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Properties

Label 1600.2.q.e.49.5
Level $1600$
Weight $2$
Character 1600.49
Analytic conductor $12.776$
Analytic rank $0$
Dimension $12$
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(49,1600)] 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("1600.49"); 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, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.4767670494822400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 49.5
Root \(1.22306 - 0.710021i\) of defining polynomial
Character \(\chi\) \(=\) 1600.49
Dual form 1600.2.q.e.849.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.09156 + 1.09156i) q^{3} +0.973926 q^{7} -0.616985i q^{9} +(-1.40810 - 1.40810i) q^{11} +(-4.60317 - 4.60317i) q^{13} -0.490104i q^{17} +(4.54863 - 4.54863i) q^{19} +(1.06310 + 1.06310i) q^{21} -1.94308 q^{23} +(3.94816 - 3.94816i) q^{27} +(3.74613 - 3.74613i) q^{29} -4.29021 q^{31} -3.07405i q^{33} +(4.55320 - 4.55320i) q^{37} -10.0493i q^{39} -10.1542i q^{41} +(-1.79055 + 1.79055i) q^{43} +10.0162i q^{47} -6.05147 q^{49} +(0.534979 - 0.534979i) q^{51} +(-5.61412 + 5.61412i) q^{53} +9.93022 q^{57} +(8.44185 + 8.44185i) q^{59} +(3.01095 - 3.01095i) q^{61} -0.600897i q^{63} +(7.07504 + 7.07504i) q^{67} +(-2.12099 - 2.12099i) q^{69} -0.897891i q^{71} +9.71555 q^{73} +(-1.37138 - 1.37138i) q^{77} -14.7857 q^{79} +6.76838 q^{81} +(-0.815000 - 0.815000i) q^{83} +8.17827 q^{87} -1.12404i q^{89} +(-4.48314 - 4.48314i) q^{91} +(-4.68303 - 4.68303i) q^{93} +7.54442i q^{97} +(-0.868775 + 0.868775i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 12 q^{7} + 2 q^{11} - 4 q^{13} - 14 q^{19} - 20 q^{21} + 12 q^{23} + 10 q^{27} + 4 q^{31} + 8 q^{37} - 4 q^{49} - 10 q^{51} + 16 q^{53} + 16 q^{57} + 20 q^{59} + 4 q^{61} + 50 q^{67} - 40 q^{73}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.09156 + 1.09156i 0.630214 + 0.630214i 0.948122 − 0.317908i \(-0.102980\pi\)
−0.317908 + 0.948122i \(0.602980\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.973926 0.368109 0.184055 − 0.982916i \(-0.441078\pi\)
0.184055 + 0.982916i \(0.441078\pi\)
\(8\) 0 0
\(9\) − 0.616985i − 0.205662i
\(10\) 0 0
\(11\) −1.40810 − 1.40810i −0.424558 − 0.424558i 0.462212 − 0.886769i \(-0.347056\pi\)
−0.886769 + 0.462212i \(0.847056\pi\)
\(12\) 0 0
\(13\) −4.60317 − 4.60317i −1.27669 − 1.27669i −0.942510 − 0.334179i \(-0.891541\pi\)
−0.334179 − 0.942510i \(-0.608459\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.490104i − 0.118868i −0.998232 − 0.0594338i \(-0.981070\pi\)
0.998232 − 0.0594338i \(-0.0189295\pi\)
\(18\) 0 0
\(19\) 4.54863 − 4.54863i 1.04353 − 1.04353i 0.0445187 − 0.999009i \(-0.485825\pi\)
0.999009 − 0.0445187i \(-0.0141754\pi\)
\(20\) 0 0
\(21\) 1.06310 + 1.06310i 0.231988 + 0.231988i
\(22\) 0 0
\(23\) −1.94308 −0.405160 −0.202580 − 0.979266i \(-0.564933\pi\)
−0.202580 + 0.979266i \(0.564933\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.94816 − 3.94816i 0.759824 − 0.759824i
\(28\) 0 0
\(29\) 3.74613 − 3.74613i 0.695640 − 0.695640i −0.267827 − 0.963467i \(-0.586306\pi\)
0.963467 + 0.267827i \(0.0863057\pi\)
\(30\) 0 0
\(31\) −4.29021 −0.770545 −0.385272 − 0.922803i \(-0.625893\pi\)
−0.385272 + 0.922803i \(0.625893\pi\)
\(32\) 0 0
\(33\) − 3.07405i − 0.535124i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.55320 − 4.55320i 0.748542 − 0.748542i −0.225663 − 0.974205i \(-0.572455\pi\)
0.974205 + 0.225663i \(0.0724549\pi\)
\(38\) 0 0
\(39\) − 10.0493i − 1.60917i
