Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [28,5,Mod(11,28)] 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("28.11"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 28.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89435896635\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.3
Character \(\chi\) \(=\) 28.11
Dual form 28.5.g.a.23.3

$q$-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+(-3.31594 + 2.23708i) q^{2} +(12.8404 + 7.41339i) q^{3} +(5.99095 - 14.8361i) q^{4} +(0.212899 + 0.368751i) q^{5} +(-59.1623 + 4.14255i) q^{6} +(13.4191 + 47.1267i) q^{7} +(13.3238 + 62.5977i) q^{8} +(69.4168 + 120.233i) q^{9} +(-1.53089 - 0.746487i) q^{10} +(-153.531 - 88.6413i) q^{11} +(186.912 - 146.087i) q^{12} +109.630 q^{13} +(-149.923 - 126.250i) q^{14} +6.31321i q^{15} +(-184.217 - 177.764i) q^{16} +(108.710 - 188.291i) q^{17} +(-499.154 - 243.396i) q^{18} +(378.104 - 218.298i) q^{19} +(6.74628 - 0.949405i) q^{20} +(-177.063 + 704.606i) q^{21} +(707.399 - 49.5321i) q^{22} +(-53.8891 + 31.1129i) q^{23} +(-292.979 + 902.553i) q^{24} +(312.409 - 541.109i) q^{25} +(-363.527 + 245.251i) q^{26} +857.486i q^{27} +(779.568 + 83.2476i) q^{28} -913.437 q^{29} +(-14.1231 - 20.9342i) q^{30} +(-857.865 - 495.289i) q^{31} +(1008.53 + 177.348i) q^{32} +(-1314.27 - 2276.38i) q^{33} +(60.7461 + 867.553i) q^{34} +(-14.5211 + 14.9815i) q^{35} +(2199.66 - 309.559i) q^{36} +(694.103 + 1202.22i) q^{37} +(-765.420 + 1569.71i) q^{38} +(1407.69 + 812.730i) q^{39} +(-20.2464 + 18.2401i) q^{40} +727.721 q^{41} +(-989.129 - 2732.54i) q^{42} -1877.25i q^{43} +(-2234.89 + 1746.75i) q^{44} +(-29.5575 + 51.1951i) q^{45} +(109.091 - 223.723i) q^{46} +(-1479.49 + 854.183i) q^{47} +(-1047.58 - 3648.23i) q^{48} +(-2040.86 + 1264.80i) q^{49} +(174.572 + 2493.17i) q^{50} +(2791.74 - 1611.81i) q^{51} +(656.788 - 1626.48i) q^{52} +(63.7173 - 110.362i) q^{53} +(-1918.26 - 2843.37i) q^{54} -75.4865i q^{55} +(-2771.23 + 1467.91i) q^{56} +6473.32 q^{57} +(3028.91 - 2043.43i) q^{58} +(-2843.35 - 1641.61i) q^{59} +(93.6631 + 37.8221i) q^{60} +(1263.71 + 2188.82i) q^{61} +(3952.63 - 276.764i) q^{62} +(-4734.70 + 4884.81i) q^{63} +(-3740.95 + 1668.08i) q^{64} +(23.3401 + 40.4262i) q^{65} +(9450.47 + 4608.21i) q^{66} +(1102.52 + 636.540i) q^{67} +(-2142.22 - 2740.86i) q^{68} -922.608 q^{69} +(14.6364 - 82.1628i) q^{70} +5085.20i q^{71} +(-6601.45 + 5947.30i) q^{72} +(-4237.85 + 7340.17i) q^{73} +(-4991.07 - 2433.74i) q^{74} +(8022.91 - 4632.03i) q^{75} +(-973.484 - 6917.38i) q^{76} +(2117.13 - 8424.91i) q^{77} +(-6485.96 + 