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Properties

Label 400.2.j.b.43.4
Level $400$
Weight $2$
Character 400.43
Analytic conductor $3.194$
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] = [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 43.4
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 400.43
Dual form 400.2.j.b.307.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +0.517638i q^{3} +2.00000 q^{4} +0.732051i q^{6} +(-3.34607 + 3.34607i) q^{7} +2.82843 q^{8} +2.73205 q^{9} +(-1.09808 + 1.09808i) q^{11} +1.03528i q^{12} +4.89898 q^{13} +(-4.73205 + 4.73205i) q^{14} +4.00000 q^{16} +(-0.707107 + 0.707107i) q^{17} +3.86370 q^{18} +(2.09808 - 2.09808i) q^{19} +(-1.73205 - 1.73205i) q^{21} +(-1.55291 + 1.55291i) q^{22} +(-4.38134 - 4.38134i) q^{23} +1.46410i q^{24} +6.92820 q^{26} +2.96713i q^{27} +(-6.69213 + 6.69213i) q^{28} +(-4.73205 - 4.73205i) q^{29} -6.19615i q^{31} +5.65685 q^{32} +(-0.568406 - 0.568406i) q^{33} +(-1.00000 + 1.00000i) q^{34} +5.46410 q^{36} -6.03579 q^{37} +(2.96713 - 2.96713i) q^{38} +2.53590i q^{39} -0.464102i q^{41} +(-2.44949 - 2.44949i) q^{42} -0.656339 q^{43} +(-2.19615 + 2.19615i) q^{44} +(-6.19615 - 6.19615i) q^{46} +(-1.41421 - 1.41421i) q^{47} +2.07055i q^{48} -15.3923i q^{49} +(-0.366025 - 0.366025i) q^{51} +9.79796 q^{52} +9.89949i q^{53} +4.19615i q^{54} +(-9.46410 + 9.46410i) q^{56} +(1.08604 + 1.08604i) q^{57} +(-6.69213 - 6.69213i) q^{58} +(7.73205 + 7.73205i) q^{59} +(-3.19615 + 3.19615i) q^{61} -8.76268i q^{62} +(-9.14162 + 9.14162i) q^{63} +8.00000 q^{64} +(-0.803848 - 0.803848i) q^{66} -5.79555 q^{67} +(-1.41421 + 1.41421i) q^{68} +(2.26795 - 2.26795i) q^{69} +0.928203 q^{71} +7.72741 q^{72} +(8.81345 - 8.81345i) q^{73} -8.53590 q^{74} +(4.19615 - 4.19615i) q^{76} -7.34847i q^{77} +3.58630i q^{78} -2.19615 q^{79} +6.66025 q^{81} -0.656339i q^{82} -17.3867i q^{83} +(-3.46410 - 3.46410i) q^{84} -0.928203 q^{86} +(2.44949 - 2.44949i) q^{87} +(-3.10583 + 3.10583i) q^{88} -10.2679 q^{89} +(-16.3923 + 16.3923i) q^{91} +(-8.76268 - 8.76268i) q^{92} +3.20736 q^{93} +(-2.00000 - 2.00000i) q^{94} +2.92820i q^{96} +(11.5911 - 11.5911i) q^{97} -21.7680i q^{98} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + 8 q^{9} + 12 q^{11} - 24 q^{14} + 32 q^{16} - 4 q^{19} - 24 q^{29} - 8 q^{34} + 16 q^{36} + 24 q^{44} - 8 q^{46} + 4 q^{51} - 48 q^{56} + 48 q^{59} + 16 q^{61} + 64 q^{64} - 48 q^{66} + 32 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) \) 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
\(3\) 0.517638i 0.298858i 0.988772 + 0.149429i \(0.0477436\pi\)
−0.988772 + 0.149429i \(0.952256\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0.732051i 0.298858i
\(7\) −3.34607 + 3.34607i −1.26469 + 1.26469i −0.315902 + 0.948792i \(0.602307\pi\)
−0.948792 + 0.315902i \(0.897693\pi\)
