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Properties

Label 400.2.q.h.149.2
Level $400$
Weight $2$
Character 400.149
Analytic conductor $3.194$
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] = [400,2,Mod(149,400)] 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("400.149"); 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([0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.19401608085\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 149.2
Root \(-0.966675 + 1.03225i\) of defining polynomial
Character \(\chi\) \(=\) 400.149
Dual form 400.2.q.h.349.2

$q$-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+(-1.29751 - 0.562546i) q^{2} +(0.209571 + 0.209571i) q^{3} +(1.36708 + 1.45982i) q^{4} +(-0.154028 - 0.389815i) q^{6} -1.73696 q^{7} +(-0.952595 - 2.66319i) q^{8} -2.91216i q^{9} +(0.505430 + 0.505430i) q^{11} +(-0.0194351 + 0.592438i) q^{12} +(-1.88750 - 1.88750i) q^{13} +(2.25374 + 0.977122i) q^{14} +(-0.262159 + 3.99140i) q^{16} -4.53524i q^{17} +(-1.63822 + 3.77857i) q^{18} +(3.22022 - 3.22022i) q^{19} +(-0.364018 - 0.364018i) q^{21} +(-0.371475 - 0.940130i) q^{22} +8.85045 q^{23} +(0.358491 - 0.757764i) q^{24} +(1.38725 + 3.51086i) q^{26} +(1.23902 - 1.23902i) q^{27} +(-2.37458 - 2.53566i) q^{28} +(2.44059 - 2.44059i) q^{29} -5.70401 q^{31} +(2.58550 - 5.03142i) q^{32} +0.211847i q^{33} +(-2.55128 + 5.88454i) q^{34} +(4.25123 - 3.98117i) q^{36} +(-5.35670 + 5.35670i) q^{37} +(-5.98979 + 2.36676i) q^{38} -0.791130i q^{39} -10.0343i q^{41} +(0.267541 + 0.677095i) q^{42} +(2.10564 - 2.10564i) q^{43} +(-0.0468722 + 1.42880i) q^{44} +(-11.4836 - 4.97878i) q^{46} -4.32303i q^{47} +(-0.891424 + 0.781541i) q^{48} -3.98295 q^{49} +(0.950456 - 0.950456i) q^{51} +(0.175041 - 5.33578i) q^{52} +(1.37458 - 1.37458i) q^{53} +(-2.30465 + 0.910639i) q^{54} +(1.65462 + 4.62586i) q^{56} +1.34973 q^{57} +(-4.53964 + 1.79375i) q^{58} +(-6.64140 - 6.64140i) q^{59} +(5.26208 - 5.26208i) q^{61} +(7.40103 + 3.20877i) q^{62} +5.05832i q^{63} +(-6.18513 + 5.07388i) q^{64} +(0.119174 - 0.274875i) q^{66} +(10.5578 + 10.5578i) q^{67} +(6.62065 - 6.20006i) q^{68} +(1.85480 + 1.85480i) q^{69} +14.0437i q^{71} +(-7.75563 + 2.77411i) q^{72} -6.63830 q^{73} +(9.96378 - 3.93700i) q^{74} +(9.10325 + 0.298634i) q^{76} +(-0.877914 - 0.877914i) q^{77} +(-0.445047 + 1.02650i) q^{78} -4.27297 q^{79} -8.21715 q^{81} +(-5.64474 + 13.0196i) q^{82} +(9.15483 + 9.15483i) q^{83} +(0.0337580 - 1.02904i) q^{84} +(-3.91661 + 1.54758i) q^{86} +1.02295 q^{87} +(0.864585 - 1.82752i) q^{88} +3.23826i q^{89} +(3.27852 + 3.27852i) q^{91} +(12.0993 + 12.9201i) q^{92} +(-1.19540 - 1.19540i) q^{93} +(-2.43190 + 5.60919i) q^{94} +(1.59629 - 0.512594i) q^{96} -1.94129i q^{97} +(5.16794 + 2.24059i) q^{98} +(1.47189 - 1.47189i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 4 q^{4} - 12 q^{6} + 8 q^{7} - 8 q^{8} - 8 q^{11} + 20 q^{12} - 4 q^{14} + 16 q^{16} + 12 q^{18} + 8 q^{19} - 20 q^{22} + 24 q^{23} - 8 q^{24} - 16 q^{26} + 24 q^{27} + 20 q^{28} + 16 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29751 − 0.562546i −0.917481 − 0.397780i
\(3\) 0.209571 + 0.209571i 0.120996 + 0.120996i 0.765012 − 0.644016i \(-0.222733\pi\)
−0.644016 + 0.765012i \(0.722733\pi\)
\(4\) 1.36708 + 1.45982i 0.683542 + 0.729911i
\(5\) 0 0
\(6\) −0.154028 − 0.389815i −0.0628817 − 0.159141i
\(7\) −1.73696 −0.656511 −0.328255 − 0.944589i \(-0.606461\pi\)
−0.328255 + 0.944589i \(0.606461\pi\)
\(8\) −0.952595 − 2.66319i −0.336793 − 0.941579i
\(9\) − 2.91216i − 0.970720i
\(10\) 0 0
\(11\) 0.505430 + 0.505430i 0.152393 + 0.152393i 0.779186 − 0.626793i \(-0.215633\pi\)
−0.626793 + 0.779186i \(0.715633\pi\)
\(12\) −0.0194351 + 0.592438i −0.00561042 + 0.171022i
\(13\) −1.88750 − 1.88750i −0.523498 − 0.523498i 0.395128 − 0.918626i \(-0.370700\pi\)
−0.918626 + 0.395128i \(0.870700\pi\)
\(14\) 2.25374 + 0.977122i 0.602336 + 0.261147i
\(15\) 0 0
\(16\) −0.262159 + 3.99140i −0.0655399 + 0.997850i
\(17\) − 4.53524i − 1.09996i −0.835178 − 0.549979i \(-0.814636\pi\)
0.835178 − 0.549979i \(-0.185364\pi\)
\(18\) −1.63822 + 3.77857i −0.386133 + 0.890617i
\(19\) 3.22022 − 3.22022i 0.738768 − 0.738768i −0.233571 − 0.972340i \(-0.575041\pi\)
0.972340 + 0.233571i \(0.0750413\pi\)
\(20\) 0 0
\(21\) −0.364018 − 0.364018i −0.0794352 − 0.0794352i
\(22\) −0.371475 − 0.940130i −0.0791987 − 0.200436i
\(23\) 8.85045 1.84545 0.922723 − 0.385463i \(-0.125958\pi\)
0.922723 + 0.385463i \(0.125958\pi\)
\(24\) 0.358491 − 0.757764i 0.0731766 − 0.154678i
\(25\) 0 0
\(26\) 1.38725 + 3.51086i 0.272062 + 0.688536i
\(27\) 1.23902 − 1.23902i 0.238449 − 0.238449i
\(28\) −2.37458 − 2.53566i −0.448753 − 0.479195i
\(29\) 2.44059 − 2.44059i 0.453205 − 0.453205i −0.443212 − 0.896417i \(-0.646161\pi\)
0.896417 + 0.443212i \(0.146161\pi\)
\(30\) 0 0
\(31\) −5.70401 −1.02447 −0.512235 − 0.858845i \(-0.671182\pi\)
−0.512235 + 0.858845i \(0.671182\pi\)
\(32\) 2.58550 − 5.03142i 0.457056 − 0.889438i
\(33\) 0.211847i 0.0368779i
\(34\) −2.55128 + 5.88454i −0.437541 + 1.00919i
\(35\) 0 0
\(36\) 4.25123 − 3.98117i 0.708539 − 0.663528i
\(37\) −5.35670 + 5.35670i −0.880636 + 0.880636i −0.993599 − 0.112963i \(-0.963966\pi\)
