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Properties

Label 400.2.s.b.107.1
Level $400$
Weight $2$
Character 400.107
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(107,400)] 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("400.107"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.s (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 107.1
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 400.107
Dual form 400.2.s.b.243.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -0.517638 q^{3} -2.00000 q^{4} +0.732051i q^{6} +(3.34607 + 3.34607i) q^{7} +2.82843i q^{8} -2.73205 q^{9} +(-1.09808 + 1.09808i) q^{11} +1.03528 q^{12} +4.89898i q^{13} +(4.73205 - 4.73205i) q^{14} +4.00000 q^{16} +(0.707107 + 0.707107i) q^{17} +3.86370i q^{18} +(-2.09808 + 2.09808i) q^{19} +(-1.73205 - 1.73205i) q^{21} +(1.55291 + 1.55291i) q^{22} +(4.38134 - 4.38134i) q^{23} -1.46410i q^{24} +6.92820 q^{26} +2.96713 q^{27} +(-6.69213 - 6.69213i) q^{28} +(4.73205 + 4.73205i) q^{29} -6.19615i q^{31} -5.65685i q^{32} +(0.568406 - 0.568406i) q^{33} +(1.00000 - 1.00000i) q^{34} +5.46410 q^{36} +6.03579i q^{37} +(2.96713 + 2.96713i) q^{38} -2.53590i q^{39} -0.464102i q^{41} +(-2.44949 + 2.44949i) q^{42} -0.656339i q^{43} +(2.19615 - 2.19615i) q^{44} +(-6.19615 - 6.19615i) q^{46} +(-1.41421 + 1.41421i) q^{47} -2.07055 q^{48} +15.3923i q^{49} +(-0.366025 - 0.366025i) q^{51} -9.79796i q^{52} -9.89949 q^{53} -4.19615i q^{54} +(-9.46410 + 9.46410i) q^{56} +(1.08604 - 1.08604i) q^{57} +(6.69213 - 6.69213i) q^{58} +(-7.73205 - 7.73205i) q^{59} +(-3.19615 + 3.19615i) q^{61} -8.76268 q^{62} +(-9.14162 - 9.14162i) q^{63} -8.00000 q^{64} +(-0.803848 - 0.803848i) q^{66} +5.79555i q^{67} +(-1.41421 - 1.41421i) q^{68} +(-2.26795 + 2.26795i) q^{69} +0.928203 q^{71} -7.72741i q^{72} +(8.81345 + 8.81345i) q^{73} +8.53590 q^{74} +(4.19615 - 4.19615i) q^{76} -7.34847 q^{77} -3.58630 q^{78} +2.19615 q^{79} +6.66025 q^{81} -0.656339 q^{82} +17.3867 q^{83} +(3.46410 + 3.46410i) q^{84} -0.928203 q^{86} +(-2.44949 - 2.44949i) q^{87} +(-3.10583 - 3.10583i) q^{88} +10.2679 q^{89} +(-16.3923 + 16.3923i) q^{91} +(-8.76268 + 8.76268i) q^{92} +3.20736i q^{93} +(2.00000 + 2.00000i) q^{94} +2.92820i q^{96} +(-11.5911 - 11.5911i) q^{97} +21.7680 q^{98} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 8 q^{9} + 12 q^{11} + 24 q^{14} + 32 q^{16} + 4 q^{19} + 24 q^{29} + 8 q^{34} + 16 q^{36} - 24 q^{44} - 8 q^{46} + 4 q^{51} - 48 q^{56} - 48 q^{59} + 16 q^{61} - 64 q^{64} - 48 q^{66} - 32 q^{69}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i
\(3\) −0.517638 −0.298858 −0.149429 − 0.988772i \(-0.547744\pi\)
−0.149429 + 0.988772i \(0.547744\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0.732051i 0.298858i
\(7\) 3.34607 + 3.34607i 1.26469 + 1.26469i 0.948792 + 0.315902i \(0.102307\pi\)
0.315902 + 0.948792i \(0.397693\pi\)
\(8\) 2.82843i 1.00000i
\(9\) −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 \(0.325218\pi\)
\(12\) 1.03528 0.298858
\(13\) 4.89898i 1.35873i 0.733799 + 0.679366i \(0.237745\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) 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 \(-0.211304\pi\)
−0.616139 + 0.787638i \(0.711304\pi\)
\(18\) 3.86370i 0.910684i
\(19\) −2.09808 + 2.09808i −0.481332 + 0.481332i −0.905557 − 0.424225i \(-0.860547\pi\)
0.424225 + 0.905557i \(0.360547\pi\)
\(20\) 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 \(-0.526443\pi\)
0.996551 + 0.0829785i \(0.0264433\pi\)
\(24\) − 1.46410i − 0.298858i
\(25\) 0 0
\(26\) 6.92820 1.35873
\(27\) 2.96713 0.571024
\(28\) −6.69213 − 6.69213i −1.26469 − 1.26469i
