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Properties

Label 4600.2.e.w.4049.10
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 4049.10
Root \(2.61696i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.61696i q^{3} -3.83744i q^{7} -3.84849 q^{9} -0.508005 q^{11} +1.01106i q^{13} -1.44705i q^{17} +0.508005 q^{19} +10.0424 q^{21} -1.00000i q^{23} -2.22047i q^{27} -7.51040 q^{29} -0.439038 q^{31} -1.32943i q^{33} +7.02642i q^{37} -2.64590 q^{39} +5.47041 q^{41} +6.72592i q^{43} +2.64098i q^{47} -7.72592 q^{49} +3.78688 q^{51} +4.77648i q^{53} +1.32943i q^{57} -3.85345 q^{59} -9.05844 q^{61} +14.7683i q^{63} -3.45696i q^{67} +2.61696 q^{69} -2.73649 q^{71} -9.21300i q^{73} +1.94944i q^{77} -10.5504 q^{79} -5.73458 q^{81} -1.40211i q^{83} -19.6544i q^{87} -6.77086 q^{89} +3.87986 q^{91} -1.14895i q^{93} +0.313420i q^{97} +1.95506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 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
\(3\) 2.61696i 1.51090i 0.655204 + 0.755452i \(0.272583\pi\)
−0.655204 + 0.755452i \(0.727417\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.83744i − 1.45041i −0.688531 − 0.725207i \(-0.741744\pi\)
0.688531 − 0.725207i \(-0.258256\pi\)
\(8\) 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
\(13\) 1.01106i 0.280417i 0.990122 + 0.140208i \(0.0447772\pi\)
−0.990122 + 0.140208i \(0.955223\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.44705i − 0.350961i −0.984483 − 0.175481i \(-0.943852\pi\)
0.984483 − 0.175481i \(-0.0561479\pi\)
\(18\) 0 0
\(19\) 0.508005 0.116544 0.0582722 − 0.998301i \(-0.481441\pi\)
0.0582722 + 0.998301i \(0.481441\pi\)
\(20\) 0 0
\(21\) 10.0424 2.19144
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 2.22047i − 0.427330i
\(28\) 0 0
\(29\) −7.51040 −1.39465 −0.697323 − 0.716757i \(-0.745626\pi\)
−0.697323 + 0.716757i \(0.745626\pi\)
\(30\) 0 0
\(31\) −0.439038 −0.0788536 −0.0394268 − 0.999222i \(-0.512553\pi\)
−0.0394268 + 0.999222i \(0.512553\pi\)
\(32\) 0 0
\(33\) − 1.32943i − 0.231424i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.02642i 1.15514i 0.816343 + 0.577568i \(0.195998\pi\)
−0.816343 + 0.577568i \(0.804002\pi\)
\(38\) 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
\(43\) 6.72592i 1.02569i 0.858480 + 0.512847i \(0.171409\pi\)
−0.858480 + 0.512847i \(0.828591\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.64098i 0.385227i 0.981275 + 0.192614i \(0.0616964\pi\)
−0.981275 + 0.192614i \(0.938304\pi\)
\(48\) 0 0
\(49\) −7.72592 −1.10370
\(50\) 0 0
\(51\) 3.78688 0.530269
\(52\) 0 0
\(53\) 4.77648i 0.656100i 0.944660 + 0.328050i \(0.106391\pi\)
−0.944660 + 0.328050i \(0.893609\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.32943i 0.176087i
\(58\) 0 0
\(59\) −3.85345 −0.501676 −0.250838 − 0.968029i \(-0.580706\pi\)
−0.250838 + 0.968029i \(0.580706\pi\)
\(60\) 0 0
\(61\) −9.05844 −1.15981 −0.579907 − 0.814683i \(-0.696911\pi\)
−0.579907 + 0.814683i \(0.696911\pi\)
\(62\) 0 0
\(63\) 14.7683i 1.86064i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.45696i − 0.422335i −0.977450 − 0.211167i \(-0.932273\pi\)
0.977450 − 0.211167i \(-0.0677265\pi\)
\(68\) 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
\(73\) − 9.21300i − 1.07830i −0.842210 − 0.539150i \(-0.818746\pi\)
0.842210 − 0.539150i \(-0.181254\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.94944i 0.222159i
\(78\) 0 0
\(79\) −10.5504 −1.18702 −0.593508 − 0.804828i \(-0.702258\pi\)
−0.593508 + 0.804828i \(0.702258\pi\)
\(80\) 0 0
\(81\) −5.73458 −0.637176
\(82\) 0 0
\(83\) − 1.40211i − 0.153901i −0.997035 − 0.0769505i \(-0.975482\pi\)
0.997035 − 0.0769505i \(-0.0245183\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 19.6544i − 2.10718i
\(88\) 0 0
\(89\) −6.77086 −0.717710 −0.358855 − 0.933393i \(-0.616833\pi\)
−0.358855 + 0.933393i \(0.616833\pi\)
\(90\) 0 0
\(91\) 3.87986 0.406720
\(92\) 0 0
\(93\) − 1.14895i − 0.119140i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.313420i 0.0318230i 0.999873 + 0.0159115i \(0.00506500\pi\)
−0.999873 + 0.0159115i \(0.994935\pi\)
\(98\) 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.e.w.4049.10 10
5.2 odd 4 4600.2.a.bf.1.5 yes 5
5.3 odd 4 4600.2.a.bd.1.1 ✓ 5
5.4 even 2 inner 4600.2.e.w.4049.1 10
20.3 even 4 9200.2.a.cv.1.5 5
20.7 even 4 9200.2.a.ct.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.1 ✓ 5 5.3 odd 4
4600.2.a.bf.1.5 yes 5 5.2 odd 4
4600.2.e.w.4049.1 10 5.4 even 2 inner
4600.2.e.w.4049.10 10 1.1 even 1 trivial
9200.2.a.ct.1.1 5 20.7 even 4
9200.2.a.cv.1.5 5 20.3 even 4