0.112963 + 0.993599i \(0.463966\pi\)
\(38\) −5.98979 + 2.36676i −0.971673 + 0.383939i
\(39\) − 0.791130i − 0.126682i
\(40\) 0 0
\(41\) − 10.0343i − 1.56709i −0.621335 − 0.783545i \(-0.713409\pi\)
0.621335 − 0.783545i \(-0.286591\pi\)
\(42\) 0.267541 + 0.677095i 0.0412825 + 0.104478i
\(43\) 2.10564 − 2.10564i 0.321107 − 0.321107i −0.528085 − 0.849192i \(-0.677090\pi\)
0.849192 + 0.528085i \(0.177090\pi\)
\(44\) −0.0468722 + 1.42880i −0.00706625 + 0.215400i
\(45\) 0 0
\(46\) −11.4836 − 4.97878i −1.69316 − 0.734082i
\(47\) − 4.32303i − 0.630578i −0.948996 − 0.315289i \(-0.897899\pi\)
0.948996 − 0.315289i \(-0.102101\pi\)
\(48\) −0.891424 + 0.781541i −0.128666 + 0.112806i
\(49\) −3.98295 −0.568993
\(50\) 0 0
\(51\) 0.950456 − 0.950456i 0.133091 − 0.133091i
\(52\) 0.175041 − 5.33578i 0.0242739 − 0.739940i
\(53\) 1.37458 − 1.37458i 0.188814 − 0.188814i −0.606369 − 0.795183i \(-0.707375\pi\)
0.795183 + 0.606369i \(0.207375\pi\)
\(54\) −2.30465 + 0.910639i −0.313623 + 0.123922i
\(55\) 0 0
\(56\) 1.65462 + 4.62586i 0.221108 + 0.618157i
\(57\) 1.34973 0.178776
\(58\) −4.53964 + 1.79375i −0.596083 + 0.235531i
\(59\) −6.64140 − 6.64140i −0.864637 − 0.864637i 0.127236 − 0.991872i \(-0.459389\pi\)
−0.991872 + 0.127236i \(0.959389\pi\)
\(60\) 0 0
\(61\) 5.26208 − 5.26208i 0.673741 − 0.673741i −0.284836 − 0.958576i \(-0.591939\pi\)
0.958576 + 0.284836i \(0.0919391\pi\)
\(62\) 7.40103 + 3.20877i 0.939932 + 0.407514i
\(63\) 5.05832i 0.637288i
\(64\) −6.18513 + 5.07388i −0.773141 + 0.634234i
\(65\) 0 0
\(66\) 0.119174 − 0.274875i 0.0146693 − 0.0338347i
\(67\) 10.5578 + 10.5578i 1.28984 + 1.28984i 0.934884 + 0.354954i \(0.115503\pi\)
0.354954 + 0.934884i \(0.384497\pi\)
\(68\) 6.62065 − 6.20006i 0.802871 − 0.751868i
\(69\) 1.85480 + 1.85480i 0.223292 + 0.223292i
\(70\) 0 0
\(71\) 14.0437i 1.66668i 0.552764 + 0.833338i \(0.313573\pi\)
−0.552764 + 0.833338i \(0.686427\pi\)
\(72\) −7.75563 + 2.77411i −0.914009 + 0.326932i
\(73\) −6.63830 −0.776954 −0.388477 − 0.921458i \(-0.626999\pi\)
−0.388477 + 0.921458i \(0.626999\pi\)
\(74\) 9.96378 − 3.93700i 1.15827 − 0.457667i
\(75\) 0 0
\(76\) 9.10325 + 0.298634i 1.04421 + 0.0342557i
\(77\) −0.877914 − 0.877914i −0.100048 − 0.100048i
\(78\) −0.445047 + 1.02650i −0.0503917 + 0.116229i
\(79\) −4.27297 −0.480746 −0.240373 − 0.970681i \(-0.577270\pi\)
−0.240373 + 0.970681i \(0.577270\pi\)
\(80\) 0 0
\(81\) −8.21715 −0.913017
\(82\) −5.64474 + 13.0196i −0.623357 + 1.43778i
\(83\) 9.15483 + 9.15483i 1.00487 + 1.00487i 0.999988 + 0.00488547i \(0.00155510\pi\)
0.00488547 + 0.999988i \(0.498445\pi\)
\(84\) 0.0337580 − 1.02904i 0.00368330 − 0.112278i
