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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 43.2
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.2

$q$-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+(-3.95485 + 0.599328i) q^{2} +(1.79635 + 1.79635i) q^{3} +(15.2816 - 4.74050i) q^{4} +(-4.04343 - 4.04343i) q^{5} +(-8.18087 - 6.02767i) q^{6} +18.5203 q^{7} +(-57.5953 + 27.9066i) q^{8} -74.5463i q^{9} +(18.4145 + 13.5678i) q^{10} +(48.6761 - 48.6761i) q^{11} +(35.9666 + 18.9355i) q^{12} +(-191.638 + 191.638i) q^{13} +(-73.2448 + 11.0997i) q^{14} -14.5268i q^{15} +(211.055 - 144.885i) q^{16} +125.405 q^{17} +(44.6776 + 294.819i) q^{18} +(-389.439 - 389.439i) q^{19} +(-80.9579 - 42.6222i) q^{20} +(33.2688 + 33.2688i) q^{21} +(-163.333 + 221.679i) q^{22} +292.883 q^{23} +(-153.591 - 53.3312i) q^{24} -592.301i q^{25} +(643.046 - 872.755i) q^{26} +(279.415 - 279.415i) q^{27} +(283.019 - 87.7952i) q^{28} +(1125.64 - 1125.64i) q^{29} +(8.70631 + 57.4512i) q^{30} -1005.46i q^{31} +(-747.858 + 699.489i) q^{32} +174.878 q^{33} +(-495.958 + 75.1588i) q^{34} +(-74.8853 - 74.8853i) q^{35} +(-353.386 - 1139.19i) q^{36} +(-1332.30 - 1332.30i) q^{37} +(1773.57 + 1306.77i) q^{38} -688.498 q^{39} +(345.721 + 120.044i) q^{40} -109.259i q^{41} +(-151.512 - 111.634i) q^{42} +(316.699 - 316.699i) q^{43} +(513.100 - 974.598i) q^{44} +(-301.422 + 301.422i) q^{45} +(-1158.31 + 175.533i) q^{46} -1386.71i q^{47} +(639.392 + 118.865i) q^{48} +343.000 q^{49} +(354.983 + 2342.46i) q^{50} +(225.271 + 225.271i) q^{51} +(-2020.08 + 3837.01i) q^{52} +(3627.01 + 3627.01i) q^{53} +(-937.582 + 1272.50i) q^{54} -393.636 q^{55} +(-1066.68 + 516.838i) q^{56} -1399.14i q^{57} +(-3777.12 + 5126.37i) q^{58} +(-4697.25 + 4697.25i) q^{59} +(-68.8642 - 221.993i) q^{60} +(2287.68 - 2287.68i) q^{61} +(602.597 + 3976.42i) q^{62} -1380.62i q^{63} +(2538.44 - 3214.58i) q^{64} +1549.75 q^{65} +(-691.616 + 104.809i) q^{66} +(-4285.76 - 4285.76i) q^{67} +(1916.39 - 594.483i) q^{68} +(526.119 + 526.119i) q^{69} +(341.041 + 251.279i) q^{70} -93.6034 q^{71} +(2080.34 + 4293.52i) q^{72} +3096.04i q^{73} +(6067.54 + 4470.57i) q^{74} +(1063.98 - 1063.98i) q^{75} +(-7797.39 - 4105.12i) q^{76} +(901.494 - 901.494i) q^{77} +(2722.90 - 412.636i) q^{78} +7817.52i q^{79} +(-1439.22 - 267.556i) q^{80} -5034.40 q^{81} +(65.4822 + 432.104i) q^{82} +(-3301.94 - 3301.94i) q^{83} +(666.111 + 350.690i) q^{84} +(-507.067 - 507.067i) q^{85} +(-1062.69 + 1442.30i) q^{86} +4044.09 q^{87} +(-1445.13 + 4161.90i) q^{88} -13746.1i q^{89} +(1011.43 - 1372.73i) q^{90} +(-3549.19 + 3549.19i) q^{91} +(4475.72 - 1388.41i) q^{92} +(1806.15 - 1806.15i) q^{93} +(831.094 + 5484.23i) q^{94} +3149.34i q^{95} +(-2599.94 - 86.8879i) q^{96} +14974.1 q^{97} +(-1356.51 + 205.569i) q^{98} +(-3628.62 - 3628.62i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.95485 + 0.599328i −0.988711 + 0.149832i
\(3\) 1.79635 + 1.79635i 0.199594 + 0.199594i 0.799826 − 0.600232i \(-0.204925\pi\)
−0.600232 + 0.799826i \(0.704925\pi\)
\(4\) 15.2816 − 4.74050i 0.955101 − 0.296281i