\(29\) 4.73205 + 4.73205i 0.878720 + 0.878720i 0.993402 − 0.114682i \(-0.0365850\pi\)
−0.114682 + 0.993402i \(0.536585\pi\)
\(30\) 0 0
\(31\) − 6.19615i − 1.11286i −0.830894 − 0.556431i \(-0.812170\pi\)
0.830894 − 0.556431i \(-0.187830\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 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
\(37\) 6.03579i 0.992278i 0.868243 + 0.496139i \(0.165249\pi\)
−0.868243 + 0.496139i \(0.834751\pi\)
\(38\) 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
\(43\) − 0.656339i − 0.100091i −0.998747 − 0.0500454i \(-0.984063\pi\)
0.998747 − 0.0500454i \(-0.0159366\pi\)
\(44\) 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 −0.802686 − 0.596402i \(-0.796597\pi\)
0.596402 + 0.802686i \(0.296597\pi\)
\(48\) −2.07055 −0.298858
\(49\) 15.3923i 2.19890i
\(50\) 0 0
\(51\) −0.366025 − 0.366025i −0.0512538 − 0.0512538i
\(52\) − 9.79796i − 1.35873i
\(53\) −9.89949 −1.35980 −0.679900 − 0.733305i \(-0.737977\pi\)
−0.679900 + 0.733305i \(0.737977\pi\)
\(54\) − 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 \(-0.997883\pi\)
−0.00664938 − 0.999978i \(-0.502117\pi\)
\(60\) 0 0
\(61\) −3.19615 + 3.19615i −0.409225 + 0.409225i −0.881468 − 0.472243i \(-0.843444\pi\)
0.472243 + 0.881468i \(0.343444\pi\)
\(62\) −8.76268 −1.11286
\(63\) −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
\(67\) 5.79555i 0.708040i 0.935238 + 0.354020i \(0.115185\pi\)
−0.935238 + 0.354020i \(0.884815\pi\)
\(68\) −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 \(0.482459\pi\)
\(72\) − 7.72741i − 0.910684i
\(73\) 8.81345 + 8.81345i 1.03154 + 1.03154i 0.999486 + 0.0320501i \(0.0102036\pi\)
0.0320501 + 0.999486i \(0.489796\pi\)
\(74\) 8.53590 0.992278
\(75\) 0 0
\(76\) 4.19615 − 4.19615i 0.481332 − 0.481332i
\(77\) −7.34847 −0.837436
\(78\) −3.58630 −0.406069
\(79\) 2.19615 0.247086 0.123543 − 0.992339i \(-0.460574\pi\)
0.123543 + 0.992339i \(0.460574\pi\)
\(80\) 0 0
\(81\) 6.66025 0.740028
\(82\) −0.656339 −0.0724805
\(83\) 17.3867 1.90843 0.954217 − 0.299115i \(-0.0966913\pi\)
0.954217 + 0.299115i \(0.0966913\pi\)
\(84\) 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 \(-0.316833\pi\)
0.544200 + 0.838955i \(0.316833\pi\)
\(90\) 0 0
\(91\) −16.3923 + 16.3923i −1.71838 + 1.71838i
\(92\) −8.76268 + 8.76268i −0.913573 + 0.913573i
\(93\) 3.20736i 0.332588i
\(94\) 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 −0.980530 − 0.196369i \(-0.937085\pi\)
−0.196369 − 0.980530i \(-0.562915\pi\)
\(98\) 21.7680 2.19890
\(99\) 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.s.b.107.1 yes 8
4.3 odd 2 1600.2.s.b.207.3 8
5.2 odd 4 400.2.j.b.43.4 yes 8
5.3 odd 4 400.2.j.b.43.1 ✓ 8
5.4 even 2 inner 400.2.s.b.107.4 yes 8
16.3 odd 4 400.2.j.b.307.2 yes 8
16.13 even 4 1600.2.j.b.1007.2 8
20.3 even 4 1600.2.j.b.143.3 8
20.7 even 4 1600.2.j.b.143.2 8
20.19 odd 2 1600.2.s.b.207.2 8
80.3 even 4 inner 400.2.s.b.243.3 yes 8
80.13 odd 4 1600.2.s.b.943.3 8
80.19 odd 4 400.2.j.b.307.3 yes 8
80.29 even 4 1600.2.j.b.1007.3 8
80.67 even 4 inner 400.2.s.b.243.2 yes 8
80.77 odd 4 1600.2.s.b.943.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.b.43.1 ✓ 8 5.3 odd 4
400.2.j.b.43.4 yes 8 5.2 odd 4
400.2.j.b.307.2 yes 8 16.3 odd 4
400.2.j.b.307.3 yes 8 80.19 odd 4
400.2.s.b.107.1 yes 8 1.1 even 1 trivial
400.2.s.b.107.4 yes 8 5.4 even 2 inner
400.2.s.b.243.2 yes 8 80.67 even 4 inner
400.2.s.b.243.3 yes 8 80.3 even 4 inner
1600.2.j.b.143.2 8 20.7 even 4
1600.2.j.b.143.3 8 20.3 even 4
1600.2.j.b.1007.2 8 16.13 even 4
1600.2.j.b.1007.3 8 80.29 even 4
1600.2.s.b.207.2 8 20.19 odd 2
1600.2.s.b.207.3 8 4.3 odd 2
1600.2.s.b.943.2 8 80.77 odd 4
1600.2.s.b.943.3 8 80.13 odd 4