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Properties

Label 1600.2.s.b.207.4
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 207.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1600.207
Dual form 1600.2.s.b.943.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.93185 q^{3} +(0.896575 + 0.896575i) q^{7} +0.732051 q^{9} +(-4.09808 + 4.09808i) q^{11} +4.89898i q^{13} +(-0.707107 - 0.707107i) q^{17} +(-3.09808 + 3.09808i) q^{19} +(1.73205 + 1.73205i) q^{21} +(-2.96713 + 2.96713i) q^{23} -4.38134 q^{27} +(1.26795 + 1.26795i) q^{29} -4.19615i q^{31} +(-7.91688 + 7.91688i) q^{33} -10.9348i q^{37} +9.46410i q^{39} +6.46410i q^{41} +9.14162i q^{43} +(-1.41421 + 1.41421i) q^{47} -5.39230i q^{49} +(-1.36603 - 1.36603i) q^{51} +9.89949 q^{53} +(-5.98502 + 5.98502i) q^{57} +(4.26795 + 4.26795i) q^{59} +(7.19615 - 7.19615i) q^{61} +(0.656339 + 0.656339i) q^{63} -1.55291i q^{67} +(-5.73205 + 5.73205i) q^{69} +12.9282 q^{71} +(-3.91447 - 3.91447i) q^{73} -7.34847 q^{77} +8.19615 q^{79} -10.6603 q^{81} -4.65874 q^{83} +(2.44949 + 2.44949i) q^{87} +13.7321 q^{89} +(-4.39230 + 4.39230i) q^{91} -8.10634i q^{93} +(-3.10583 - 3.10583i) q^{97} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9} - 12 q^{11} - 4 q^{19} + 24 q^{29} - 4 q^{51} + 48 q^{59} + 16 q^{61} - 32 q^{69} + 48 q^{71} + 24 q^{79} - 16 q^{81} + 96 q^{89} + 48 q^{91} - 24 q^{99}+O(q^{100}) \) 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
\(3\) 1.93185 1.11536 0.557678 − 0.830058i \(-0.311693\pi\)
0.557678 + 0.830058i \(0.311693\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.896575 + 0.896575i 0.338874 + 0.338874i 0.855943 − 0.517070i \(-0.172977\pi\)
−0.517070 + 0.855943i \(0.672977\pi\)
\(8\) 0 0
\(9\) 0.732051 0.244017
\(10\) 0 0
\(11\) −4.09808 + 4.09808i −1.23562 + 1.23562i −0.273842 + 0.961775i \(0.588294\pi\)
−0.961775 + 0.273842i \(0.911706\pi\)
\(12\) 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 0.616139 − 0.787638i \(-0.288696\pi\)
−0.787638 + 0.616139i \(0.788696\pi\)
\(18\) 0 0
\(19\) −3.09808 + 3.09808i −0.710747 + 0.710747i −0.966692 − 0.255944i \(-0.917614\pi\)
0.255944 + 0.966692i \(0.417614\pi\)
\(20\) 0 0
\(21\) 1.73205 + 1.73205i 0.377964 + 0.377964i
\(22\) 0 0
\(23\) −2.96713 + 2.96713i −0.618689 + 0.618689i −0.945195 − 0.326506i \(-0.894129\pi\)
0.326506 + 0.945195i \(0.394129\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.38134 −0.843190
\(28\) 0 0
\(29\) 1.26795 + 1.26795i 0.235452 + 0.235452i 0.814964 − 0.579512i \(-0.196757\pi\)
−0.579512 + 0.814964i \(0.696757\pi\)
\(30\) 0 0
\(31\) − 4.19615i − 0.753651i −0.926284 − 0.376826i \(-0.877016\pi\)
0.926284 − 0.376826i \(-0.122984\pi\)
\(32\) 0 0
\(33\) −7.91688 + 7.91688i −1.37815 + 1.37815i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.9348i − 1.79767i −0.438292 − 0.898833i \(-0.644416\pi\)
0.438292 − 0.898833i \(-0.355584\pi\)
\(38\) 0 0
\(39\) 9.46410i 1.51547i
\(40\) 0 0
\(41\) 6.46410i 1.00952i 0.863259 + 0.504762i \(0.168420\pi\)
−0.863259 + 0.504762i \(0.831580\pi\)
\(42\) 0 0
\(43\) 9.14162i 1.39408i 0.717030 + 0.697042i \(0.245501\pi\)
