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Properties

Label 1600.2.s.b.207.3
Level $1600$
Weight $2$
Character 1600.207
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [1600,2,Mod(207,1600)] 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("1600.207"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.s (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 207.3
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1600.207
Dual form 1600.2.s.b.943.3

$q$-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)
 
\(f(q)\) \(=\) \(q+0.517638 q^{3} +(-3.34607 - 3.34607i) q^{7} -2.73205 q^{9} +(1.09808 - 1.09808i) q^{11} +4.89898i q^{13} +(0.707107 + 0.707107i) q^{17} +(2.09808 - 2.09808i) q^{19} +(-1.73205 - 1.73205i) q^{21} +(-4.38134 + 4.38134i) q^{23} -2.96713 q^{27} +(4.73205 + 4.73205i) q^{29} +6.19615i q^{31} +(0.568406 - 0.568406i) q^{33} +6.03579i q^{37} +2.53590i q^{39} -0.464102i q^{41} +0.656339i q^{43} +(1.41421 - 1.41421i) q^{47} +15.3923i q^{49} +(0.366025 + 0.366025i) q^{51} -9.89949 q^{53} +(1.08604 - 1.08604i) q^{57} +(7.73205 + 7.73205i) q^{59} +(-3.19615 + 3.19615i) q^{61} +(9.14162 + 9.14162i) q^{63} -5.79555i q^{67} +(-2.26795 + 2.26795i) q^{69} -0.928203 q^{71} +(8.81345 + 8.81345i) q^{73} -7.34847 q^{77} -2.19615 q^{79} +6.66025 q^{81} -17.3867 q^{83} +(2.44949 + 2.44949i) q^{87} +10.2679 q^{89} +(16.3923 - 16.3923i) q^{91} +3.20736i q^{93} +(-11.5911 - 11.5911i) q^{97} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9} - 12 q^{11} - 4 q^{19} + 24 q^{29} - 4 q^{51} + 48 q^{59} + 16 q^{61} - 32 q^{69} + 48 q^{71} + 24 q^{79} - 16 q^{81} + 96 q^{89} + 48 q^{91} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) 0.517638 0.298858 0.149429 − 0.988772i \(-0.452256\pi\)
0.149429 + 0.988772i \(0.452256\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.34607 − 3.34607i −1.26469 − 1.26469i −0.948792 − 0.315902i \(-0.897693\pi\)
−0.315902 − 0.948792i \(-0.602307\pi\)
\(8\) 0 0
\(9\) −2.73205 −0.910684
\(10\) 0 0
\(11\) 1.09808 − 1.09808i 0.331082 − 0.331082i −0.521915 − 0.852997i \(-0.674782\pi\)
0.852997 + 0.521915i \(0.174782\pi\)
\(12\) 0 0
\(13\) 4.89898i 1.35873i 0.733799 + 0.679366i \(0.237745\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.707107 + 0.707107i 0.171499 + 0.171499i 0.787638 − 0.616139i \(-0.211304\pi\)
−0.616139 + 0.787638i \(0.711304\pi\)
\(18\) 0 0
\(19\) 2.09808 − 2.09808i 0.481332 − 0.481332i −0.424225 − 0.905557i \(-0.639453\pi\)
0.905557 + 0.424225i \(0.139453\pi\)
\(20\) 0 0
\(21\) −1.73205 − 1.73205i −0.377964 − 0.377964i
\(22\) 0 0
\(23\) −4.38134 + 4.38134i −0.913573 + 0.913573i −0.996551 − 0.0829785i \(-0.973557\pi\)
0.0829785 + 0.996551i \(0.473557\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.96713 −0.571024
\(28\) 0 0
\(29\) 4.73205 + 4.73205i 0.878720 + 0.878720i 0.993402 − 0.114682i \(-0.0365850\pi\)
−0.114682 + 0.993402i \(0.536585\pi\)
\(30\) 0 0
\(31\) 6.19615i 1.11286i 0.830894 + 0.556431i \(0.187830\pi\)
−0.830894 + 0.556431i \(0.812170\pi\)
\(32\) 0 0
\(33\) 0.568406 − 0.568406i 0.0989468 − 0.0989468i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.03579i 0.992278i 0.868243 + 0.496139i \(0.165249\pi\)
−0.868243 + 0.496139i \(0.834751\pi\)
\(38\) 0 0
\(39\) 2.53590i 0.406069i
\(40\) 0 0
\(41\) − 0.464102i − 0.0724805i −0.999343 − 0.0362402i \(-0.988462\pi\)
0.999343 − 0.0362402i \(-0.0115382\pi\)