\(8\) 2.82843 1.00000
\(9\) 2.73205 0.910684
\(10\) 0 0
\(11\) −1.09808 + 1.09808i −0.331082 + 0.331082i −0.852997 − 0.521915i \(-0.825218\pi\)
0.521915 + 0.852997i \(0.325218\pi\)
\(12\) 1.03528i 0.298858i
\(13\) 4.89898 1.35873 0.679366 − 0.733799i \(-0.262255\pi\)
0.679366 + 0.733799i \(0.262255\pi\)
\(14\) −4.73205 + 4.73205i −1.26469 + 1.26469i
\(15\) 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 \(0.288696\pi\)
\(18\) 3.86370 0.910684
\(19\) 2.09808 − 2.09808i 0.481332 − 0.481332i −0.424225 − 0.905557i \(-0.639453\pi\)
0.905557 + 0.424225i \(0.139453\pi\)
\(20\) 0 0
\(21\) −1.73205 − 1.73205i −0.377964 − 0.377964i
\(22\) −1.55291 + 1.55291i −0.331082 + 0.331082i
\(23\) −4.38134 − 4.38134i −0.913573 − 0.913573i 0.0829785 − 0.996551i \(-0.473557\pi\)
−0.996551 + 0.0829785i \(0.973557\pi\)
\(24\) 1.46410i 0.298858i
\(25\) 0 0
\(26\) 6.92820 1.35873
\(27\) 2.96713i 0.571024i
\(28\) −6.69213 + 6.69213i −1.26469 + 1.26469i
\(29\) −4.73205 − 4.73205i −0.878720 − 0.878720i 0.114682 − 0.993402i \(-0.463415\pi\)
−0.993402 + 0.114682i \(0.963415\pi\)
\(30\) 0 0
\(31\) − 6.19615i − 1.11286i −0.830894 − 0.556431i \(-0.812170\pi\)
0.830894 − 0.556431i \(-0.187830\pi\)
\(32\) 5.65685 1.00000
\(33\) −0.568406 − 0.568406i −0.0989468 − 0.0989468i
\(34\) −1.00000 + 1.00000i −0.171499 + 0.171499i
\(35\) 0 0
\(36\) 5.46410 0.910684
\(37\) −6.03579 −0.992278 −0.496139 − 0.868243i \(-0.665249\pi\)
−0.496139 + 0.868243i \(0.665249\pi\)
\(38\) 2.96713 − 2.96713i 0.481332 − 0.481332i
\(39\) 2.53590i 0.406069i
\(40\) 0 0
\(41\) − 0.464102i − 0.0724805i −0.999343 − 0.0362402i \(-0.988462\pi\)
0.999343 − 0.0362402i \(-0.0115382\pi\)
\(42\) −2.44949 − 2.44949i −0.377964 − 0.377964i
\(43\) −0.656339 −0.100091 −0.0500454 − 0.998747i \(-0.515937\pi\)
−0.0500454 + 0.998747i \(0.515937\pi\)
\(44\) −2.19615 + 2.19615i −0.331082 + 0.331082i
\(45\) 0 0
\(46\) −6.19615 − 6.19615i −0.913573 − 0.913573i
\(47\) −1.41421 − 1.41421i −0.206284 − 0.206284i 0.596402 − 0.802686i \(-0.296597\pi\)
−0.802686 + 0.596402i \(0.796597\pi\)
\(48\) 2.07055i 0.298858i
\(49\) − 15.3923i − 2.19890i
\(50\) 0 0
\(51\) −0.366025 − 0.366025i −0.0512538 − 0.0512538i
\(52\) 9.79796 1.35873
\(53\) 9.89949i 1.35980i 0.733305 + 0.679900i \(0.237977\pi\)
−0.733305 + 0.679900i \(0.762023\pi\)
\(54\) 4.19615i 0.571024i
\(55\) 0 0
\(56\) −9.46410 + 9.46410i −1.26469 + 1.26469i
\(57\) 1.08604 + 1.08604i 0.143850 + 0.143850i
\(58\) −6.69213 − 6.69213i −0.878720 − 0.878720i
\(59\) 7.73205 + 7.73205i 1.00663 + 1.00663i 0.999978 + 0.00664938i \(0.00211658\pi\)
0.00664938 + 0.999978i \(0.497883\pi\)
\(60\) 0 0
\(61\) −3.19615 + 3.19615i −0.409225 + 0.409225i −0.881468 − 0.472243i \(-0.843444\pi\)
0.472243 + 0.881468i \(0.343444\pi\)
\(62\) − 8.76268i − 1.11286i
