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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,4,Mod(29,200)] 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("200.29"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.o (of order \(10\), degree \(4\), 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: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(88\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 109.3
Character \(\chi\) \(=\) 200.109
Dual form 200.4.o.a.189.3

$q$-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+(-2.79309 - 0.445727i) q^{2} +(-0.406468 - 1.25098i) q^{3} +(7.60265 + 2.48991i) q^{4} +(3.97062 + 10.4515i) q^{5} +(0.577704 + 3.67527i) q^{6} -16.0266i q^{7} +(-20.1250 - 10.3432i) q^{8} +(20.4437 - 14.8532i) q^{9} +(-6.43175 - 30.9618i) q^{10} +(5.93935 - 8.17481i) q^{11} +(0.0245896 - 10.5228i) q^{12} +(-50.2559 + 36.5130i) q^{13} +(-7.14351 + 44.7638i) q^{14} +(11.4607 - 9.21537i) q^{15} +(51.6007 + 37.8598i) q^{16} +(44.4648 + 14.4475i) q^{17} +(-63.7216 + 32.3740i) q^{18} +(52.8449 + 17.1704i) q^{19} +(4.16392 + 89.3457i) q^{20} +(-20.0490 + 6.51431i) q^{21} +(-20.2329 + 20.1856i) q^{22} +(63.9687 - 88.0453i) q^{23} +(-4.75900 + 29.3802i) q^{24} +(-93.4684 + 82.9980i) q^{25} +(156.644 - 79.5836i) q^{26} +(-55.6228 - 40.4123i) q^{27} +(39.9048 - 121.845i) q^{28} +(136.309 - 44.2894i) q^{29} +(-36.1183 + 20.6310i) q^{30} +(38.9241 - 119.796i) q^{31} +(-127.250 - 128.746i) q^{32} +(-12.6407 - 4.10721i) q^{33} +(-117.754 - 60.1723i) q^{34} +(167.503 - 63.6356i) q^{35} +(192.410 - 62.0210i) q^{36} +(-27.8402 + 20.2271i) q^{37} +(-139.947 - 71.5127i) q^{38} +(66.1045 + 48.0277i) q^{39} +(28.1936 - 251.406i) q^{40} +(387.830 - 281.775i) q^{41} +(58.9022 - 9.25865i) q^{42} +477.611 q^{43} +(65.5094 - 47.3618i) q^{44} +(236.413 + 154.691i) q^{45} +(-217.914 + 217.406i) q^{46} +(61.3776 - 19.9428i) q^{47} +(26.3879 - 79.9403i) q^{48} +86.1471 q^{49} +(298.060 - 190.159i) q^{50} -61.4970i q^{51} +(-472.992 + 152.463i) q^{52} +(-42.3621 - 130.377i) q^{53} +(137.346 + 137.668i) q^{54} +(109.022 + 29.6161i) q^{55} +(-165.767 + 322.537i) q^{56} -73.0872i q^{57} +(-400.463 + 62.9476i) q^{58} +(132.420 + 182.260i) q^{59} +(110.077 - 41.5252i) q^{60} +(-265.324 + 365.187i) q^{61} +(-162.115 + 317.251i) q^{62} +(-238.047 - 327.644i) q^{63} +(298.035 + 416.316i) q^{64} +(-581.164 - 380.271i) q^{65} +(33.4758 + 17.1061i) q^{66} +(12.5224 - 38.5400i) q^{67} +(302.078 + 220.553i) q^{68} +(-136.144 - 