↑

Properties

Label 400.2.j.b.43.3
Level $400$
Weight $2$
Character 400.43
Analytic conductor $3.194$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,2,Mod(43,400)] 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("400.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.j (of order \(4\), degree \(2\), 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: \(3.19401608085\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.3
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 400.43
Dual form 400.2.j.b.307.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.93185i q^{3} +2.00000 q^{4} -2.73205i q^{6} +(-0.896575 + 0.896575i) q^{7} +2.82843 q^{8} -0.732051 q^{9} +(4.09808 - 4.09808i) q^{11} -3.86370i q^{12} -4.89898 q^{13} +(-1.26795 + 1.26795i) q^{14} +4.00000 q^{16} +(-0.707107 + 0.707107i) q^{17} -1.03528 q^{18} +(-3.09808 + 3.09808i) q^{19} +(1.73205 + 1.73205i) q^{21} +(5.79555 - 5.79555i) q^{22} +(2.96713 + 2.96713i) q^{23} -5.46410i q^{24} -6.92820 q^{26} -4.38134i q^{27} +(-1.79315 + 1.79315i) q^{28} +(-1.26795 - 1.26795i) q^{29} +4.19615i q^{31} +5.65685 q^{32} +(-7.91688 - 7.91688i) q^{33} +(-1.00000 + 1.00000i) q^{34} -1.46410 q^{36} -10.9348 q^{37} +(-4.38134 + 4.38134i) q^{38} +9.46410i q^{39} +6.46410i q^{41} +(2.44949 + 2.44949i) q^{42} +9.14162 q^{43} +(8.19615 - 8.19615i) q^{44} +(4.19615 + 4.19615i) q^{46} +(-1.41421 - 1.41421i) q^{47} -7.72741i q^{48} +5.39230i q^{49} +(1.36603 + 1.36603i) q^{51} -9.79796 q^{52} +9.89949i q^{53} -6.19615i q^{54} +(-2.53590 + 2.53590i) q^{56} +(5.98502 + 5.98502i) q^{57} +(-1.79315 - 1.79315i) q^{58} +(4.26795 + 4.26795i) q^{59} +(7.19615 - 7.19615i) q^{61} +5.93426i q^{62} +(0.656339 - 0.656339i) q^{63} +8.00000 q^{64} +(-11.1962 - 11.1962i) q^{66} +1.55291 q^{67} +(-1.41421 + 1.41421i) q^{68} +(5.73205 - 5.73205i) q^{69} -12.9282 q^{71} -2.07055 q^{72} +(3.91447 - 3.91447i) q^{73} -15.4641 q^{74} +(-6.19615 + 6.19615i) q^{76} +7.34847i q^{77} +13.3843i q^{78} +8.19615 q^{79} -10.6603 q^{81} +9.14162i q^{82} +4.65874i q^{83} +(3.46410 + 3.46410i) q^{84} +12.9282 q^{86} +(-2.44949 + 2.44949i) q^{87} +(11.5911 - 11.5911i) q^{88} -13.7321 q^{89} +(4.39230 - 4.39230i) q^{91} +(5.93426 + 5.93426i) q^{92} +8.10634 q^{93} +(-2.00000 - 2.00000i) q^{94} -10.9282i q^{96} +(-3.10583 + 3.10583i) q^{97} +7.62587i q^{98} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + 8 q^{9} + 12 q^{11} - 24 q^{14} + 32 q^{16} - 4 q^{19} - 24 q^{29} - 8 q^{34} + 16 q^{36} + 24 q^{44} - 8 q^{46} + 4 q^{51} - 48 q^{56} + 48 q^{59} + 16 q^{61} + 64 q^{64} - 48 q^{66} + 32 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000
\(3\) − 1.93185i − 1.11536i −0.830058 − 0.557678i \(-0.811693\pi\)