\(40\) 0 0
\(41\) − 10.1542i − 1.58582i −0.609341 − 0.792908i \(-0.708566\pi\)
0.609341 − 0.792908i \(-0.291434\pi\)
\(42\) 0 0
\(43\) −1.79055 + 1.79055i −0.273057 + 0.273057i −0.830329 − 0.557273i \(-0.811848\pi\)
0.557273 + 0.830329i \(0.311848\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0162i 1.46102i 0.682902 + 0.730510i \(0.260717\pi\)
−0.682902 + 0.730510i \(0.739283\pi\)
\(48\) 0 0
\(49\) −6.05147 −0.864495
\(50\) 0 0
\(51\) 0.534979 − 0.534979i 0.0749120 − 0.0749120i
\(52\) 0 0
\(53\) −5.61412 + 5.61412i −0.771158 + 0.771158i −0.978309 − 0.207151i \(-0.933581\pi\)
0.207151 + 0.978309i \(0.433581\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.93022 1.31529
\(58\) 0 0
\(59\) 8.44185 + 8.44185i 1.09904 + 1.09904i 0.994524 + 0.104512i \(0.0333281\pi\)
0.104512 + 0.994524i \(0.466672\pi\)
\(60\) 0 0
\(61\) 3.01095 − 3.01095i 0.385513 − 0.385513i −0.487571 − 0.873084i \(-0.662117\pi\)
0.873084 + 0.487571i \(0.162117\pi\)
\(62\) 0 0
\(63\) − 0.600897i − 0.0757060i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.07504 + 7.07504i 0.864354 + 0.864354i 0.991840 − 0.127486i \(-0.0406908\pi\)
−0.127486 + 0.991840i \(0.540691\pi\)
\(68\) 0 0
\(69\) −2.12099 − 2.12099i −0.255337 − 0.255337i
\(70\) 0 0
\(71\) − 0.897891i − 0.106560i −0.998580 − 0.0532800i \(-0.983032\pi\)
0.998580 − 0.0532800i \(-0.0169676\pi\)
\(72\) 0 0
\(73\) 9.71555 1.13712 0.568559 − 0.822642i \(-0.307501\pi\)
0.568559 + 0.822642i \(0.307501\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.37138 − 1.37138i −0.156284 − 0.156284i
\(78\) 0 0
\(79\) −14.7857 −1.66352 −0.831760 − 0.555135i \(-0.812666\pi\)
−0.831760 + 0.555135i \(0.812666\pi\)
\(80\) 0 0
\(81\) 6.76838 0.752042
\(82\) 0 0
\(83\) −0.815000 − 0.815000i −0.0894579 − 0.0894579i 0.660962 − 0.750420i \(-0.270149\pi\)
−0.750420 + 0.660962i \(0.770149\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.17827 0.876803
\(88\) 0 0
\(89\) − 1.12404i − 0.119148i −0.998224 − 0.0595739i \(-0.981026\pi\)
0.998224 − 0.0595739i \(-0.0189742\pi\)
\(90\) 0 0
\(91\) −4.48314 − 4.48314i −0.469961 − 0.469961i
\(92\) 0 0
\(93\) −4.68303 − 4.68303i −0.485608 − 0.485608i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.54442i 0.766019i 0.923744 + 0.383010i \(0.125112\pi\)
−0.923744 + 0.383010i \(0.874888\pi\)
\(98\) 0 0
\(99\) −0.868775 + 0.868775i −0.0873152 + 0.0873152i
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.q.e.49.5 12
4.3 odd 2 400.2.q.e.149.5 12
5.2 odd 4 1600.2.l.f.1201.5 12
5.3 odd 4 1600.2.l.g.1201.2 12
5.4 even 2 1600.2.q.f.49.2 12
16.3 odd 4 400.2.q.f.349.2 12
16.13 even 4 1600.2.q.f.849.2 12
20.3 even 4 400.2.l.f.101.2 ✓ 12
20.7 even 4 400.2.l.g.101.5 yes 12
20.19 odd 2 400.2.q.f.149.2 12
80.3 even 4 400.2.l.f.301.2 yes 12
80.13 odd 4 1600.2.l.g.401.2 12
80.19 odd 4 400.2.q.e.349.5 12
80.29 even 4 inner 1600.2.q.e.849.5 12
80.67 even 4 400.2.l.g.301.5 yes 12
80.77 odd 4 1600.2.l.f.401.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.l.f.101.2 ✓ 12 20.3 even 4
400.2.l.f.301.2 yes 12 80.3 even 4
400.2.l.g.101.5 yes 12 20.7 even 4
400.2.l.g.301.5 yes 12 80.67 even 4
400.2.q.e.149.5 12 4.3 odd 2
400.2.q.e.349.5 12 80.19 odd 4
400.2.q.f.149.2 12 20.19 odd 2
400.2.q.f.349.2 12 16.3 odd 4
1600.2.l.f.401.5 12 80.77 odd 4
1600.2.l.f.1201.5 12 5.2 odd 4
1600.2.l.g.401.2 12 80.13 odd 4
1600.2.l.g.1201.2 12 5.3 odd 4
1600.2.q.e.49.5 12 1.1 even 1 trivial
1600.2.q.e.849.5 12 80.29 even 4 inner
1600.2.q.f.49.2 12 5.4 even 2
1600.2.q.f.849.2 12 16.13 even 4