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Properties

Label 1600.2.l.i.1201.5
Level $1600$
Weight $2$
Character 1600.1201
Analytic conductor $12.776$
Analytic rank $0$
Dimension $16$
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(401,1600)] 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("1600.401"); 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([0, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1201.5
Root \(-0.966675 - 1.03225i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1201
Dual form 1600.2.l.i.401.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.209571 - 0.209571i) q^{3} -1.73696i q^{7} +2.91216i q^{9} +(-0.505430 - 0.505430i) q^{11} +(1.88750 - 1.88750i) q^{13} -4.53524 q^{17} +(3.22022 - 3.22022i) q^{19} +(-0.364018 - 0.364018i) q^{21} -8.85045i q^{23} +(1.23902 + 1.23902i) q^{27} +(-2.44059 + 2.44059i) q^{29} +5.70401 q^{31} -0.211847 q^{33} +(5.35670 + 5.35670i) q^{37} -0.791130i q^{39} -10.0343i q^{41} +(-2.10564 - 2.10564i) q^{43} +4.32303 q^{47} +3.98295 q^{49} +(-0.950456 + 0.950456i) q^{51} +(1.37458 + 1.37458i) q^{53} -1.34973i q^{57} +(-6.64140 - 6.64140i) q^{59} +(5.26208 - 5.26208i) q^{61} +5.05832 q^{63} +(-10.5578 + 10.5578i) q^{67} +(-1.85480 - 1.85480i) q^{69} -14.0437i q^{71} -6.63830i q^{73} +(-0.877914 + 0.877914i) q^{77} -4.27297 q^{79} -8.21715 q^{81} +(9.15483 - 9.15483i) q^{83} +1.02295i q^{87} -3.23826i q^{89} +(-3.27852 - 3.27852i) q^{91} +(1.19540 - 1.19540i) q^{93} -1.94129 q^{97} +(1.47189 - 1.47189i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{11} + 8 q^{19} + 24 q^{27} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 40 q^{47} - 16 q^{49} + 32 q^{51} - 16 q^{53} + 8 q^{59} + 16 q^{61} + 40 q^{63} + 40 q^{67} + 16 q^{69} - 16 q^{77} - 16 q^{79}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.209571 − 0.209571i 0.120996 − 0.120996i −0.644016 − 0.765012i \(-0.722733\pi\)
0.765012 + 0.644016i \(0.222733\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.73696i − 0.656511i −0.944589 − 0.328255i \(-0.893539\pi\)
0.944589 − 0.328255i \(-0.106461\pi\)
\(8\) 0 0
\(9\) 2.91216i 0.970720i
\(10\) 0 0
\(11\) −0.505430 − 0.505430i −0.152393 − 0.152393i 0.626793 − 0.779186i \(-0.284367\pi\)
−0.779186 + 0.626793i \(0.784367\pi\)
\(12\) 0 0
\(13\) 1.88750 − 1.88750i 0.523498 − 0.523498i −0.395128 − 0.918626i \(-0.629300\pi\)
0.918626 + 0.395128i \(0.129300\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.53524 −1.09996 −0.549979 − 0.835178i \(-0.685364\pi\)
−0.549979 + 0.835178i \(0.685364\pi\)
\(18\) 0 0
\(19\) 3.22022 − 3.22022i 0.738768 − 0.738768i −0.233571 − 0.972340i \(-0.575041\pi\)
0.972340 + 0.233571i \(0.0750413\pi\)
\(20\) 0 0
\(21\) −0.364018 − 0.364018i −0.0794352 − 0.0794352i
\(22\) 0 0
\(23\) − 8.85045i − 1.84545i −0.385463 − 0.922723i \(-0.625958\pi\)
0.385463 − 0.922723i \(-0.374042\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.23902 + 1.23902i 0.238449 + 0.238449i
\(28\) 0 0
\(29\) −2.44059 + 2.44059i −0.453205 + 0.453205i −0.896417 − 0.443212i \(-0.853839\pi\)
0.443212 + 0.896417i \(0.353839\pi\)
\(30\) 0 0
\(31\) 5.70401 1.02447 0.512235 − 0.858845i \(-0.328818\pi\)
0.512235 + 0.858845i \(0.328818\pi\)
\(32\) 0 0
\(33\) −0.211847 −0.0368779
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.35670 + 5.35670i 0.880636 + 0.880636i 0.993599 − 0.112963i \(-0.0360342\pi\)
−0.112963 + 0.993599i \(0.536034\pi\)
\(38\) 0 0
\(39\) − 0.791130i − 0.126682i
\(40\) 0 0
\(41\) − 10.0343i − 1.56709i −0.621335 − 0.783545i \(-0.713409\pi\)
0.621335 − 0.783545i \(-0.286591\pi\)
\(42\) 0 0
\(43\) −2.10564 − 2.10564i −0.321107 − 0.321107i 0.528085 − 0.849192i \(-0.322910\pi\)
−0.849192 + 0.528085i \(0.822910\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.32303 0.630578 0.315289 − 0.948996i \(-0.397899\pi\)
0.315289 + 0.948996i \(0.397899\pi\)