0.830058 − 0.557678i \(-0.188307\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) − 2.73205i − 1.11536i
\(7\) −0.896575 + 0.896575i −0.338874 + 0.338874i −0.855943 − 0.517070i \(-0.827023\pi\)
0.517070 + 0.855943i \(0.327023\pi\)
\(8\) 2.82843 1.00000
\(9\) −0.732051 −0.244017
\(10\) 0 0
\(11\) 4.09808 − 4.09808i 1.23562 − 1.23562i 0.273842 − 0.961775i \(-0.411706\pi\)
0.961775 − 0.273842i \(-0.0882944\pi\)
\(12\) − 3.86370i − 1.11536i
\(13\) −4.89898 −1.35873 −0.679366 − 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) −1.26795 + 1.26795i −0.338874 + 0.338874i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −0.707107 + 0.707107i −0.171499 + 0.171499i −0.787638 − 0.616139i \(-0.788696\pi\)
0.616139 + 0.787638i \(0.288696\pi\)
\(18\) −1.03528 −0.244017
\(19\) −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
\(22\) 5.79555 − 5.79555i 1.23562 − 1.23562i
\(23\) 2.96713 + 2.96713i 0.618689 + 0.618689i 0.945195 − 0.326506i \(-0.105871\pi\)
−0.326506 + 0.945195i \(0.605871\pi\)
\(24\) − 5.46410i − 1.11536i
\(25\) 0 0
\(26\) −6.92820 −1.35873
\(27\) − 4.38134i − 0.843190i
\(28\) −1.79315 + 1.79315i −0.338874 + 0.338874i
\(29\) −1.26795 − 1.26795i −0.235452 − 0.235452i 0.579512 − 0.814964i \(-0.303243\pi\)
−0.814964 + 0.579512i \(0.803243\pi\)
\(30\) 0 0
\(31\) 4.19615i 0.753651i 0.926284 + 0.376826i \(0.122984\pi\)
−0.926284 + 0.376826i \(0.877016\pi\)
\(32\) 5.65685 1.00000
\(33\) −7.91688 − 7.91688i −1.37815 − 1.37815i
\(34\) −1.00000 + 1.00000i −0.171499 + 0.171499i
\(35\) 0 0
\(36\) −1.46410 −0.244017
\(37\) −10.9348 −1.79767 −0.898833 − 0.438292i \(-0.855584\pi\)
−0.898833 + 0.438292i \(0.855584\pi\)
\(38\) −4.38134 + 4.38134i −0.710747 + 0.710747i
\(39\) 9.46410i 1.51547i
\(40\) 0 0
\(41\) 6.46410i 1.00952i 0.863259 + 0.504762i \(0.168420\pi\)
−0.863259 + 0.504762i \(0.831580\pi\)
\(42\) 2.44949 + 2.44949i 0.377964 + 0.377964i
\(43\) 9.14162 1.39408 0.697042 − 0.717030i \(-0.254499\pi\)
0.697042 + 0.717030i \(0.254499\pi\)
\(44\) 8.19615 − 8.19615i 1.23562 − 1.23562i
\(45\) 0 0
\(46\) 4.19615 + 4.19615i 0.618689 + 0.618689i
\(47\) −1.41421 − 1.41421i −0.206284 − 0.206284i 0.596402 − 0.802686i \(-0.296597\pi\)
−0.802686 + 0.596402i \(0.796597\pi\)
\(48\) − 7.72741i − 1.11536i
\(49\) 5.39230i 0.770329i
\(50\) 0 0
\(51\) 1.36603 + 1.36603i 0.191282 + 0.191282i
\(52\) −9.79796 −1.35873
\(53\) 9.89949i 1.35980i 0.733305 + 0.679900i \(0.237977\pi\)
−0.733305 + 0.679900i \(0.762023\pi\)
\(54\) − 6.19615i − 0.843190i
\(55\) 0 0
\(56\) −2.53590 + 2.53590i −0.338874 + 0.338874i
\(57\) 5.98502 + 5.98502i 0.792736 + 0.792736i
\(58\) −1.79315 − 1.79315i −0.235452 − 0.235452i
\(59\) 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 \(0.0241363\pi\)
\(62\) 5.93426i 0.753651i
