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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,5,Mod(193,400)] 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("400.193"); S:= CuspForms(chi, 5); 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, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-8,0,0,0,-40,0,0,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3479852335\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.2
Root \(1.13432i\) of defining polynomial
Character \(\chi\) \(=\) 400.193
Dual form 400.5.p.o.257.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.958314 - 0.958314i) q^{3} +(26.0617 - 26.0617i) q^{7} -79.1633i q^{9} -75.9566 q^{11} +(99.1428 + 99.1428i) q^{13} +(-77.9171 + 77.9171i) q^{17} -640.445i q^{19} -49.9505 q^{21} +(117.134 + 117.134i) q^{23} +(-153.487 + 153.487i) q^{27} +752.534i q^{29} -708.205 q^{31} +(72.7903 + 72.7903i) q^{33} +(-1170.51 + 1170.51i) q^{37} -190.020i q^{39} +1822.21 q^{41} +(-2372.12 - 2372.12i) q^{43} +(2458.79 - 2458.79i) q^{47} +1042.58i q^{49} +149.338 q^{51} +(-2118.26 - 2118.26i) q^{53} +(-613.748 + 613.748i) q^{57} -5251.96i q^{59} -1312.89 q^{61} +(-2063.13 - 2063.13i) q^{63} +(-5569.35 + 5569.35i) q^{67} -224.503i q^{69} -7554.66 q^{71} +(-1206.35 - 1206.35i) q^{73} +(-1979.55 + 1979.55i) q^{77} -3397.97i q^{79} -6118.05 q^{81} +(928.421 + 928.421i) q^{83} +(721.164 - 721.164i) q^{87} -11981.9i q^{89} +5167.65 q^{91} +(678.683 + 678.683i) q^{93} +(-4683.76 + 4683.76i) q^{97} +6012.97i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 40 q^{7} - 72 q^{11} + 62 q^{13} - 418 q^{17} + 1736 q^{21} - 1760 q^{23} + 2248 q^{27} - 2864 q^{31} + 8 q^{33} - 5366 q^{37} + 1656 q^{41} - 2760 q^{43} - 432 q^{47} - 15840 q^{51} - 534 q^{53}+ \cdots + 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.958314 − 0.958314i −0.106479 − 0.106479i 0.651860 − 0.758339i \(-0.273989\pi\)
−0.758339 + 0.651860i \(0.773989\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 26.0617 − 26.0617i 0.531871 − 0.531871i −0.389258 − 0.921129i \(-0.627269\pi\)
0.921129 + 0.389258i \(0.127269\pi\)
\(8\) 0 0
\(9\) − 79.1633i − 0.977324i
\(10\) 0 0
\(11\) −75.9566 −0.627740 −0.313870 − 0.949466i \(-0.601626\pi\)
−0.313870 + 0.949466i \(0.601626\pi\)
\(12\) 0 0
\(13\) 99.1428 + 99.1428i 0.586644 + 0.586644i 0.936721 − 0.350077i \(-0.113845\pi\)
−0.350077 + 0.936721i \(0.613845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −77.9171 + 77.9171i −0.269609 + 0.269609i −0.828943 − 0.559333i \(-0.811057\pi\)
0.559333 + 0.828943i \(0.311057\pi\)
\(18\) 0 0
\(19\) − 640.445i − 1.77409i −0.461686 − 0.887044i \(-0.652755\pi\)
0.461686 − 0.887044i \(-0.347245\pi\)
\(20\) 0 0
\(21\) −49.9505 −0.113267
\(22\) 0 0
\(23\) 117.134 + 117.134i 0.221426 + 0.221426i 0.809099 − 0.587673i \(-0.199956\pi\)
−0.587673 + 0.809099i \(0.699956\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −153.487 + 153.487i −0.210544 + 0.210544i
\(28\) 0 0
\(29\) 752.534i 0.894809i 0.894332 + 0.447404i \(0.147652\pi\)
−0.894332 + 0.447404i \(0.852348\pi\)
\(30\) 0 0
\(31\) −708.205 −0.736946 −0.368473 − 0.929638i \(-0.620119\pi\)
−0.368473 + 0.929638i \(0.620119\pi\)
\(32\) 0 0
\(33\) 72.7903 + 72.7903i 0.0668414 + 0.0668414i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1170.51 + 1170.51i −0.855013 + 0.855013i −0.990745 − 0.135733i \(-0.956661\pi\)