\(5\) −4.04343 − 4.04343i −0.161737 − 0.161737i 0.621599 − 0.783336i \(-0.286483\pi\)
−0.783336 + 0.621599i \(0.786483\pi\)
\(6\) −8.18087 − 6.02767i −0.227246 − 0.167435i
\(7\) 18.5203 0.377964
\(8\) −57.5953 + 27.9066i −0.899927 + 0.436041i
\(9\) − 74.5463i − 0.920324i
\(10\) 18.4145 + 13.5678i 0.184145 + 0.135678i
\(11\) 48.6761 − 48.6761i 0.402282 − 0.402282i −0.476755 − 0.879036i \(-0.658187\pi\)
0.879036 + 0.476755i \(0.158187\pi\)
\(12\) 35.9666 + 18.9355i 0.249768 + 0.131496i
\(13\) −191.638 + 191.638i −1.13396 + 1.13396i −0.144442 + 0.989513i \(0.546139\pi\)
−0.989513 + 0.144442i \(0.953861\pi\)
\(14\) −73.2448 + 11.0997i −0.373698 + 0.0566311i
\(15\) − 14.5268i − 0.0645635i
\(16\) 211.055 − 144.885i 0.824435 − 0.565957i
\(17\) 125.405 0.433928 0.216964 − 0.976180i \(-0.430385\pi\)
0.216964 + 0.976180i \(0.430385\pi\)
\(18\) 44.6776 + 294.819i 0.137894 + 0.909935i
\(19\) −389.439 − 389.439i −1.07878 − 1.07878i −0.996619 − 0.0821596i \(-0.973818\pi\)
−0.0821596 − 0.996619i \(-0.526182\pi\)
\(20\) −80.9579 − 42.6222i −0.202395 − 0.106556i
\(21\) 33.2688 + 33.2688i 0.0754394 + 0.0754394i
\(22\) −163.333 + 221.679i −0.337466 + 0.458015i
\(23\) 292.883 0.553654 0.276827 − 0.960920i \(-0.410717\pi\)
0.276827 + 0.960920i \(0.410717\pi\)
\(24\) −153.591 − 53.3312i −0.266651 − 0.0925888i
\(25\) − 592.301i − 0.947682i
\(26\) 643.046 − 872.755i 0.951252 − 1.29106i
\(27\) 279.415 − 279.415i 0.383285 − 0.383285i
\(28\) 283.019 − 87.7952i 0.360994 − 0.111984i
\(29\) 1125.64 − 1125.64i 1.33846 − 1.33846i 0.440903 − 0.897555i \(-0.354658\pi\)
0.897555 − 0.440903i \(-0.145342\pi\)
\(30\) 8.70631 + 57.4512i 0.00967367 + 0.0638347i
\(31\) − 1005.46i − 1.04626i −0.852253 − 0.523130i \(-0.824764\pi\)
0.852253 − 0.523130i \(-0.175236\pi\)
\(32\) −747.858 + 699.489i −0.730330 + 0.683094i
\(33\) 174.878 0.160586
\(34\) −495.958 + 75.1588i −0.429030 + 0.0650163i
\(35\) −74.8853 − 74.8853i −0.0611309 − 0.0611309i
\(36\) −353.386 − 1139.19i −0.272675 − 0.879003i
\(37\) −1332.30 − 1332.30i −0.973194 − 0.973194i 0.0264561 − 0.999650i \(-0.491578\pi\)
−0.999650 + 0.0264561i \(0.991578\pi\)
\(38\) 1773.57 + 1306.77i 1.22824 + 0.904965i
\(39\) −688.498 −0.452661
\(40\) 345.721 + 120.044i 0.216076 + 0.0750275i
\(41\) − 109.259i − 0.0649967i −0.999472 − 0.0324983i \(-0.989654\pi\)
0.999472 − 0.0324983i \(-0.0103464\pi\)
\(42\) −151.512 − 111.634i −0.0858911 − 0.0632846i
\(43\) 316.699 − 316.699i 0.171281 − 0.171281i −0.616261 − 0.787542i \(-0.711353\pi\)
0.787542 + 0.616261i \(0.211353\pi\)
\(44\) 513.100 − 974.598i 0.265031 − 0.503408i
\(45\) −301.422 + 301.422i −0.148851 + 0.148851i
\(46\) −1158.31 + 175.533i −0.547404 + 0.0829550i
\(47\) − 1386.71i − 0.627755i −0.949464 − 0.313877i \(-0.898372\pi\)
0.949464 − 0.313877i \(-0.101628\pi\)
\(48\) 639.392 + 118.865i 0.277514 + 0.0515908i
\(49\) 343.000 0.142857
\(50\) 354.983 + 2342.46i 0.141993 + 0.936984i
\(51\) 225.271 + 225.271i 0.0866095 + 0.0866095i
\(52\) −2020.08 + 3837.01i −0.747072 + 1.41901i
\(53\) 3627.01 + 3627.01i 1.29121 + 1.29121i 0.934039 + 0.357171i \(0.116259\pi\)