\(63\) 0.656339 − 0.656339i 0.0826909 − 0.0826909i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) −11.1962 − 11.1962i −1.37815 − 1.37815i
\(67\) 1.55291 0.189719 0.0948593 − 0.995491i \(-0.469760\pi\)
0.0948593 + 0.995491i \(0.469760\pi\)
\(68\) −1.41421 + 1.41421i −0.171499 + 0.171499i
\(69\) 5.73205 − 5.73205i 0.690058 − 0.690058i
\(70\) 0 0
\(71\) −12.9282 −1.53430 −0.767148 − 0.641470i \(-0.778325\pi\)
−0.767148 + 0.641470i \(0.778325\pi\)
\(72\) −2.07055 −0.244017
\(73\) 3.91447 − 3.91447i 0.458154 − 0.458154i −0.439895 − 0.898049i \(-0.644984\pi\)
0.898049 + 0.439895i \(0.144984\pi\)
\(74\) −15.4641 −1.79767
\(75\) 0 0
\(76\) −6.19615 + 6.19615i −0.710747 + 0.710747i
\(77\) 7.34847i 0.837436i
\(78\) 13.3843i 1.51547i
\(79\) 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
\(82\) 9.14162i 1.00952i
\(83\) 4.65874i 0.511363i 0.966761 + 0.255682i \(0.0822999\pi\)
−0.966761 + 0.255682i \(0.917700\pi\)
\(84\) 3.46410 + 3.46410i 0.377964 + 0.377964i
\(85\) 0 0
\(86\) 12.9282 1.39408
\(87\) −2.44949 + 2.44949i −0.262613 + 0.262613i
\(88\) 11.5911 − 11.5911i 1.23562 − 1.23562i
\(89\) −13.7321 −1.45559 −0.727797 − 0.685792i \(-0.759456\pi\)
−0.727797 + 0.685792i \(0.759456\pi\)
\(90\) 0 0
\(91\) 4.39230 − 4.39230i 0.460439 − 0.460439i
\(92\) 5.93426 + 5.93426i 0.618689 + 0.618689i
\(93\) 8.10634 0.840589
\(94\) −2.00000 − 2.00000i −0.206284 − 0.206284i
\(95\) 0 0
\(96\) − 10.9282i − 1.11536i
\(97\) −3.10583 + 3.10583i −0.315349 + 0.315349i −0.846978 − 0.531629i \(-0.821580\pi\)
0.531629 + 0.846978i \(0.321580\pi\)
\(98\) 7.62587i 0.770329i
\(99\) −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 400.2.j.b.43.3 yes 8
4.3 odd 2 1600.2.j.b.143.4 8
5.2 odd 4 400.2.s.b.107.3 yes 8
5.3 odd 4 400.2.s.b.107.2 yes 8
5.4 even 2 inner 400.2.j.b.43.2 ✓ 8
16.3 odd 4 400.2.s.b.243.1 yes 8
16.13 even 4 1600.2.s.b.943.4 8
20.3 even 4 1600.2.s.b.207.1 8
20.7 even 4 1600.2.s.b.207.4 8
20.19 odd 2 1600.2.j.b.143.1 8
80.3 even 4 inner 400.2.j.b.307.1 yes 8
80.13 odd 4 1600.2.j.b.1007.4 8
80.19 odd 4 400.2.s.b.243.4 yes 8
80.29 even 4 1600.2.s.b.943.1 8
80.67 even 4 inner 400.2.j.b.307.4 yes 8
80.77 odd 4 1600.2.j.b.1007.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.b.43.2 ✓ 8 5.4 even 2 inner
400.2.j.b.43.3 yes 8 1.1 even 1 trivial
400.2.j.b.307.1 yes 8 80.3 even 4 inner
400.2.j.b.307.4 yes 8 80.67 even 4 inner
400.2.s.b.107.2 yes 8 5.3 odd 4
400.2.s.b.107.3 yes 8 5.2 odd 4
400.2.s.b.243.1 yes 8 16.3 odd 4
400.2.s.b.243.4 yes 8 80.19 odd 4
1600.2.j.b.143.1 8 20.19 odd 2
1600.2.j.b.143.4 8 4.3 odd 2
1600.2.j.b.1007.1 8 80.77 odd 4
1600.2.j.b.1007.4 8 80.13 odd 4
1600.2.s.b.207.1 8 20.3 even 4
1600.2.s.b.207.4 8 20.7 even 4
1600.2.s.b.943.1 8 80.29 even 4
1600.2.s.b.943.4 8 16.13 even 4