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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] 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("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (of order \(4\), 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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.7
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.7

$q$-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+(-3.74845 + 1.39612i) q^{2} +(-8.86492 - 8.86492i) q^{3} +(12.1017 - 10.4665i) q^{4} +(8.20272 + 8.20272i) q^{5} +(45.6061 + 20.8532i) q^{6} +18.5203 q^{7} +(-30.7502 + 56.1287i) q^{8} +76.1735i q^{9} +(-42.1994 - 19.2955i) q^{10} +(-163.942 + 163.942i) q^{11} +(-200.066 - 14.4958i) q^{12} +(109.488 - 109.488i) q^{13} +(-69.4222 + 25.8564i) q^{14} -145.433i q^{15} +(36.9033 - 253.326i) q^{16} +308.420 q^{17} +(-106.347 - 285.532i) q^{18} +(297.428 + 297.428i) q^{19} +(185.121 + 13.4130i) q^{20} +(-164.181 - 164.181i) q^{21} +(385.647 - 843.412i) q^{22} -98.5873 q^{23} +(770.174 - 224.978i) q^{24} -490.431i q^{25} +(-257.552 + 563.267i) q^{26} +(-42.7864 + 42.7864i) q^{27} +(224.127 - 193.843i) q^{28} +(719.430 - 719.430i) q^{29} +(203.041 + 545.147i) q^{30} -1034.35i q^{31} +(215.343 + 1001.10i) q^{32} +2906.67 q^{33} +(-1156.10 + 430.590i) q^{34} +(151.916 + 151.916i) q^{35} +(797.273 + 921.831i) q^{36} +(1169.53 + 1169.53i) q^{37} +(-1530.14 - 699.648i) q^{38} -1941.20 q^{39} +(-712.642 + 208.173i) q^{40} -2038.00i q^{41} +(844.637 + 386.207i) q^{42} +(1723.35 - 1723.35i) q^{43} +(-268.076 + 3699.89i) q^{44} +(-624.830 + 624.830i) q^{45} +(369.550 - 137.639i) q^{46} -483.752i q^{47} +(-2572.86 + 1918.57i) q^{48} +343.000 q^{49} +(684.698 + 1838.35i) q^{50} +(-2734.12 - 2734.12i) q^{51} +(179.033 - 2470.95i) q^{52} +(-385.642 - 385.642i) q^{53} +(100.648 - 220.118i) q^{54} -2689.55 q^{55} +(-569.501 + 1039.52i) q^{56} -5273.34i q^{57} +(-1692.34 + 3701.15i) q^{58} +(-280.046 + 280.046i) q^{59} +(-1522.18 - 1759.99i) q^{60} +(-1593.12 + 1593.12i) q^{61} +(1444.08 + 3877.22i) q^{62} +1410.75i q^{63} +(-2204.85 - 3451.93i) q^{64} +1796.19 q^{65} +(-10895.5 + 4058.05i) q^{66} +(2409.95 + 2409.95i) q^{67} +(3732.41 - 3228.09i) q^{68} +(873.969 + 873.969i) q^{69} +(-781.544 - 357.358i) q^{70} +4500.32 q^{71} +(-4275.52 - 2342.35i) q^{72} +3614.24i q^{73} +(-6016.74 - 2751.13i) q^{74} +(-4347.63 + 4347.63i) q^{75} +(6712.42 + 486.349i) q^{76} +(-3036.26 + 3036.26i) q^{77} +(7276.49 - 2710.14i) q^{78} -12253.3i q^{79} +(2380.67 - 1775.26i) q^{80} +6928.65 q^{81} +(2845.28 + 7639.33i) q^{82} +(6206.38 + 6206.38i) q^{83} +(-3705.27 - 268.466i) q^{84} +(2529.88 + 2529.88i) q^{85} +(-4053.90 + 8865.89i) q^{86} -12755.4 q^{87} +(-4160.61 - 14243.1i) q^{88} +6811.06i q^{89} +(1469.81 - 3214.48i) q^{90} +(2027.74 - 2027.74i) q^{91} +(-1193.08 + 1031.87i) q^{92} +(-9169.46 + 9169.46i) q^{93} +(675.374 + 1813.32i) q^{94} +4879.43i q^{95} +(6965.68 - 10783.7i) q^{96} -5780.13 q^{97} +(-1285.72 + 478.868i) q^{98} +(-12488.1 - 12488.1i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.74845 + 1.39612i −0.937112 + 0.349029i