44.2359i) q^{69} +(-496.213 + 103.079i) q^{70} +(95.1861 + 292.953i) q^{71} +(-565.061 + 87.4677i) q^{72} +(200.412 - 275.843i) q^{73} +(86.7758 - 44.0869i) q^{74} +(141.821 + 83.1911i) q^{75} +(359.009 + 262.119i) q^{76} +(-131.015 - 95.1878i) q^{77} +(-163.228 - 163.610i) q^{78} +(208.430 + 641.482i) q^{79} +(-190.806 + 689.633i) q^{80} +(182.892 - 562.883i) q^{81} +(-1208.84 + 614.155i) q^{82} +(399.979 - 1231.01i) q^{83} +(-168.646 - 0.394088i) q^{84} +(25.5546 + 522.090i) q^{85} +(-1334.01 - 212.884i) q^{86} +(-110.810 - 152.517i) q^{87} +(-204.084 + 103.086i) q^{88} +(-894.063 - 649.575i) q^{89} +(-591.372 - 537.442i) q^{90} +(585.181 + 805.433i) q^{91} +(705.557 - 510.102i) q^{92} -165.684 q^{93} +(-180.322 + 28.3443i) q^{94} +(30.3708 + 620.487i) q^{95} +(-109.335 + 211.518i) q^{96} +(-1553.29 + 504.694i) q^{97} +(-240.616 - 38.3981i) q^{98} -255.342i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 352 q - 5 q^{2} - 3 q^{4} + 13 q^{6} - 110 q^{8} - 762 q^{9} + 43 q^{10} - 5 q^{12} - 45 q^{14} + 46 q^{15} - 111 q^{16} - 10 q^{17} + 191 q^{20} + 320 q^{22} - 10 q^{23} + 32 q^{24} + 14 q^{25} + 18 q^{26}+ \cdots - 5120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{7}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.79309 − 0.445727i −0.987505 − 0.157588i
\(3\) −0.406468 − 1.25098i −0.0782248 − 0.240751i 0.904295 − 0.426907i \(-0.140397\pi\)
−0.982520 + 0.186156i \(0.940397\pi\)
\(4\) 7.60265 + 2.48991i 0.950332 + 0.311239i
\(5\) 3.97062 + 10.4515i 0.355143 + 0.934812i
\(6\) 0.577704 + 3.67527i 0.0393078 + 0.250070i
\(7\) − 16.0266i − 0.865357i −0.901548 − 0.432678i \(-0.857569\pi\)
0.901548 − 0.432678i \(-0.142431\pi\)
\(8\) −20.1250 − 10.3432i −0.889410 − 0.457111i
\(9\) 20.4437 − 14.8532i 0.757175 − 0.550120i
\(10\) −6.43175 − 30.9618i −0.203390 − 0.979098i
\(11\) 5.93935 − 8.17481i 0.162798 − 0.224073i −0.719823 − 0.694158i \(-0.755777\pi\)
0.882621 + 0.470085i \(0.155777\pi\)
\(12\) 0.0245896 − 10.5228i 0.000591533 − 0.253140i
\(13\) −50.2559 + 36.5130i −1.07219 + 0.778992i −0.976305 − 0.216400i \(-0.930568\pi\)
−0.0958857 + 0.995392i \(0.530568\pi\)
\(14\) −7.14351 + 44.7638i −0.136370 + 0.854544i
\(15\) 11.4607 − 9.21537i 0.197276 − 0.158627i
\(16\) 51.6007 + 37.8598i 0.806261 + 0.591560i
\(17\) 44.4648 + 14.4475i 0.634371 + 0.206120i 0.608510 − 0.793546i \(-0.291767\pi\)
0.0258604 + 0.999666i \(0.491767\pi\)
\(18\) −63.7216 + 32.3740i −0.834406 + 0.423924i
\(19\) 52.8449 + 17.1704i 0.638077 + 0.207324i 0.610150 − 0.792286i \(-0.291109\pi\)