\(63\) −9.14162 + 9.14162i −1.15174 + 1.15174i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) −0.803848 − 0.803848i −0.0989468 − 0.0989468i
\(67\) −5.79555 −0.708040 −0.354020 − 0.935238i \(-0.615185\pi\)
−0.354020 + 0.935238i \(0.615185\pi\)
\(68\) −1.41421 + 1.41421i −0.171499 + 0.171499i
\(69\) 2.26795 − 2.26795i 0.273029 − 0.273029i
\(70\) 0 0
\(71\) 0.928203 0.110157 0.0550787 − 0.998482i \(-0.482459\pi\)
0.0550787 + 0.998482i \(0.482459\pi\)
\(72\) 7.72741 0.910684
\(73\) 8.81345 − 8.81345i 1.03154 − 1.03154i 0.0320501 − 0.999486i \(-0.489796\pi\)
0.999486 − 0.0320501i \(-0.0102036\pi\)
\(74\) −8.53590 −0.992278
\(75\) 0 0
\(76\) 4.19615 − 4.19615i 0.481332 − 0.481332i
\(77\) − 7.34847i − 0.837436i
\(78\) 3.58630i 0.406069i
\(79\) −2.19615 −0.247086 −0.123543 − 0.992339i \(-0.539426\pi\)
−0.123543 + 0.992339i \(0.539426\pi\)
\(80\) 0 0
\(81\) 6.66025 0.740028
\(82\) − 0.656339i − 0.0724805i
\(83\) − 17.3867i − 1.90843i −0.299115 − 0.954217i \(-0.596691\pi\)
0.299115 − 0.954217i \(-0.403309\pi\)
\(84\) −3.46410 − 3.46410i −0.377964 − 0.377964i
\(85\) 0 0
\(86\) −0.928203 −0.100091
\(87\) 2.44949 − 2.44949i 0.262613 − 0.262613i
\(88\) −3.10583 + 3.10583i −0.331082 + 0.331082i
\(89\) −10.2679 −1.08840 −0.544200 − 0.838955i \(-0.683167\pi\)
−0.544200 + 0.838955i \(0.683167\pi\)
\(90\) 0 0
\(91\) −16.3923 + 16.3923i −1.71838 + 1.71838i
\(92\) −8.76268 − 8.76268i −0.913573 − 0.913573i
\(93\) 3.20736 0.332588
\(94\) −2.00000 − 2.00000i −0.206284 − 0.206284i
\(95\) 0 0
\(96\) 2.92820i 0.298858i
\(97\) 11.5911 − 11.5911i 1.17690 − 1.17690i 0.196369 − 0.980530i \(-0.437085\pi\)
0.980530 − 0.196369i \(-0.0629150\pi\)
\(98\) − 21.7680i − 2.19890i
\(99\) −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.43.4 yes 8
4.3 odd 2 1600.2.j.b.143.2 8
5.2 odd 4 400.2.s.b.107.4 yes 8
5.3 odd 4 400.2.s.b.107.1 yes 8
5.4 even 2 inner 400.2.j.b.43.1 ✓ 8
16.3 odd 4 400.2.s.b.243.2 yes 8
16.13 even 4 1600.2.s.b.943.2 8
20.3 even 4 1600.2.s.b.207.3 8
20.7 even 4 1600.2.s.b.207.2 8
20.19 odd 2 1600.2.j.b.143.3 8
80.3 even 4 inner 400.2.j.b.307.2 yes 8
80.13 odd 4 1600.2.j.b.1007.2 8
80.19 odd 4 400.2.s.b.243.3 yes 8
80.29 even 4 1600.2.s.b.943.3 8
80.67 even 4 inner 400.2.j.b.307.3 yes 8
80.77 odd 4 1600.2.j.b.1007.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.b.43.1 ✓ 8 5.4 even 2 inner
400.2.j.b.43.4 yes 8 1.1 even 1 trivial
400.2.j.b.307.2 yes 8 80.3 even 4 inner
400.2.j.b.307.3 yes 8 80.67 even 4 inner
400.2.s.b.107.1 yes 8 5.3 odd 4
400.2.s.b.107.4 yes 8 5.2 odd 4
400.2.s.b.243.2 yes 8 16.3 odd 4
400.2.s.b.243.3 yes 8 80.19 odd 4
1600.2.j.b.143.2 8 4.3 odd 2
1600.2.j.b.143.3 8 20.19 odd 2
1600.2.j.b.1007.2 8 80.13 odd 4
1600.2.j.b.1007.3 8 80.77 odd 4
1600.2.s.b.207.2 8 20.7 even 4
1600.2.s.b.207.3 8 20.3 even 4
1600.2.s.b.943.2 8 16.13 even 4
1600.2.s.b.943.3 8 80.29 even 4