454.147i) q^{78} +(6221.39 - 3591.92i) q^{79} +(26.3312 - 105.776i) q^{80} +(-734.120 + 1271.53i) q^{81} +(-2413.08 + 1627.97i) q^{82} +5786.37i q^{83} +(9392.79 + 6848.17i) q^{84} +92.5766 q^{85} +(4199.55 + 6224.85i) q^{86} +(-11728.9 - 6771.67i) q^{87} +(3503.13 - 10791.8i) q^{88} +(777.594 + 1346.83i) q^{89} +(-16.5165 - 235.882i) q^{90} +(1471.13 + 5166.50i) q^{91} +(138.745 + 985.897i) q^{92} +(-7343.54 - 12719.4i) q^{93} +(2995.02 - 6142.15i) q^{94} +(160.996 + 92.9509i) q^{95} +(11635.1 + 9753.81i) q^{96} -7310.51 q^{97} +(3937.91 - 8759.55i) q^{98} -24612.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 2 q^{2} - 4 q^{4} - 2 q^{5} - 36 q^{6} - 184 q^{8} + 268 q^{9} + 130 q^{10} - 48 q^{12} + 344 q^{13} - 474 q^{14} - 432 q^{16} - 2 q^{17} - 568 q^{18} + 664 q^{20} + 426 q^{21} + 196 q^{22} - 692 q^{24}+ \cdots - 9682 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.31594 + 2.23708i −0.828986 + 0.559270i
\(3\) 12.8404 + 7.41339i 1.42671 + 0.823710i 0.996859 − 0.0791907i \(-0.0252336\pi\)
0.429849 + 0.902901i \(0.358567\pi\)
\(4\) 5.99095 − 14.8361i 0.374434 − 0.927253i
\(5\) 0.212899 + 0.368751i 0.00851595 + 0.0147501i 0.870252 − 0.492607i \(-0.163956\pi\)
−0.861736 + 0.507357i \(0.830623\pi\)
\(6\) −59.1623 + 4.14255i −1.64340 + 0.115071i
\(7\) 13.4191 + 47.1267i 0.273859 + 0.961770i
\(8\) 13.3238 + 62.5977i 0.208184 + 0.978090i
\(9\) 69.4168 + 120.233i 0.856997 + 1.48436i
\(10\) −1.53089 − 0.746487i −0.0153089 − 0.00746487i
\(11\) −153.531 − 88.6413i −1.26885 − 0.732573i −0.294083 − 0.955780i \(-0.595014\pi\)
−0.974771 + 0.223207i \(0.928347\pi\)
\(12\) 186.912 − 146.087i 1.29800 − 1.01449i
\(13\) 109.630 0.648698 0.324349 − 0.945937i \(-0.394855\pi\)
0.324349 + 0.945937i \(0.394855\pi\)
\(14\) −149.923 − 126.250i −0.764914 − 0.644132i
\(15\) 6.31321i 0.0280587i
\(16\) −184.217 − 177.764i −0.719598 − 0.694391i
\(17\) 108.710 − 188.291i 0.376158 − 0.651525i −0.614342 − 0.789040i \(-0.710578\pi\)
0.990500 + 0.137516i \(0.0439117\pi\)
\(18\) −499.154 − 243.396i −1.54060 − 0.751223i
\(19\) 378.104 − 218.298i 1.04738 − 0.604704i 0.125464 − 0.992098i \(-0.459958\pi\)
0.921914 + 0.387394i \(0.126625\pi\)
\(20\) 6.74628 − 0.949405i 0.0168657 − 0.00237351i
\(21\) −177.063 + 704.606i −0.401503 + 1.59775i
\(22\) 707.399 − 49.5321i 1.46157 − 0.102339i
\(23\) −53.8891 + 31.1129i −0.101870 + 0.0588145i −0.550069 − 0.835119i \(-0.685399\pi\)
0.448200 + 0.893934i \(0.352065\pi\)
\(24\) −292.979 + 902.553i −0.508645 + 1.56693i
\(25\) 312.409 − 541.109i 0.499855 − 0.865774i
\(26\) −363.527 + 245.251i −0.537761 + 0.362797i