\(3\) −8.86492 − 8.86492i −0.984991 − 0.984991i 0.0148982 − 0.999889i \(-0.495258\pi\)
−0.999889 + 0.0148982i \(0.995258\pi\)
\(4\) 12.1017 − 10.4665i 0.756358 − 0.654158i
\(5\) 8.20272 + 8.20272i 0.328109 + 0.328109i 0.851867 − 0.523758i \(-0.175470\pi\)
−0.523758 + 0.851867i \(0.675470\pi\)
\(6\) 45.6061 + 20.8532i 1.26684 + 0.579256i
\(7\) 18.5203 0.377964
\(8\) −30.7502 + 56.1287i −0.480471 + 0.877010i
\(9\) 76.1735i 0.940414i
\(10\) −42.1994 − 19.2955i −0.421994 − 0.192955i
\(11\) −163.942 + 163.942i −1.35490 + 1.35490i −0.474804 + 0.880091i \(0.657481\pi\)
−0.880091 + 0.474804i \(0.842519\pi\)
\(12\) −200.066 − 14.4958i −1.38935 − 0.100665i
\(13\) 109.488 − 109.488i 0.647857 − 0.647857i −0.304618 − 0.952475i \(-0.598529\pi\)
0.952475 + 0.304618i \(0.0985288\pi\)
\(14\) −69.4222 + 25.8564i −0.354195 + 0.131921i
\(15\) − 145.433i − 0.646368i
\(16\) 36.9033 − 253.326i 0.144153 − 0.989555i
\(17\) 308.420 1.06720 0.533598 − 0.845738i \(-0.320839\pi\)
0.533598 + 0.845738i \(0.320839\pi\)
\(18\) −106.347 − 285.532i −0.328232 − 0.881273i
\(19\) 297.428 + 297.428i 0.823899 + 0.823899i 0.986665 − 0.162766i \(-0.0520414\pi\)
−0.162766 + 0.986665i \(0.552041\pi\)
\(20\) 185.121 + 13.4130i 0.462802 + 0.0335324i
\(21\) −164.181 − 164.181i −0.372292 − 0.372292i
\(22\) 385.647 − 843.412i 0.796791 − 1.74259i
\(23\) −98.5873 −0.186365 −0.0931827 − 0.995649i \(-0.529704\pi\)
−0.0931827 + 0.995649i \(0.529704\pi\)
\(24\) 770.174 − 224.978i 1.33711 − 0.390587i
\(25\) − 490.431i − 0.784689i
\(26\) −257.552 + 563.267i −0.380993 + 0.833235i
\(27\) −42.7864 + 42.7864i −0.0586919 + 0.0586919i
\(28\) 224.127 − 193.843i 0.285876 − 0.247249i
\(29\) 719.430 − 719.430i 0.855446 − 0.855446i −0.135351 − 0.990798i \(-0.543216\pi\)
0.990798 + 0.135351i \(0.0432164\pi\)
\(30\) 203.041 + 545.147i 0.225601 + 0.605719i
\(31\) − 1034.35i − 1.07633i −0.842839 − 0.538165i \(-0.819118\pi\)
0.842839 − 0.538165i \(-0.180882\pi\)
\(32\) 215.343 + 1001.10i 0.210296 + 0.977638i
\(33\) 2906.67 2.66912
\(34\) −1156.10 + 430.590i −1.00008 + 0.372483i
\(35\) 151.916 + 151.916i 0.124013 + 0.124013i
\(36\) 797.273 + 921.831i 0.615180 + 0.711289i
\(37\) 1169.53 + 1169.53i 0.854298 + 0.854298i 0.990659 − 0.136361i \(-0.0435409\pi\)
−0.136361 + 0.990659i \(0.543541\pi\)
\(38\) −1530.14 − 699.648i −1.05965 − 0.484521i
\(39\) −1941.20 −1.27627
\(40\) −712.642 + 208.173i −0.445401 + 0.130108i
\(41\) − 2038.00i − 1.21237i −0.795323 − 0.606186i \(-0.792699\pi\)
0.795323 − 0.606186i \(-0.207301\pi\)
\(42\) 844.637 + 386.207i 0.478819 + 0.218938i
\(43\) 1723.35 − 1723.35i 0.932046 − 0.932046i −0.0657881 − 0.997834i \(-0.520956\pi\)
0.997834 + 0.0657881i \(0.0209561\pi\)
\(44\) −268.076 + 3699.89i −0.138469 + 1.91110i
\(45\) −624.830 + 624.830i −0.308558 + 0.308558i
\(46\) 369.550 − 137.639i 0.174645 − 0.0650470i
\(47\) − 483.752i − 0.218991i −0.993987 − 0.109496i \(-0.965076\pi\)
0.993987 − 0.109496i \(-0.0349236\pi\)
\(48\) −2572.86 + 1918.57i −1.11669 + 0.832713i
\(49\) 343.000 0.142857
\(50\) 684.698 + 1838.35i 0.273879 + 0.735342i
\(51\) −2734.12 − 2734.12i −1.05118 − 1.05118i
\(52\) 179.033 − 2470.95i 0.0662103 − 0.913812i
