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Properties

Label 1600.2.q.h.49.4
Level $1600$
Weight $2$
Character 1600.49
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(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 = [16,0,0,0,0,0,8] 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: \(16\)
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 49.4
Root \(-0.296075 + 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 1600.49
Dual form 1600.2.q.h.849.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.120009 - 0.120009i) q^{3} -2.66881 q^{7} -2.97120i q^{9} +(3.49714 + 3.49714i) q^{11} +(-2.94072 - 2.94072i) q^{13} +1.85116i q^{17} +(-3.44856 + 3.44856i) q^{19} +(0.320281 + 0.320281i) q^{21} -0.707288 q^{23} +(-0.716597 + 0.716597i) q^{27} +(3.49909 - 3.49909i) q^{29} -6.84272 q^{31} -0.839377i q^{33} +(0.0975060 - 0.0975060i) q^{37} +0.705826i q^{39} +10.2052i q^{41} +(-4.43844 + 4.43844i) q^{43} +1.89428i q^{47} +0.122561 q^{49} +(0.222155 - 0.222155i) q^{51} +(-7.43897 + 7.43897i) q^{53} +0.827717 q^{57} +(0.959574 + 0.959574i) q^{59} +(6.49825 - 6.49825i) q^{61} +7.92956i q^{63} +(-3.49691 - 3.49691i) q^{67} +(0.0848809 + 0.0848809i) q^{69} +7.86777i q^{71} -15.6564 q^{73} +(-9.33322 - 9.33322i) q^{77} -6.70212 q^{79} -8.74159 q^{81} +(-3.87327 - 3.87327i) q^{83} -0.839845 q^{87} +10.5055i q^{89} +(7.84824 + 7.84824i) q^{91} +(0.821187 + 0.821187i) q^{93} +4.79937i q^{97} +(10.3907 - 10.3907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7} + 8 q^{11} - 8 q^{19} + 24 q^{23} + 24 q^{27} + 16 q^{29} + 16 q^{37} - 8 q^{43} + 16 q^{49} + 32 q^{51} + 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{69} + 16 q^{77} + 16 q^{79}+ \cdots - 8 q^{99}+O(q^{100}) \) 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
\(3\) −0.120009 − 0.120009i −0.0692872 − 0.0692872i 0.671614 − 0.740901i \(-0.265601\pi\)
−0.740901 + 0.671614i \(0.765601\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.66881 −1.00872 −0.504358 − 0.863495i \(-0.668271\pi\)
−0.504358 + 0.863495i \(0.668271\pi\)
\(8\) 0 0
\(9\) − 2.97120i − 0.990399i
\(10\) 0 0
\(11\) 3.49714 + 3.49714i 1.05443 + 1.05443i 0.998431 + 0.0559977i \(0.0178339\pi\)
0.0559977 + 0.998431i \(0.482166\pi\)
\(12\) 0 0
\(13\) −2.94072 − 2.94072i −0.815610 − 0.815610i 0.169858 − 0.985468i \(-0.445669\pi\)
−0.985468 + 0.169858i \(0.945669\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.85116i 0.448971i 0.974477 + 0.224486i \(0.0720702\pi\)
−0.974477 + 0.224486i \(0.927930\pi\)
\(18\) 0 0
\(19\) −3.44856 + 3.44856i −0.791155 + 0.791155i −0.981682 − 0.190527i \(-0.938980\pi\)
0.190527 + 0.981682i \(0.438980\pi\)
\(20\) 0 0
\(21\) 0.320281 + 0.320281i 0.0698911 + 0.0698911i
\(22\) 0 0
\(23\) −0.707288 −0.147480 −0.0737399 − 0.997278i \(-0.523493\pi\)
−0.0737399 + 0.997278i \(0.523493\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.716597 + 0.716597i −0.137909 + 0.137909i
\(28\) 0 0
\(29\) 3.49909 − 3.49909i 0.649766 − 0.649766i −0.303171 − 0.952936i \(-0.598045\pi\)
0.952936 + 0.303171i \(0.0980452\pi\)
\(30\) 0 0
\(31\) −6.84272 −1.22899 −0.614494 − 0.788921i \(-0.710640\pi\)
−0.614494 + 0.788921i \(0.710640\pi\)
\(32\) 0 0
\(33\) − 0.839377i − 0.146117i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.0975060 − 0.0975060i 0.0160299 − 0.0160299i −0.699046 − 0.715076i \(-0.746392\pi\)
0.715076 + 0.699046i \(0.246392\pi\)
\(38\) 0 0
\(39\) 0.705826i 0.113023i
\(40\) 0 0
\(41\) 10.2052i 1.59379i 0.604117 + 0.796896i \(0.293526\pi\)
−0.604117 + 0.796896i \(0.706474\pi\)
\(42\) 0 0
\(43\) −4.43844 + 4.43844i −0.676855 + 0.676855i −0.959287 − 0.282432i \(-0.908859\pi\)
0.282432 + 0.959287i \(0.408859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.89428i 0.276310i 0.990411 + 0.138155i \(0.0441172\pi\)
−0.990411 + 0.138155i \(0.955883\pi\)