0.135733 + 0.990745i \(0.456661\pi\)
\(38\) 0 0
\(39\) − 190.020i − 0.124931i
\(40\) 0 0
\(41\) 1822.21 1.08400 0.542001 − 0.840378i \(-0.317667\pi\)
0.542001 + 0.840378i \(0.317667\pi\)
\(42\) 0 0
\(43\) −2372.12 − 2372.12i −1.28292 − 1.28292i −0.938998 − 0.343921i \(-0.888245\pi\)
−0.343921 − 0.938998i \(-0.611755\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2458.79 − 2458.79i 1.11308 − 1.11308i 0.120344 − 0.992732i \(-0.461600\pi\)
0.992732 − 0.120344i \(-0.0383997\pi\)
\(48\) 0 0
\(49\) 1042.58i 0.434227i
\(50\) 0 0
\(51\) 149.338 0.0574157
\(52\) 0 0
\(53\) −2118.26 − 2118.26i −0.754097 − 0.754097i 0.221144 − 0.975241i \(-0.429021\pi\)
−0.975241 + 0.221144i \(0.929021\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −613.748 + 613.748i −0.188904 + 0.188904i
\(58\) 0 0
\(59\) − 5251.96i − 1.50875i −0.656443 − 0.754376i \(-0.727940\pi\)
0.656443 − 0.754376i \(-0.272060\pi\)
\(60\) 0 0
\(61\) −1312.89 −0.352831 −0.176416 − 0.984316i \(-0.556450\pi\)
−0.176416 + 0.984316i \(0.556450\pi\)
\(62\) 0 0
\(63\) −2063.13 − 2063.13i −0.519810 − 0.519810i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5569.35 + 5569.35i −1.24067 + 1.24067i −0.280940 + 0.959725i \(0.590646\pi\)
−0.959725 + 0.280940i \(0.909354\pi\)
\(68\) 0 0
\(69\) − 224.503i − 0.0471545i
\(70\) 0 0
\(71\) −7554.66 −1.49864 −0.749322 − 0.662206i \(-0.769620\pi\)
−0.749322 + 0.662206i \(0.769620\pi\)
\(72\) 0 0
\(73\) −1206.35 − 1206.35i −0.226374 − 0.226374i 0.584802 − 0.811176i \(-0.301172\pi\)
−0.811176 + 0.584802i \(0.801172\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1979.55 + 1979.55i −0.333877 + 0.333877i
\(78\) 0 0
\(79\) − 3397.97i − 0.544460i −0.962232 − 0.272230i \(-0.912239\pi\)
0.962232 − 0.272230i \(-0.0877612\pi\)
\(80\) 0 0
\(81\) −6118.05 −0.932487
\(82\) 0 0
\(83\) 928.421 + 928.421i 0.134769 + 0.134769i 0.771273 − 0.636504i \(-0.219620\pi\)
−0.636504 + 0.771273i \(0.719620\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 721.164 − 721.164i 0.0952787 − 0.0952787i
\(88\) 0 0
\(89\) − 11981.9i − 1.51268i −0.654180 − 0.756339i \(-0.726986\pi\)
0.654180 − 0.756339i \(-0.273014\pi\)
\(90\) 0 0
\(91\) 5167.65 0.624037
\(92\) 0 0
\(93\) 678.683 + 678.683i 0.0784695 + 0.0784695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4683.76 + 4683.76i −0.497796 + 0.497796i −0.910751 − 0.412955i \(-0.864497\pi\)
0.412955 + 0.910751i \(0.364497\pi\)
\(98\) 0 0
\(99\) 6012.97i 0.613506i
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 400.5.p.o.193.2 6
4.3 odd 2 200.5.l.e.193.2 6
5.2 odd 4 inner 400.5.p.o.257.2 6
5.3 odd 4 80.5.p.g.17.2 6
5.4 even 2 80.5.p.g.33.2 6
20.3 even 4 40.5.l.c.17.2 ✓ 6
20.7 even 4 200.5.l.e.57.2 6
20.19 odd 2 40.5.l.c.33.2 yes 6
40.3 even 4 320.5.p.p.257.2 6
40.13 odd 4 320.5.p.o.257.2 6
40.19 odd 2 320.5.p.p.193.2 6
40.29 even 2 320.5.p.o.193.2 6
60.23 odd 4 360.5.v.c.217.2 6
60.59 even 2 360.5.v.c.73.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.2 ✓ 6 20.3 even 4
40.5.l.c.33.2 yes 6 20.19 odd 2
80.5.p.g.17.2 6 5.3 odd 4
80.5.p.g.33.2 6 5.4 even 2
200.5.l.e.57.2 6 20.7 even 4
200.5.l.e.193.2 6 4.3 odd 2
320.5.p.o.193.2 6 40.29 even 2
320.5.p.o.257.2 6 40.13 odd 4
320.5.p.p.193.2 6 40.19 odd 2
320.5.p.p.257.2 6 40.3 even 4
360.5.v.c.73.2 6 60.59 even 2
360.5.v.c.217.2 6 60.23 odd 4
400.5.p.o.193.2 6 1.1 even 1 trivial
400.5.p.o.257.2 6 5.2 odd 4 inner