\(85\) 0 0
\(86\) −3.91661 + 1.54758i −0.422339 + 0.166879i
\(87\) 1.02295 0.109672
\(88\) 0.864585 − 1.82752i 0.0921650 − 0.194815i
\(89\) 3.23826i 0.343255i 0.985162 + 0.171627i \(0.0549025\pi\)
−0.985162 + 0.171627i \(0.945097\pi\)
\(90\) 0 0
\(91\) 3.27852 + 3.27852i 0.343682 + 0.343682i
\(92\) 12.0993 + 12.9201i 1.26144 + 1.34701i
\(93\) −1.19540 − 1.19540i −0.123957 − 0.123957i
\(94\) −2.43190 + 5.60919i −0.250831 + 0.578543i
\(95\) 0 0
\(96\) 1.59629 − 0.512594i 0.162920 − 0.0523164i
\(97\) − 1.94129i − 0.197108i −0.995132 − 0.0985541i \(-0.968578\pi\)
0.995132 − 0.0985541i \(-0.0314217\pi\)
\(98\) 5.16794 + 2.24059i 0.522041 + 0.226334i
\(99\) 1.47189 − 1.47189i 0.147931 − 0.147931i
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.q.h.149.2 16
4.3 odd 2 1600.2.q.g.49.4 16
5.2 odd 4 80.2.l.a.21.6 ✓ 16
5.3 odd 4 400.2.l.h.101.3 16
5.4 even 2 400.2.q.g.149.7 16
15.2 even 4 720.2.t.c.181.3 16
16.3 odd 4 1600.2.q.h.849.5 16
16.13 even 4 400.2.q.g.349.7 16
20.3 even 4 1600.2.l.i.1201.5 16
20.7 even 4 320.2.l.a.241.4 16
20.19 odd 2 1600.2.q.h.49.5 16
40.27 even 4 640.2.l.a.481.5 16
40.37 odd 4 640.2.l.b.481.4 16
60.47 odd 4 2880.2.t.c.2161.4 16
80.3 even 4 1600.2.l.i.401.5 16
80.13 odd 4 400.2.l.h.301.3 16
80.19 odd 4 1600.2.q.g.849.4 16
80.27 even 4 640.2.l.a.161.5 16
80.29 even 4 inner 400.2.q.h.349.2 16
80.37 odd 4 640.2.l.b.161.4 16
80.67 even 4 320.2.l.a.81.4 16
80.77 odd 4 80.2.l.a.61.6 yes 16
160.67 even 8 5120.2.a.u.1.3 8
160.77 odd 8 5120.2.a.v.1.3 8
160.147 even 8 5120.2.a.t.1.6 8
160.157 odd 8 5120.2.a.s.1.6 8
240.77 even 4 720.2.t.c.541.3 16
240.227 odd 4 2880.2.t.c.721.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.6 ✓ 16 5.2 odd 4
80.2.l.a.61.6 yes 16 80.77 odd 4
320.2.l.a.81.4 16 80.67 even 4
320.2.l.a.241.4 16 20.7 even 4
400.2.l.h.101.3 16 5.3 odd 4
400.2.l.h.301.3 16 80.13 odd 4
400.2.q.g.149.7 16 5.4 even 2
400.2.q.g.349.7 16 16.13 even 4
400.2.q.h.149.2 16 1.1 even 1 trivial
400.2.q.h.349.2 16 80.29 even 4 inner
640.2.l.a.161.5 16 80.27 even 4
640.2.l.a.481.5 16 40.27 even 4
640.2.l.b.161.4 16 80.37 odd 4
640.2.l.b.481.4 16 40.37 odd 4
720.2.t.c.181.3 16 15.2 even 4
720.2.t.c.541.3 16 240.77 even 4
1600.2.l.i.401.5 16 80.3 even 4
1600.2.l.i.1201.5 16 20.3 even 4
1600.2.q.g.49.4 16 4.3 odd 2
1600.2.q.g.849.4 16 80.19 odd 4
1600.2.q.h.49.5 16 20.19 odd 2
1600.2.q.h.849.5 16 16.3 odd 4
2880.2.t.c.721.1 16 240.227 odd 4
2880.2.t.c.2161.4 16 60.47 odd 4
5120.2.a.s.1.6 8 160.157 odd 8
5120.2.a.t.1.6 8 160.147 even 8
5120.2.a.u.1.3 8 160.67 even 8
5120.2.a.v.1.3 8 160.77 odd 8