\(48\) 0 0
\(49\) 0.122561 0.0175087
\(50\) 0 0
\(51\) 0.222155 − 0.222155i 0.0311079 − 0.0311079i
\(52\) 0 0
\(53\) −7.43897 + 7.43897i −1.02182 + 1.02182i −0.0220650 + 0.999757i \(0.507024\pi\)
−0.999757 + 0.0220650i \(0.992976\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.827717 0.109634
\(58\) 0 0
\(59\) 0.959574 + 0.959574i 0.124926 + 0.124926i 0.766805 − 0.641880i \(-0.221845\pi\)
−0.641880 + 0.766805i \(0.721845\pi\)
\(60\) 0 0
\(61\) 6.49825 − 6.49825i 0.832015 − 0.832015i −0.155777 − 0.987792i \(-0.549788\pi\)
0.987792 + 0.155777i \(0.0497881\pi\)
\(62\) 0 0
\(63\) 7.92956i 0.999031i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.49691 − 3.49691i −0.427216 − 0.427216i 0.460463 − 0.887679i \(-0.347683\pi\)
−0.887679 + 0.460463i \(0.847683\pi\)
\(68\) 0 0
\(69\) 0.0848809 + 0.0848809i 0.0102185 + 0.0102185i
\(70\) 0 0
\(71\) 7.86777i 0.933733i 0.884328 + 0.466866i \(0.154617\pi\)
−0.884328 + 0.466866i \(0.845383\pi\)
\(72\) 0 0
\(73\) −15.6564 −1.83244 −0.916220 − 0.400675i \(-0.868776\pi\)
−0.916220 + 0.400675i \(0.868776\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.33322 − 9.33322i −1.06362 − 1.06362i
\(78\) 0 0
\(79\) −6.70212 −0.754047 −0.377024 − 0.926204i \(-0.623052\pi\)
−0.377024 + 0.926204i \(0.623052\pi\)
\(80\) 0 0
\(81\) −8.74159 −0.971288
\(82\) 0 0
\(83\) −3.87327 − 3.87327i −0.425147 − 0.425147i 0.461825 − 0.886971i \(-0.347195\pi\)
−0.886971 + 0.461825i \(0.847195\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.839845 −0.0900408
\(88\) 0 0
\(89\) 10.5055i 1.11358i 0.830653 + 0.556790i \(0.187967\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(90\) 0 0
\(91\) 7.84824 + 7.84824i 0.822719 + 0.822719i
\(92\) 0 0
\(93\) 0.821187 + 0.821187i 0.0851531 + 0.0851531i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.79937i 0.487303i 0.969863 + 0.243651i \(0.0783453\pi\)
−0.969863 + 0.243651i \(0.921655\pi\)
\(98\) 0 0
\(99\) 10.3907 − 10.3907i 1.04430 − 1.04430i
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.h.49.4 16
4.3 odd 2 400.2.q.g.149.3 16
5.2 odd 4 1600.2.l.i.1201.4 16
5.3 odd 4 320.2.l.a.241.5 16
5.4 even 2 1600.2.q.g.49.5 16
15.8 even 4 2880.2.t.c.2161.7 16
16.3 odd 4 400.2.q.h.349.6 16
16.13 even 4 1600.2.q.g.849.5 16
20.3 even 4 80.2.l.a.21.7 ✓ 16
20.7 even 4 400.2.l.h.101.2 16
20.19 odd 2 400.2.q.h.149.6 16
40.3 even 4 640.2.l.b.481.5 16
40.13 odd 4 640.2.l.a.481.4 16
60.23 odd 4 720.2.t.c.181.2 16
80.3 even 4 80.2.l.a.61.7 yes 16
80.13 odd 4 320.2.l.a.81.5 16
80.19 odd 4 400.2.q.g.349.3 16
80.29 even 4 inner 1600.2.q.h.849.4 16
80.43 even 4 640.2.l.b.161.5 16
80.53 odd 4 640.2.l.a.161.4 16
80.67 even 4 400.2.l.h.301.2 16
80.77 odd 4 1600.2.l.i.401.4 16
160.3 even 8 5120.2.a.v.1.4 8
160.13 odd 8 5120.2.a.u.1.4 8
160.83 even 8 5120.2.a.s.1.5 8
160.93 odd 8 5120.2.a.t.1.5 8
240.83 odd 4 720.2.t.c.541.2 16
240.173 even 4 2880.2.t.c.721.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 20.3 even 4
80.2.l.a.61.7 yes 16 80.3 even 4
320.2.l.a.81.5 16 80.13 odd 4
320.2.l.a.241.5 16 5.3 odd 4
400.2.l.h.101.2 16 20.7 even 4
400.2.l.h.301.2 16 80.67 even 4
400.2.q.g.149.3 16 4.3 odd 2
400.2.q.g.349.3 16 80.19 odd 4
400.2.q.h.149.6 16 20.19 odd 2
400.2.q.h.349.6 16 16.3 odd 4
640.2.l.a.161.4 16 80.53 odd 4
640.2.l.a.481.4 16 40.13 odd 4
640.2.l.b.161.5 16 80.43 even 4
640.2.l.b.481.5 16 40.3 even 4
720.2.t.c.181.2 16 60.23 odd 4
720.2.t.c.541.2 16 240.83 odd 4
1600.2.l.i.401.4 16 80.77 odd 4
1600.2.l.i.1201.4 16 5.2 odd 4
1600.2.q.g.49.5 16 5.4 even 2
1600.2.q.g.849.5 16 16.13 even 4
1600.2.q.h.49.4 16 1.1 even 1 trivial
1600.2.q.h.849.4 16 80.29 even 4 inner
2880.2.t.c.721.6 16 240.173 even 4
2880.2.t.c.2161.7 16 15.8 even 4
5120.2.a.s.1.5 8 160.83 even 8
5120.2.a.t.1.5 8 160.93 odd 8
5120.2.a.u.1.4 8 160.13 odd 8
5120.2.a.v.1.4 8 160.3 even 8