151 + 0.857099i \(0.327736\pi\)
\(90\) 0 0
\(91\) −7.90452 −0.828619
\(92\) 0 0
\(93\) − 11.1229i − 1.15339i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.9952i 1.21793i 0.793197 + 0.608966i \(0.208415\pi\)
−0.793197 + 0.608966i \(0.791585\pi\)
\(98\) 0 0
\(99\) 6.33643 0.636835
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.w.4049.9 10
5.2 odd 4 4600.2.a.bd.1.5 ✓ 5
5.3 odd 4 4600.2.a.bf.1.1 yes 5
5.4 even 2 inner 4600.2.e.w.4049.2 10
20.3 even 4 9200.2.a.ct.1.5 5
20.7 even 4 9200.2.a.cv.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.5 ✓ 5 5.2 odd 4
4600.2.a.bf.1.1 yes 5 5.3 odd 4
4600.2.e.w.4049.2 10 5.4 even 2 inner
4600.2.e.w.4049.9 10 1.1 even 1 trivial
9200.2.a.ct.1.5 5 20.3 even 4
9200.2.a.cv.1.1 5 20.7 even 4