\(27\) 857.486i 1.17625i
\(28\) 779.568 + 83.2476i 0.994347 + 0.106183i
\(29\) −913.437 −1.08613 −0.543066 − 0.839690i \(-0.682737\pi\)
−0.543066 + 0.839690i \(0.682737\pi\)
\(30\) −14.1231 − 20.9342i −0.0156924 − 0.0232603i
\(31\) −857.865 − 495.289i −0.892679 − 0.515389i −0.0178615 − 0.999840i \(-0.505686\pi\)
−0.874818 + 0.484452i \(0.839019\pi\)
\(32\) 1008.53 + 177.348i 0.984888 + 0.173191i
\(33\) −1314.27 − 2276.38i −1.20686 − 2.09034i
\(34\) 60.7461 + 867.553i 0.0525485 + 0.750478i
\(35\) −14.5211 + 14.9815i −0.0118540 + 0.0122298i
\(36\) 2199.66 − 309.559i 1.69727 − 0.238857i
\(37\) 694.103 + 1202.22i 0.507015 + 0.878176i 0.999967 + 0.00811920i \(0.00258445\pi\)
−0.492952 + 0.870056i \(0.664082\pi\)
\(38\) −765.420 + 1569.71i −0.530069 + 1.08706i
\(39\) 1407.69 + 812.730i 0.925503 + 0.534339i
\(40\) −20.2464 + 18.2401i −0.0126540 + 0.0114001i
\(41\) 727.721 0.432910 0.216455 − 0.976293i \(-0.430551\pi\)
0.216455 + 0.976293i \(0.430551\pi\)
\(42\) −989.129 − 2732.54i −0.560731 − 1.54906i
\(43\) − 1877.25i − 1.01528i −0.861570 − 0.507639i \(-0.830518\pi\)
0.861570 − 0.507639i \(-0.169482\pi\)
\(44\) −2234.89 + 1746.75i −1.15438 + 0.902248i
\(45\) −29.5575 + 51.1951i −0.0145963 + 0.0252815i
\(46\) 109.091 − 223.723i 0.0515554 − 0.105729i
\(47\) −1479.49 + 854.183i −0.669755 + 0.386683i −0.795984 − 0.605318i \(-0.793046\pi\)
0.126229 + 0.992001i \(0.459713\pi\)
\(48\) −1047.58 − 3648.23i −0.454679 − 1.58343i
\(49\) −2040.86 + 1264.80i −0.850003 + 0.526779i
\(50\) 174.572 + 2493.17i 0.0698288 + 0.997268i
\(51\) 2791.74 − 1611.81i 1.07333 − 0.619690i
\(52\) 656.788 − 1626.48i 0.242895 − 0.601508i
\(53\) 63.7173 − 110.362i 0.0226833 − 0.0392886i −0.854461 − 0.519516i \(-0.826112\pi\)
0.877144 + 0.480227i \(0.159446\pi\)
\(54\) −1918.26 − 2843.37i −0.657841 − 0.975094i
\(55\) − 75.4865i − 0.0249542i
\(56\) −2771.23 + 1467.91i −0.883684 + 0.468084i
\(57\) 6473.32 1.99240
\(58\) 3028.91 − 2043.43i 0.900388 − 0.607441i
\(59\) −2843.35 − 1641.61i −0.816821 − 0.471592i 0.0324982 − 0.999472i \(-0.489654\pi\)
−0.849319 + 0.527880i \(0.822987\pi\)
\(60\) 93.6631 + 37.8221i 0.0260175 + 0.0105061i
\(61\) 1263.71 + 2188.82i 0.339617 + 0.588234i 0.984361 − 0.176165i \(-0.0563693\pi\)
−0.644744 + 0.764399i \(0.723036\pi\)
\(62\) 3952.63 − 276.764i 1.02826 − 0.0719988i
\(63\) −4734.70 + 4884.81i −1.19292 + 1.23074i
\(64\) −3740.95 + 1668.08i −0.913319 + 0.407245i
\(65\) 23.3401 + 40.4262i 0.00552428 + 0.00956834i
\(66\) 9450.47 + 4608.21i 2.16953 + 1.05790i
\(67\) 1102.52 + 636.540i 0.245605 + 0.141800i 0.617750 − 0.786375i \(-0.288044\pi\)