\(42\) 0 0
\(43\) 0.656339i 0.100091i 0.998747 + 0.0500454i \(0.0159366\pi\)
−0.998747 + 0.0500454i \(0.984063\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41421 − 1.41421i 0.206284 − 0.206284i −0.596402 − 0.802686i \(-0.703403\pi\)
0.802686 + 0.596402i \(0.203403\pi\)
\(48\) 0 0
\(49\) 15.3923i 2.19890i
\(50\) 0 0
\(51\) 0.366025 + 0.366025i 0.0512538 + 0.0512538i
\(52\) 0 0
\(53\) −9.89949 −1.35980 −0.679900 − 0.733305i \(-0.737977\pi\)
−0.679900 + 0.733305i \(0.737977\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.08604 − 1.08604i 0.143850 − 0.143850i
\(58\) 0 0
\(59\) 7.73205 + 7.73205i 1.00663 + 1.00663i 0.999978 + 0.00664938i \(0.00211658\pi\)
0.00664938 + 0.999978i \(0.497883\pi\)
\(60\) 0 0
\(61\) −3.19615 + 3.19615i −0.409225 + 0.409225i −0.881468 − 0.472243i \(-0.843444\pi\)
0.472243 + 0.881468i \(0.343444\pi\)
\(62\) 0 0
\(63\) 9.14162 + 9.14162i 1.15174 + 1.15174i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.79555i − 0.708040i −0.935238 − 0.354020i \(-0.884815\pi\)
0.935238 − 0.354020i \(-0.115185\pi\)
\(68\) 0 0
\(69\) −2.26795 + 2.26795i −0.273029 + 0.273029i
\(70\) 0 0
\(71\) −0.928203 −0.110157 −0.0550787 − 0.998482i \(-0.517541\pi\)
−0.0550787 + 0.998482i \(0.517541\pi\)
\(72\) 0 0
\(73\) 8.81345 + 8.81345i 1.03154 + 1.03154i 0.999486 + 0.0320501i \(0.0102036\pi\)
0.0320501 + 0.999486i \(0.489796\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.34847 −0.837436
\(78\) 0 0
\(79\) −2.19615 −0.247086 −0.123543 − 0.992339i \(-0.539426\pi\)
−0.123543 + 0.992339i \(0.539426\pi\)
\(80\) 0 0
\(81\) 6.66025 0.740028
\(82\) 0 0
\(83\) −17.3867 −1.90843 −0.954217 − 0.299115i \(-0.903309\pi\)
−0.954217 + 0.299115i \(0.903309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.44949 + 2.44949i 0.262613 + 0.262613i
\(88\) 0 0
\(89\) 10.2679 1.08840 0.544200 − 0.838955i \(-0.316833\pi\)
0.544200 + 0.838955i \(0.316833\pi\)
\(90\) 0 0
\(91\) 16.3923 − 16.3923i 1.71838 − 1.71838i
\(92\) 0 0
\(93\) 3.20736i 0.332588i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.5911 − 11.5911i −1.17690 − 1.17690i −0.980530 − 0.196369i \(-0.937085\pi\)
−0.196369 − 0.980530i \(-0.562915\pi\)
\(98\) 0 0
\(99\) −3.00000 + 3.00000i −0.301511 + 0.301511i
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 1600.2.s.b.207.3 8
4.3 odd 2 400.2.s.b.107.1 yes 8
5.2 odd 4 1600.2.j.b.143.2 8
5.3 odd 4 1600.2.j.b.143.3 8
5.4 even 2 inner 1600.2.s.b.207.2 8
16.3 odd 4 1600.2.j.b.1007.2 8
16.13 even 4 400.2.j.b.307.2 yes 8
20.3 even 4 400.2.j.b.43.1 ✓ 8
20.7 even 4 400.2.j.b.43.4 yes 8
20.19 odd 2 400.2.s.b.107.4 yes 8
80.3 even 4 inner 1600.2.s.b.943.3 8
80.13 odd 4 400.2.s.b.243.3 yes 8
80.19 odd 4 1600.2.j.b.1007.3 8
80.29 even 4 400.2.j.b.307.3 yes 8
80.67 even 4 inner 1600.2.s.b.943.2 8
80.77 odd 4 400.2.s.b.243.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.b.43.1 ✓ 8 20.3 even 4
400.2.j.b.43.4 yes 8 20.7 even 4
400.2.j.b.307.2 yes 8 16.13 even 4
400.2.j.b.307.3 yes 8 80.29 even 4
400.2.s.b.107.1 yes 8 4.3 odd 2
400.2.s.b.107.4 yes 8 20.19 odd 2
400.2.s.b.243.2 yes 8 80.77 odd 4
400.2.s.b.243.3 yes 8 80.13 odd 4
1600.2.j.b.143.2 8 5.2 odd 4
1600.2.j.b.143.3 8 5.3 odd 4
1600.2.j.b.1007.2 8 16.3 odd 4
1600.2.j.b.1007.3 8 80.19 odd 4
1600.2.s.b.207.2 8 5.4 even 2 inner
1600.2.s.b.207.3 8 1.1 even 1 trivial
1600.2.s.b.943.2 8 80.67 even 4 inner
1600.2.s.b.943.3 8 80.3 even 4 inner