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Properties

Label 1600.2.s.a.207.1
Level $1600$
Weight $2$
Character 1600.207
Analytic conductor $12.776$
Analytic rank $1$
Dimension $2$
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] = [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 = [2] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 207.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.207
Dual form 1600.2.s.a.943.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +(-3.00000 - 3.00000i) q^{7} +1.00000 q^{9} +(1.00000 - 1.00000i) q^{11} -2.00000i q^{13} +(-1.00000 - 1.00000i) q^{17} +(3.00000 - 3.00000i) q^{19} +(6.00000 + 6.00000i) q^{21} +(-1.00000 + 1.00000i) q^{23} +4.00000 q^{27} +(-7.00000 - 7.00000i) q^{29} +2.00000i q^{31} +(-2.00000 + 2.00000i) q^{33} +6.00000i q^{37} +4.00000i q^{39} +4.00000i q^{41} -4.00000i q^{43} +(-7.00000 + 7.00000i) q^{47} +11.0000i q^{49} +(2.00000 + 2.00000i) q^{51} +8.00000 q^{53} +(-6.00000 + 6.00000i) q^{57} +(-3.00000 - 3.00000i) q^{59} +(-1.00000 + 1.00000i) q^{61} +(-3.00000 - 3.00000i) q^{63} +4.00000i q^{67} +(2.00000 - 2.00000i) q^{69} +(-3.00000 - 3.00000i) q^{73} -6.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -2.00000 q^{83} +(14.0000 + 14.0000i) q^{87} -6.00000 q^{89} +(-6.00000 + 6.00000i) q^{91} -4.00000i q^{93} +(11.0000 + 11.0000i) q^{97} +(1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 2 q^{23} + 8 q^{27} - 14 q^{29} - 4 q^{33} - 14 q^{47} + 4 q^{51} + 16 q^{53} - 12 q^{57} - 6 q^{59} - 2 q^{61} - 6 q^{63}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.00000 − 3.00000i −1.13389 − 1.13389i −0.989524 − 0.144370i \(-0.953885\pi\)
−0.144370 − 0.989524i \(-0.546115\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 − 1.00000i 0.301511 − 0.301511i −0.540094 − 0.841605i \(-0.681611\pi\)
0.841605 + 0.540094i \(0.181611\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 − 1.00000i −0.242536 − 0.242536i 0.575363 − 0.817898i \(-0.304861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) 0 0
\(19\) 3.00000 − 3.00000i 0.688247 − 0.688247i −0.273597 − 0.961844i \(-0.588214\pi\)
0.961844 + 0.273597i \(0.0882135\pi\)
\(20\) 0 0
\(21\) 6.00000 + 6.00000i 1.30931 + 1.30931i
\(22\) 0 0
\(23\) −1.00000 + 1.00000i −0.208514 + 0.208514i −0.803636 − 0.595121i \(-0.797104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −7.00000 − 7.00000i −1.29987 − 1.29987i −0.928477 − 0.371391i \(-0.878881\pi\)
−0.371391 − 0.928477i \(-0.621119\pi\)
\(30\) 0 0
\(31\) 2.00000i 0.359211i 0.983739 + 0.179605i \(0.0574821\pi\)
−0.983739 + 0.179605i \(0.942518\pi\)
\(32\) 0 0
\(33\) −2.00000 + 2.00000i −0.348155 + 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 4.00000i 0.640513i
\(40\) 0 0
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.00000 + 7.00000i −1.02105 + 1.02105i −0.0212814 + 0.999774i \(0.506775\pi\)
−0.999774 + 0.0212814i \(0.993225\pi\)
\(48\) 0 0
\(49\) 11.0000i 1.57143i