\(48\) 0 0
\(49\) 3.98295 0.568993
\(50\) 0 0
\(51\) −0.950456 + 0.950456i −0.133091 + 0.133091i
\(52\) 0 0
\(53\) 1.37458 + 1.37458i 0.188814 + 0.188814i 0.795183 − 0.606369i \(-0.207375\pi\)
−0.606369 + 0.795183i \(0.707375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.34973i − 0.178776i
\(58\) 0 0
\(59\) −6.64140 − 6.64140i −0.864637 − 0.864637i 0.127236 − 0.991872i \(-0.459389\pi\)
−0.991872 + 0.127236i \(0.959389\pi\)
\(60\) 0 0
\(61\) 5.26208 − 5.26208i 0.673741 − 0.673741i −0.284836 − 0.958576i \(-0.591939\pi\)
0.958576 + 0.284836i \(0.0919391\pi\)
\(62\) 0 0
\(63\) 5.05832 0.637288
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.5578 + 10.5578i −1.28984 + 1.28984i −0.354954 + 0.934884i \(0.615503\pi\)
−0.934884 + 0.354954i \(0.884497\pi\)
\(68\) 0 0
\(69\) −1.85480 − 1.85480i −0.223292 − 0.223292i
\(70\) 0 0
\(71\) − 14.0437i − 1.66668i −0.552764 − 0.833338i \(-0.686427\pi\)
0.552764 − 0.833338i \(-0.313573\pi\)
\(72\) 0 0
\(73\) − 6.63830i − 0.776954i −0.921458 − 0.388477i \(-0.873001\pi\)
0.921458 − 0.388477i \(-0.126999\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.877914 + 0.877914i −0.100048 + 0.100048i
\(78\) 0 0
\(79\) −4.27297 −0.480746 −0.240373 − 0.970681i \(-0.577270\pi\)
−0.240373 + 0.970681i \(0.577270\pi\)
\(80\) 0 0
\(81\) −8.21715 −0.913017
\(82\) 0 0
\(83\) 9.15483 − 9.15483i 1.00487 − 1.00487i 0.00488547 − 0.999988i \(-0.498445\pi\)
0.999988 − 0.00488547i \(-0.00155510\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.02295i 0.109672i
\(88\) 0 0
\(89\) − 3.23826i − 0.343255i −0.985162 − 0.171627i \(-0.945097\pi\)
0.985162 − 0.171627i \(-0.0549025\pi\)
\(90\) 0 0
\(91\) −3.27852 − 3.27852i −0.343682 − 0.343682i
\(92\) 0 0
\(93\) 1.19540 − 1.19540i 0.123957 − 0.123957i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.94129 −0.197108 −0.0985541 − 0.995132i \(-0.531422\pi\)
−0.0985541 + 0.995132i \(0.531422\pi\)
\(98\) 0 0
\(99\) 1.47189 − 1.47189i 0.147931 − 0.147931i
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.l.i.1201.5 16
4.3 odd 2 400.2.l.h.101.3 16
5.2 odd 4 1600.2.q.g.49.4 16
5.3 odd 4 1600.2.q.h.49.5 16
5.4 even 2 320.2.l.a.241.4 16
15.14 odd 2 2880.2.t.c.2161.4 16
16.3 odd 4 400.2.l.h.301.3 16
16.13 even 4 inner 1600.2.l.i.401.5 16
20.3 even 4 400.2.q.g.149.7 16
20.7 even 4 400.2.q.h.149.2 16
20.19 odd 2 80.2.l.a.21.6 ✓ 16
40.19 odd 2 640.2.l.b.481.4 16
40.29 even 2 640.2.l.a.481.5 16
60.59 even 2 720.2.t.c.181.3 16
80.3 even 4 400.2.q.h.349.2 16
80.13 odd 4 1600.2.q.g.849.4 16
80.19 odd 4 80.2.l.a.61.6 yes 16
80.29 even 4 320.2.l.a.81.4 16
80.59 odd 4 640.2.l.b.161.4 16
80.67 even 4 400.2.q.g.349.7 16
80.69 even 4 640.2.l.a.161.5 16
80.77 odd 4 1600.2.q.h.849.5 16
160.19 odd 8 5120.2.a.v.1.3 8
160.29 even 8 5120.2.a.u.1.3 8
160.99 odd 8 5120.2.a.s.1.6 8
160.109 even 8 5120.2.a.t.1.6 8
240.29 odd 4 2880.2.t.c.721.1 16
240.179 even 4 720.2.t.c.541.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.6 ✓ 16 20.19 odd 2
80.2.l.a.61.6 yes 16 80.19 odd 4
320.2.l.a.81.4 16 80.29 even 4
320.2.l.a.241.4 16 5.4 even 2
400.2.l.h.101.3 16 4.3 odd 2
400.2.l.h.301.3 16 16.3 odd 4
400.2.q.g.149.7 16 20.3 even 4
400.2.q.g.349.7 16 80.67 even 4
400.2.q.h.149.2 16 20.7 even 4
400.2.q.h.349.2 16 80.3 even 4
640.2.l.a.161.5 16 80.69 even 4
640.2.l.a.481.5 16 40.29 even 2
640.2.l.b.161.4 16 80.59 odd 4
640.2.l.b.481.4 16 40.19 odd 2
720.2.t.c.181.3 16 60.59 even 2
720.2.t.c.541.3 16 240.179 even 4
1600.2.l.i.401.5 16 16.13 even 4 inner
1600.2.l.i.1201.5 16 1.1 even 1 trivial
1600.2.q.g.49.4 16 5.2 odd 4
1600.2.q.g.849.4 16 80.13 odd 4
1600.2.q.h.49.5 16 5.3 odd 4
1600.2.q.h.849.5 16 80.77 odd 4
2880.2.t.c.721.1 16 240.29 odd 4
2880.2.t.c.2161.4 16 15.14 odd 2
5120.2.a.s.1.6 8 160.99 odd 8
5120.2.a.t.1.6 8 160.109 even 8
5120.2.a.u.1.3 8 160.29 even 8
5120.2.a.v.1.3 8 160.19 odd 8