\(53\) −385.642 − 385.642i −0.137288 − 0.137288i 0.635123 − 0.772411i \(-0.280949\pi\)
−0.772411 + 0.635123i \(0.780949\pi\)
\(54\) 100.648 − 220.118i 0.0345157 − 0.0754861i
\(55\) −2689.55 −0.889106
\(56\) −569.501 + 1039.52i −0.181601 + 0.331479i
\(57\) − 5273.34i − 1.62307i
\(58\) −1692.34 + 3701.15i −0.503073 + 1.10022i
\(59\) −280.046 + 280.046i −0.0804499 + 0.0804499i −0.746187 − 0.665737i \(-0.768117\pi\)
0.665737 + 0.746187i \(0.268117\pi\)
\(60\) −1522.18 − 1759.99i −0.422827 − 0.488885i
\(61\) −1593.12 + 1593.12i −0.428144 + 0.428144i −0.887996 − 0.459852i \(-0.847903\pi\)
0.459852 + 0.887996i \(0.347903\pi\)
\(62\) 1444.08 + 3877.22i 0.375670 + 1.00864i
\(63\) 1410.75i 0.355443i
\(64\) −2204.85 − 3451.93i −0.538294 − 0.842757i
\(65\) 1796.19 0.425135
\(66\) −10895.5 + 4058.05i −2.50126 + 0.931600i
\(67\) 2409.95 + 2409.95i 0.536857 + 0.536857i 0.922604 − 0.385748i \(-0.126056\pi\)
−0.385748 + 0.922604i \(0.626056\pi\)
\(68\) 3732.41 − 3228.09i 0.807182 − 0.698116i
\(69\) 873.969 + 873.969i 0.183568 + 0.183568i
\(70\) −781.544 − 357.358i −0.159499 − 0.0729302i
\(71\) 4500.32 0.892743 0.446371 − 0.894848i \(-0.352716\pi\)
0.446371 + 0.894848i \(0.352716\pi\)
\(72\) −4275.52 − 2342.35i −0.824753 − 0.451842i
\(73\) 3614.24i 0.678221i 0.940747 + 0.339110i \(0.110126\pi\)
−0.940747 + 0.339110i \(0.889874\pi\)
\(74\) −6016.74 − 2751.13i −1.09875 − 0.502398i
\(75\) −4347.63 + 4347.63i −0.772912 + 0.772912i
\(76\) 6712.42 + 486.349i 1.16212 + 0.0842017i
\(77\) −3036.26 + 3036.26i −0.512102 + 0.512102i
\(78\) 7276.49 − 2710.14i 1.19600 − 0.445454i
\(79\) − 12253.3i − 1.96336i −0.190529 − 0.981682i \(-0.561020\pi\)
0.190529 − 0.981682i \(-0.438980\pi\)
\(80\) 2380.67 − 1775.26i 0.371980 − 0.277384i
\(81\) 6928.65 1.05604
\(82\) 2845.28 + 7639.33i 0.423153 + 1.13613i
\(83\) 6206.38 + 6206.38i 0.900912 + 0.900912i 0.995515 − 0.0946032i \(-0.0301582\pi\)
−0.0946032 + 0.995515i \(0.530158\pi\)
\(84\) −3705.27 − 268.466i −0.525123 − 0.0380479i
\(85\) 2529.88 + 2529.88i 0.350156 + 0.350156i
\(86\) −4053.90 + 8865.89i −0.548120 + 1.19874i
\(87\) −12755.4 −1.68521
\(88\) −4160.61 − 14243.1i −0.537269 − 1.83925i
\(89\) 6811.06i 0.859873i 0.902859 + 0.429937i \(0.141464\pi\)
−0.902859 + 0.429937i \(0.858536\pi\)
\(90\) 1469.81 − 3214.48i 0.181458 − 0.396849i
\(91\) 2027.74 − 2027.74i 0.244867 − 0.244867i
\(92\) −1193.08 + 1031.87i −0.140959 + 0.121913i
\(93\) −9169.46 + 9169.46i −1.06018 + 1.06018i
\(94\) 675.374 + 1813.32i 0.0764343 + 0.205219i
\(95\) 4879.43i 0.540657i
\(96\) 6965.68 − 10783.7i 0.755825 − 1.17010i
\(97\) −5780.13 −0.614319 −0.307160 − 0.951658i \(-0.599379\pi\)
−0.307160 + 0.951658i \(0.599379\pi\)
\(98\) −1285.72 + 478.868i −0.133873 + 0.0498613i
\(99\) −12488.1 − 12488.1i −1.27416 − 1.27416i
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 112.5.k.a.43.7 ✓ 96
4.3 odd 2 448.5.k.a.15.41 96
16.3 odd 4 inner 112.5.k.a.99.7 yes 96
16.13 even 4 448.5.k.a.239.41 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.7 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.7 yes 96 16.3 odd 4 inner
448.5.k.a.15.41 96 4.3 odd 2
448.5.k.a.239.41 96 16.13 even 4