\(50\) 0 0
\(51\) 2.00000 + 2.00000i 0.280056 + 0.280056i
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 + 6.00000i −0.794719 + 0.794719i
\(58\) 0 0
\(59\) −3.00000 − 3.00000i −0.390567 − 0.390567i 0.484323 − 0.874889i \(-0.339066\pi\)
−0.874889 + 0.484323i \(0.839066\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.00000i −0.128037 + 0.128037i −0.768221 − 0.640184i \(-0.778858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −3.00000 − 3.00000i −0.377964 − 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 2.00000 − 2.00000i 0.240772 − 0.240772i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.00000 − 3.00000i −0.351123 − 0.351123i 0.509404 − 0.860527i \(-0.329866\pi\)
−0.860527 + 0.509404i \(0.829866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 14.0000 + 14.0000i 1.50096 + 1.50096i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 + 6.00000i −0.628971 + 0.628971i
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000 + 11.0000i 1.11688 + 1.11688i 0.992196 + 0.124684i \(0.0397918\pi\)
0.124684 + 0.992196i \(0.460208\pi\)
\(98\) 0 0
\(99\) 1.00000 − 1.00000i 0.100504 − 0.100504i
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.a.207.1 2
4.3 odd 2 400.2.s.a.107.1 2
5.2 odd 4 320.2.j.a.143.1 2
5.3 odd 4 1600.2.j.a.143.1 2
5.4 even 2 320.2.s.a.207.1 2
16.3 odd 4 1600.2.j.a.1007.1 2
16.13 even 4 400.2.j.a.307.1 2
20.3 even 4 400.2.j.a.43.1 2
20.7 even 4 80.2.j.a.43.1 ✓ 2
20.19 odd 2 80.2.s.a.27.1 yes 2
40.19 odd 2 640.2.s.b.287.1 2
40.27 even 4 640.2.j.a.543.1 2
40.29 even 2 640.2.s.a.287.1 2
40.37 odd 4 640.2.j.b.543.1 2
60.47 odd 4 720.2.bd.a.523.1 2
60.59 even 2 720.2.z.d.667.1 2
80.3 even 4 inner 1600.2.s.a.943.1 2
80.13 odd 4 400.2.s.a.243.1 2
80.19 odd 4 320.2.j.a.47.1 2
80.27 even 4 640.2.s.a.223.1 2
80.29 even 4 80.2.j.a.67.1 yes 2
80.37 odd 4 640.2.s.b.223.1 2
80.59 odd 4 640.2.j.b.607.1 2
80.67 even 4 320.2.s.a.303.1 2
80.69 even 4 640.2.j.a.607.1 2
80.77 odd 4 80.2.s.a.3.1 yes 2
240.29 odd 4 720.2.bd.a.307.1 2
240.77 even 4 720.2.z.d.163.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.a.43.1 ✓ 2 20.7 even 4
80.2.j.a.67.1 yes 2 80.29 even 4
80.2.s.a.3.1 yes 2 80.77 odd 4
80.2.s.a.27.1 yes 2 20.19 odd 2
320.2.j.a.47.1 2 80.19 odd 4
320.2.j.a.143.1 2 5.2 odd 4
320.2.s.a.207.1 2 5.4 even 2
320.2.s.a.303.1 2 80.67 even 4
400.2.j.a.43.1 2 20.3 even 4
400.2.j.a.307.1 2 16.13 even 4
400.2.s.a.107.1 2 4.3 odd 2
400.2.s.a.243.1 2 80.13 odd 4
640.2.j.a.543.1 2 40.27 even 4
640.2.j.a.607.1 2 80.69 even 4
640.2.j.b.543.1 2 40.37 odd 4
640.2.j.b.607.1 2 80.59 odd 4
640.2.s.a.223.1 2 80.27 even 4
640.2.s.a.287.1 2 40.29 even 2
640.2.s.b.223.1 2 80.37 odd 4
640.2.s.b.287.1 2 40.19 odd 2
720.2.z.d.163.1 2 240.77 even 4
720.2.z.d.667.1 2 60.59 even 2
720.2.bd.a.307.1 2 240.29 odd 4
720.2.bd.a.523.1 2 60.47 odd 4
1600.2.j.a.143.1 2 5.3 odd 4
1600.2.j.a.1007.1 2 16.3 odd 4
1600.2.s.a.207.1 2 1.1 even 1 trivial
1600.2.s.a.943.1 2 80.3 even 4 inner