Show commands: Magma / Pari/GP / SageMath

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 = [2,0,2,0,0,0,-38,0,0,0,-404] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.193
Dual form 400.5.p.b.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.00000i) q^{3} +(-19.0000 + 19.0000i) q^{7} -79.0000i q^{9} -202.000 q^{11} +(99.0000 + 99.0000i) q^{13} +(239.000 - 239.000i) q^{17} +40.0000i q^{19} -38.0000 q^{21} +(541.000 + 541.000i) q^{23} +(160.000 - 160.000i) q^{27} +200.000i q^{29} +758.000 q^{31} +(-202.000 - 202.000i) q^{33} +(-141.000 + 141.000i) q^{37} +198.000i q^{39} +1042.00 q^{41} +(-759.000 - 759.000i) q^{43} +(-459.000 + 459.000i) q^{47} +1679.00i q^{49} +478.000 q^{51} +(1819.00 + 1819.00i) q^{53} +(-40.0000 + 40.0000i) q^{57} +4600.00i q^{59} +2082.00 q^{61} +(1501.00 + 1501.00i) q^{63} +(5081.00 - 5081.00i) q^{67} +1082.00i q^{69} +3478.00 q^{71} +(3479.00 + 3479.00i) q^{73} +(3838.00 - 3838.00i) q^{77} -7680.00i q^{79} -6079.00 q^{81} +(6081.00 + 6081.00i) q^{83} +(-200.000 + 200.000i) q^{87} +5680.00i q^{89} -3762.00 q^{91} +(758.000 + 758.000i) q^{93} +(-561.000 + 561.000i) q^{97} +15958.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 38 q^{7} - 404 q^{11} + 198 q^{13} + 478 q^{17} - 76 q^{21} + 1082 q^{23} + 320 q^{27} + 1516 q^{31} - 404 q^{33} - 282 q^{37} + 2084 q^{41} - 1518 q^{43} - 918 q^{47} + 956 q^{51} + 3638 q^{53}+ \cdots - 1122 q^{97}+O(q^{100}) \) 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
\(3\) 1.00000 + 1.00000i 0.111111 + 0.111111i 0.760477 − 0.649365i \(-0.224965\pi\)
−0.649365 + 0.760477i \(0.724965\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −19.0000 + 19.0000i −0.387755 + 0.387755i −0.873886 − 0.486131i \(-0.838408\pi\)
0.486131 + 0.873886i \(0.338408\pi\)
\(8\) 0 0
\(9\) − 79.0000i − 0.975309i
\(10\) 0 0
\(11\) −202.000 −1.66942 −0.834711 − 0.550689i \(-0.814365\pi\)
−0.834711 + 0.550689i \(0.814365\pi\)
\(12\) 0 0
\(13\) 99.0000 + 99.0000i 0.585799 + 0.585799i 0.936491 − 0.350692i \(-0.114054\pi\)
−0.350692 + 0.936491i \(0.614054\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 239.000 − 239.000i 0.826990 − 0.826990i −0.160110 − 0.987099i \(-0.551185\pi\)
0.987099 + 0.160110i \(0.0511848\pi\)
\(18\) 0 0
\(19\) 40.0000i 0.110803i 0.998464 + 0.0554017i \(0.0176439\pi\)
−0.998464 + 0.0554017i \(0.982356\pi\)
\(20\) 0 0
\(21\) −38.0000 −0.0861678
\(22\) 0 0
\(23\) 541.000 + 541.000i 1.02268 + 1.02268i 0.999737 + 0.0229476i \(0.00730510\pi\)
0.0229476 + 0.999737i \(0.492695\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 160.000 − 160.000i 0.219479 − 0.219479i
\(28\) 0 0
\(29\) 200.000i 0.237812i 0.992906 + 0.118906i \(0.0379387\pi\)
−0.992906 + 0.118906i \(0.962061\pi\)
\(30\) 0 0
\(31\) 758.000 0.788762 0.394381 − 0.918947i \(-0.370959\pi\)
0.394381 + 0.918947i \(0.370959\pi\)
\(32\) 0 0
\(33\) −202.000 − 202.000i −0.185491 − 0.185491i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −141.000 + 141.000i −0.102995 + 0.102995i −0.756726 − 0.653732i \(-0.773203\pi\)
0.653732 + 0.756726i \(0.273203\pi\)
\(38\) 0 0
\(39\) 198.000i 0.130178i
\(40\) 0 0
\(41\) 1042.00 0.619869 0.309935 − 0.950758i \(-0.399693\pi\)
0.309935 + 0.950758i \(0.399693\pi\)
\(42\) 0 0