0.357171 + 0.934039i \(0.383741\pi\)
\(54\) −937.582 + 1272.50i −0.321530 + 0.436387i
\(55\) −393.636 −0.130128
\(56\) −1066.68 + 516.838i −0.340140 + 0.164808i
\(57\) − 1399.14i − 0.430636i
\(58\) −3777.12 + 5126.37i −1.12281 + 1.52389i
\(59\) −4697.25 + 4697.25i −1.34940 + 1.34940i −0.463080 + 0.886316i \(0.653256\pi\)
−0.886316 + 0.463080i \(0.846744\pi\)
\(60\) −68.8642 − 221.993i −0.0191289 − 0.0616647i
\(61\) 2287.68 − 2287.68i 0.614801 − 0.614801i −0.329392 − 0.944193i \(-0.606844\pi\)
0.944193 + 0.329392i \(0.106844\pi\)
\(62\) 602.597 + 3976.42i 0.156763 + 1.03445i
\(63\) − 1380.62i − 0.347850i
\(64\) 2538.44 − 3214.58i 0.619736 − 0.784810i
\(65\) 1549.75 0.366805
\(66\) −691.616 + 104.809i −0.158773 + 0.0240609i
\(67\) −4285.76 − 4285.76i −0.954725 − 0.954725i 0.0442932 − 0.999019i \(-0.485896\pi\)
−0.999019 + 0.0442932i \(0.985896\pi\)
\(68\) 1916.39 − 594.483i 0.414445 − 0.128565i
\(69\) 526.119 + 526.119i 0.110506 + 0.110506i
\(70\) 341.041 + 251.279i 0.0696002 + 0.0512814i
\(71\) −93.6034 −0.0185684 −0.00928421 − 0.999957i \(-0.502955\pi\)
−0.00928421 + 0.999957i \(0.502955\pi\)
\(72\) 2080.34 + 4293.52i 0.401299 + 0.828225i
\(73\) 3096.04i 0.580980i 0.956878 + 0.290490i \(0.0938184\pi\)
−0.956878 + 0.290490i \(0.906182\pi\)
\(74\) 6067.54 + 4470.57i 1.10802 + 0.816392i
\(75\) 1063.98 − 1063.98i 0.189152 − 0.189152i
\(76\) −7797.39 − 4105.12i −1.34996 − 0.710721i
\(77\) 901.494 − 901.494i 0.152048 − 0.152048i
\(78\) 2722.90 − 412.636i 0.447551 − 0.0678231i
\(79\) 7817.52i 1.25261i 0.779579 + 0.626304i \(0.215433\pi\)
−0.779579 + 0.626304i \(0.784567\pi\)
\(80\) −1439.22 − 267.556i −0.224878 − 0.0418056i
\(81\) −5034.40 −0.767322
\(82\) 65.4822 + 432.104i 0.00973857 + 0.0642629i
\(83\) −3301.94 − 3301.94i −0.479306 − 0.479306i 0.425603 − 0.904910i \(-0.360062\pi\)
−0.904910 + 0.425603i \(0.860062\pi\)
\(84\) 666.111 + 350.690i 0.0944036 + 0.0497010i
\(85\) −507.067 − 507.067i −0.0701823 − 0.0701823i
\(86\) −1062.69 + 1442.30i −0.143685 + 0.195011i
\(87\) 4044.09 0.534296
\(88\) −1445.13 + 4161.90i −0.186613 + 0.537435i
\(89\) − 13746.1i − 1.73540i −0.497090 − 0.867699i \(-0.665598\pi\)
0.497090 − 0.867699i \(-0.334402\pi\)
\(90\) 1011.43 − 1372.73i 0.124868 − 0.169473i
\(91\) −3549.19 + 3549.19i −0.428595 + 0.428595i
\(92\) 4475.72 − 1388.41i 0.528795 − 0.164037i
\(93\) 1806.15 − 1806.15i 0.208827 − 0.208827i
\(94\) 831.094 + 5484.23i 0.0940577 + 0.620668i
\(95\) 3149.34i 0.348957i
\(96\) −2599.94 − 86.8879i −0.282111 − 0.00942794i
\(97\) 14974.1 1.59146 0.795731 − 0.605651i \(-0.207087\pi\)
0.795731 + 0.605651i \(0.207087\pi\)
\(98\) −1356.51 + 205.569i −0.141244 + 0.0214046i
\(99\) −3628.62 − 3628.62i −0.370230 − 0.370230i
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.43.2 ✓ 96
4.3 odd 2 448.5.k.a.15.22 96
16.3 odd 4 inner 112.5.k.a.99.2 yes 96
16.13 even 4 448.5.k.a.239.22 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.2 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.2 yes 96 16.3 odd 4 inner
448.5.k.a.15.22 96 4.3 odd 2
448.5.k.a.239.22 96 16.13 even 4