0.0279272 + 0.999610i \(0.491109\pi\)
\(20\) 4.16392 + 89.3457i 0.0465540 + 0.998916i
\(21\) −20.0490 + 6.51431i −0.208336 + 0.0676924i
\(22\) −20.2329 + 20.1856i −0.196075 + 0.195618i
\(23\) 63.9687 − 88.0453i 0.579930 − 0.798205i −0.413758 − 0.910387i \(-0.635784\pi\)
0.993688 + 0.112182i \(0.0357839\pi\)
\(24\) −4.75900 + 29.3802i −0.0404761 + 0.249884i
\(25\) −93.4684 + 82.9980i −0.747747 + 0.663984i
\(26\) 156.644 − 79.5836i 1.18155 − 0.600294i
\(27\) −55.6228 − 40.4123i −0.396467 − 0.288050i
\(28\) 39.9048 − 121.845i 0.269332 − 0.822376i
\(29\) 136.309 − 44.2894i 0.872825 − 0.283598i 0.161850 − 0.986815i \(-0.448254\pi\)
0.710975 + 0.703218i \(0.248254\pi\)
\(30\) −36.1183 + 20.6310i −0.219809 + 0.125556i
\(31\) 38.9241 − 119.796i 0.225515 − 0.694065i −0.772724 − 0.634743i \(-0.781106\pi\)
0.998239 − 0.0593221i \(-0.0188939\pi\)
\(32\) −127.250 − 128.746i −0.702964 − 0.711226i
\(33\) −12.6407 − 4.10721i −0.0666806 − 0.0216659i
\(34\) −117.754 − 60.1723i −0.593962 − 0.303513i
\(35\) 167.503 − 63.6356i 0.808946 − 0.307325i
\(36\) 192.410 − 62.0210i 0.890786 − 0.287134i
\(37\) −27.8402 + 20.2271i −0.123700 + 0.0898733i −0.647915 − 0.761713i \(-0.724359\pi\)
0.524215 + 0.851586i \(0.324359\pi\)
\(38\) −139.947 − 71.5127i −0.597432 − 0.305287i
\(39\) 66.1045 + 48.0277i 0.271415 + 0.197195i
\(40\) 28.1936 − 251.406i 0.111445 − 0.993771i
\(41\) 387.830 − 281.775i 1.47729 − 1.07331i 0.498870 − 0.866677i \(-0.333749\pi\)
0.978418 − 0.206636i \(-0.0662515\pi\)
\(42\) 58.9022 − 9.25865i 0.216400 − 0.0340153i
\(43\) 477.611 1.69384 0.846919 − 0.531721i \(-0.178455\pi\)
0.846919 + 0.531721i \(0.178455\pi\)
\(44\) 65.5094 − 47.3618i 0.224452 − 0.162274i
\(45\) 236.413 + 154.691i 0.783164 + 0.512445i
\(46\) −217.914 + 217.406i −0.698472 + 0.696841i
\(47\) 61.3776 − 19.9428i 0.190486 − 0.0618927i −0.212221 − 0.977222i \(-0.568070\pi\)
0.402707 + 0.915329i \(0.368070\pi\)
\(48\) 26.3879 − 79.9403i 0.0793491 − 0.240383i
\(49\) 86.1471 0.251158
\(50\) 298.060 − 190.159i 0.843040 − 0.537851i
\(51\) − 61.4970i − 0.168849i
\(52\) −472.992 + 152.463i −1.26139 + 0.406594i
\(53\) −42.3621 − 130.377i −0.109790 − 0.337900i 0.881035 − 0.473052i \(-0.156848\pi\)
−0.990825 + 0.135152i \(0.956848\pi\)
\(54\) 137.346 + 137.668i 0.346120 + 0.346930i
\(55\) 109.022 + 29.6161i 0.267282 + 0.0726080i
\(56\) −165.767 + 322.537i −0.395564 + 0.769657i
\(57\) − 73.0872i − 0.169836i
\(58\) −400.463 + 62.9476i −0.906610 + 0.142507i