\(43\) −759.000 − 759.000i −0.410492 − 0.410492i 0.471418 − 0.881910i \(-0.343742\pi\)
−0.881910 + 0.471418i \(0.843742\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −459.000 + 459.000i −0.207786 + 0.207786i −0.803326 − 0.595540i \(-0.796938\pi\)
0.595540 + 0.803326i \(0.296938\pi\)
\(48\) 0 0
\(49\) 1679.00i 0.699292i
\(50\) 0 0
\(51\) 478.000 0.183775
\(52\) 0 0
\(53\) 1819.00 + 1819.00i 0.647561 + 0.647561i 0.952403 − 0.304842i \(-0.0986035\pi\)
−0.304842 + 0.952403i \(0.598604\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −40.0000 + 40.0000i −0.0123115 + 0.0123115i
\(58\) 0 0
\(59\) 4600.00i 1.32146i 0.750624 + 0.660730i \(0.229753\pi\)
−0.750624 + 0.660730i \(0.770247\pi\)
\(60\) 0 0
\(61\) 2082.00 0.559527 0.279764 − 0.960069i \(-0.409744\pi\)
0.279764 + 0.960069i \(0.409744\pi\)
\(62\) 0 0
\(63\) 1501.00 + 1501.00i 0.378181 + 0.378181i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5081.00 − 5081.00i 1.13188 − 1.13188i 0.142013 − 0.989865i \(-0.454642\pi\)
0.989865 − 0.142013i \(-0.0453575\pi\)
\(68\) 0 0
\(69\) 1082.00i 0.227263i
\(70\) 0 0
\(71\) 3478.00 0.689942 0.344971 − 0.938613i \(-0.387889\pi\)
0.344971 + 0.938613i \(0.387889\pi\)
\(72\) 0 0
\(73\) 3479.00 + 3479.00i 0.652843 + 0.652843i 0.953677 − 0.300834i \(-0.0972649\pi\)
−0.300834 + 0.953677i \(0.597265\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3838.00 − 3838.00i 0.647327 − 0.647327i
\(78\) 0 0
\(79\) − 7680.00i − 1.23057i −0.788304 − 0.615286i \(-0.789041\pi\)
0.788304 − 0.615286i \(-0.210959\pi\)
\(80\) 0 0
\(81\) −6079.00 −0.926536
\(82\) 0 0
\(83\) 6081.00 + 6081.00i 0.882712 + 0.882712i 0.993809 − 0.111098i \(-0.0354367\pi\)
−0.111098 + 0.993809i \(0.535437\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −200.000 + 200.000i −0.0264236 + 0.0264236i
\(88\) 0 0
\(89\) 5680.00i 0.717081i 0.933514 + 0.358541i \(0.116726\pi\)
−0.933514 + 0.358541i \(0.883274\pi\)
\(90\) 0 0
\(91\) −3762.00 −0.454293
\(92\) 0 0
\(93\) 758.000 + 758.000i 0.0876402 + 0.0876402i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −561.000 + 561.000i −0.0596238 + 0.0596238i −0.736290 − 0.676666i \(-0.763424\pi\)
0.676666 + 0.736290i \(0.263424\pi\)
\(98\) 0 0
\(99\) 15958.0i 1.62820i
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.b.193.1 2
4.3 odd 2 50.5.c.a.43.1 2
5.2 odd 4 inner 400.5.p.b.257.1 2
5.3 odd 4 80.5.p.c.17.1 2
5.4 even 2 80.5.p.c.33.1 2
12.11 even 2 450.5.g.b.343.1 2
20.3 even 4 10.5.c.b.7.1 yes 2
20.7 even 4 50.5.c.a.7.1 2
20.19 odd 2 10.5.c.b.3.1 ✓ 2
40.3 even 4 320.5.p.d.257.1 2
40.13 odd 4 320.5.p.g.257.1 2
40.19 odd 2 320.5.p.d.193.1 2
40.29 even 2 320.5.p.g.193.1 2
60.23 odd 4 90.5.g.a.37.1 2
60.47 odd 4 450.5.g.b.307.1 2
60.59 even 2 90.5.g.a.73.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.5.c.b.3.1 ✓ 2 20.19 odd 2
10.5.c.b.7.1 yes 2 20.3 even 4
50.5.c.a.7.1 2 20.7 even 4
50.5.c.a.43.1 2 4.3 odd 2
80.5.p.c.17.1 2 5.3 odd 4
80.5.p.c.33.1 2 5.4 even 2
90.5.g.a.37.1 2 60.23 odd 4
90.5.g.a.73.1 2 60.59 even 2
320.5.p.d.193.1 2 40.19 odd 2
320.5.p.d.257.1 2 40.3 even 4
320.5.p.g.193.1 2 40.29 even 2
320.5.p.g.257.1 2 40.13 odd 4
400.5.p.b.193.1 2 1.1 even 1 trivial
400.5.p.b.257.1 2 5.2 odd 4 inner
450.5.g.b.307.1 2 60.47 odd 4
450.5.g.b.343.1 2 12.11 even 2