−0.372145 + 0.928175i \(0.621378\pi\)
\(68\) −2142.22 − 2740.86i −0.463282 − 0.592747i
\(69\) −922.608 −0.193784
\(70\) 14.6364 − 82.1628i 0.00298702 − 0.0167679i
\(71\) 5085.20i 1.00877i 0.863479 + 0.504384i \(0.168281\pi\)
−0.863479 + 0.504384i \(0.831719\pi\)
\(72\) −6601.45 + 5947.30i −1.27343 + 1.14724i
\(73\) −4237.85 + 7340.17i −0.795243 + 1.37740i 0.127441 + 0.991846i \(0.459324\pi\)
−0.922685 + 0.385556i \(0.874010\pi\)
\(74\) −4991.07 − 2433.74i −0.911445 − 0.444437i
\(75\) 8022.91 − 4632.03i 1.42629 − 0.823471i
\(76\) −973.484 − 6917.38i −0.168539 − 1.19761i
\(77\) 2117.13 − 8424.91i 0.357080 − 1.42097i
\(78\) −6485.96 + 454.147i −1.06607 + 0.0746462i
\(79\) 6221.39 − 3591.92i 0.996858 − 0.575536i 0.0895407 − 0.995983i \(-0.471460\pi\)
0.907317 + 0.420447i \(0.138127\pi\)
\(80\) 26.3312 − 105.776i 0.00411425 − 0.0165275i
\(81\) −734.120 + 1271.53i −0.111892 + 0.193802i
\(82\) −2413.08 + 1627.97i −0.358876 + 0.242113i
\(83\) 5786.37i 0.839943i 0.907537 + 0.419971i \(0.137960\pi\)
−0.907537 + 0.419971i \(0.862040\pi\)
\(84\) 9392.79 + 6848.17i 1.33118 + 0.970546i
\(85\) 92.5766 0.0128134
\(86\) 4199.55 + 6224.85i 0.567814 + 0.841651i
\(87\) −11728.9 − 6771.67i −1.54959 − 0.894658i
\(88\) 3503.13 − 10791.8i 0.452367 − 1.39356i
\(89\) 777.594 + 1346.83i 0.0981687 + 0.170033i 0.910927 − 0.412568i \(-0.135368\pi\)
−0.812758 + 0.582601i \(0.802035\pi\)
\(90\) −16.5165 − 235.882i −0.00203907 − 0.0291213i
\(91\) 1471.13 + 5166.50i 0.177652 + 0.623898i
\(92\) 138.745 + 985.897i 0.0163924 + 0.116481i
\(93\) −7343.54 − 12719.4i −0.849062 − 1.47062i
\(94\) 2995.02 − 6142.15i 0.338957 − 0.695128i
\(95\) 160.996 + 92.9509i 0.0178388 + 0.0102993i
\(96\) 11635.1 + 9753.81i 1.26249 + 1.05836i
\(97\) −7310.51 −0.776970 −0.388485 − 0.921455i \(-0.627001\pi\)
−0.388485 + 0.921455i \(0.627001\pi\)
\(98\) 3937.91 − 8759.55i 0.410029 − 0.912073i
\(99\) − 24612.8i − 2.51125i
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 28.5.g.a.11.3 ✓ 28
4.3 odd 2 inner 28.5.g.a.11.13 yes 28
7.2 even 3 inner 28.5.g.a.23.13 yes 28
7.3 odd 6 196.5.c.g.99.7 14
7.4 even 3 196.5.c.h.99.7 14
28.3 even 6 196.5.c.g.99.8 14
28.11 odd 6 196.5.c.h.99.8 14
28.23 odd 6 inner 28.5.g.a.23.3 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.g.a.11.3 ✓ 28 1.1 even 1 trivial
28.5.g.a.11.13 yes 28 4.3 odd 2 inner
28.5.g.a.23.3 yes 28 28.23 odd 6 inner
28.5.g.a.23.13 yes 28 7.2 even 3 inner
196.5.c.g.99.7 14 7.3 odd 6
196.5.c.g.99.8 14 28.3 even 6
196.5.c.h.99.7 14 7.4 even 3
196.5.c.h.99.8 14 28.11 odd 6