−0.717030 + 0.697042i \(0.754499\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.41421 + 1.41421i −0.206284 + 0.206284i −0.802686 − 0.596402i \(-0.796597\pi\)
0.596402 + 0.802686i \(0.296597\pi\)
\(48\) 0 0
\(49\) − 5.39230i − 0.770329i
\(50\) 0 0
\(51\) −1.36603 − 1.36603i −0.191282 − 0.191282i
\(52\) 0 0
\(53\) 9.89949 1.35980 0.679900 − 0.733305i \(-0.262023\pi\)
0.679900 + 0.733305i \(0.262023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.98502 + 5.98502i −0.792736 + 0.792736i
\(58\) 0 0
\(59\) 4.26795 + 4.26795i 0.555640 + 0.555640i 0.928063 − 0.372423i \(-0.121473\pi\)
−0.372423 + 0.928063i \(0.621473\pi\)
\(60\) 0 0
\(61\) 7.19615 − 7.19615i 0.921373 − 0.921373i −0.0757537 − 0.997127i \(-0.524136\pi\)
0.997127 + 0.0757537i \(0.0241363\pi\)
\(62\) 0 0
\(63\) 0.656339 + 0.656339i 0.0826909 + 0.0826909i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.55291i − 0.189719i −0.995491 − 0.0948593i \(-0.969760\pi\)
0.995491 − 0.0948593i \(-0.0302401\pi\)
\(68\) 0 0
\(69\) −5.73205 + 5.73205i −0.690058 + 0.690058i
\(70\) 0 0
\(71\) 12.9282 1.53430 0.767148 − 0.641470i \(-0.221675\pi\)
0.767148 + 0.641470i \(0.221675\pi\)
\(72\) 0 0
\(73\) −3.91447 − 3.91447i −0.458154 − 0.458154i 0.439895 − 0.898049i \(-0.355016\pi\)
−0.898049 + 0.439895i \(0.855016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.34847 −0.837436
\(78\) 0 0
\(79\) 8.19615 0.922139 0.461070 − 0.887364i \(-0.347466\pi\)
0.461070 + 0.887364i \(0.347466\pi\)
\(80\) 0 0
\(81\) −10.6603 −1.18447
\(82\) 0 0
\(83\) −4.65874 −0.511363 −0.255682 − 0.966761i \(-0.582300\pi\)
−0.255682 + 0.966761i \(0.582300\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.44949 + 2.44949i 0.262613 + 0.262613i
\(88\) 0 0
\(89\) 13.7321 1.45559 0.727797 − 0.685792i \(-0.240544\pi\)
0.727797 + 0.685792i \(0.240544\pi\)
\(90\) 0 0
\(91\) −4.39230 + 4.39230i −0.460439 + 0.460439i
\(92\) 0 0
\(93\) − 8.10634i − 0.840589i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.10583 − 3.10583i −0.315349 − 0.315349i 0.531629 − 0.846978i \(-0.321580\pi\)
−0.846978 + 0.531629i \(0.821580\pi\)
\(98\) 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.207.4 8
4.3 odd 2 400.2.s.b.107.3 yes 8
5.2 odd 4 1600.2.j.b.143.1 8
5.3 odd 4 1600.2.j.b.143.4 8
5.4 even 2 inner 1600.2.s.b.207.1 8
16.3 odd 4 1600.2.j.b.1007.1 8
16.13 even 4 400.2.j.b.307.4 yes 8
20.3 even 4 400.2.j.b.43.3 yes 8
20.7 even 4 400.2.j.b.43.2 ✓ 8
20.19 odd 2 400.2.s.b.107.2 yes 8
80.3 even 4 inner 1600.2.s.b.943.4 8
80.13 odd 4 400.2.s.b.243.1 yes 8
80.19 odd 4 1600.2.j.b.1007.4 8
80.29 even 4 400.2.j.b.307.1 yes 8
80.67 even 4 inner 1600.2.s.b.943.1 8
80.77 odd 4 400.2.s.b.243.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.b.43.2 ✓ 8 20.7 even 4
400.2.j.b.43.3 yes 8 20.3 even 4
400.2.j.b.307.1 yes 8 80.29 even 4
400.2.j.b.307.4 yes 8 16.13 even 4
400.2.s.b.107.2 yes 8 20.19 odd 2
400.2.s.b.107.3 yes 8 4.3 odd 2
400.2.s.b.243.1 yes 8 80.13 odd 4
400.2.s.b.243.4 yes 8 80.77 odd 4
1600.2.j.b.143.1 8 5.2 odd 4
1600.2.j.b.143.4 8 5.3 odd 4
1600.2.j.b.1007.1 8 16.3 odd 4
1600.2.j.b.1007.4 8 80.19 odd 4
1600.2.s.b.207.1 8 5.4 even 2 inner
1600.2.s.b.207.4 8 1.1 even 1 trivial
1600.2.s.b.943.1 8 80.67 even 4 inner
1600.2.s.b.943.4 8 80.3 even 4 inner