\(59\) 132.420 + 182.260i 0.292197 + 0.402174i 0.929726 − 0.368252i \(-0.120044\pi\)
−0.637529 + 0.770426i \(0.720044\pi\)
\(60\) 110.077 − 41.5252i 0.236849 − 0.0893480i
\(61\) −265.324 + 365.187i −0.556905 + 0.766514i −0.990929 − 0.134387i \(-0.957093\pi\)
0.434024 + 0.900901i \(0.357093\pi\)
\(62\) −162.115 + 317.251i −0.332074 + 0.649854i
\(63\) −238.047 − 327.644i −0.476050 − 0.655226i
\(64\) 298.035 + 416.316i 0.582099 + 0.813118i
\(65\) −581.164 − 380.271i −1.10899 − 0.725643i
\(66\) 33.4758 + 17.1061i 0.0624332 + 0.0319032i
\(67\) 12.5224 − 38.5400i 0.0228337 − 0.0702748i −0.938990 − 0.343943i \(-0.888237\pi\)
0.961824 + 0.273669i \(0.0882372\pi\)
\(68\) 302.078 + 220.553i 0.538710 + 0.393323i
\(69\) −136.144 − 44.2359i −0.237534 − 0.0771794i
\(70\) −496.213 + 103.079i −0.847269 + 0.176005i
\(71\) 95.1861 + 292.953i 0.159106 + 0.489677i 0.998554 − 0.0537629i \(-0.0171215\pi\)
−0.839448 + 0.543440i \(0.817122\pi\)
\(72\) −565.061 + 87.4677i −0.924904 + 0.143169i
\(73\) 200.412 − 275.843i 0.321321 − 0.442260i −0.617549 − 0.786532i \(-0.711874\pi\)
0.938870 + 0.344272i \(0.111874\pi\)
\(74\) 86.7758 − 44.0869i 0.136317 − 0.0692567i
\(75\) 141.821 + 83.1911i 0.218347 + 0.128081i
\(76\) 359.009 + 262.119i 0.541858 + 0.395620i
\(77\) −131.015 − 95.1878i −0.193903 − 0.140879i
\(78\) −163.228 − 163.610i −0.236948 − 0.237503i
\(79\) 208.430 + 641.482i 0.296838 + 0.913575i 0.982598 + 0.185746i \(0.0594702\pi\)
−0.685759 + 0.727828i \(0.740530\pi\)
\(80\) −190.806 + 689.633i −0.266659 + 0.963791i
\(81\) 182.892 − 562.883i 0.250880 − 0.772130i
\(82\) −1208.84 + 614.155i −1.62797 + 0.827098i
\(83\) 399.979 − 1231.01i 0.528956 − 1.62796i −0.227401 − 0.973801i \(-0.573023\pi\)
0.756357 − 0.654159i \(-0.226977\pi\)
\(84\) −168.646 − 0.394088i −0.219057 − 0.000511887i
\(85\) 25.5546 + 522.090i 0.0326092 + 0.666219i
\(86\) −1334.01 − 212.884i −1.67267 − 0.266929i
\(87\) −110.810 − 152.517i −0.136553 − 0.187949i
\(88\) −204.084 + 103.086i −0.247220 + 0.124876i
\(89\) −894.063 − 649.575i −1.06484 − 0.773650i −0.0898601 − 0.995954i \(-0.528642\pi\)
−0.974977 + 0.222305i \(0.928642\pi\)
\(90\) −591.372 − 537.442i −0.692623 − 0.629460i
\(91\) 585.181 + 805.433i 0.674106 + 0.927827i
\(92\) 705.557 − 510.102i 0.799558 − 0.578063i
\(93\) −165.684 −0.184738
\(94\) −180.322 + 28.3443i −0.197859 + 0.0311009i
\(95\) 30.3708 + 620.487i 0.0327997 + 0.670111i
\(96\) −109.335 + 211.518i −0.116239 + 0.224875i
\(97\) −1553.29 + 504.694i −1.62590 + 0.528287i −0.973324 − 0.229435i \(-0.926312\pi\)
−0.652577 + 0.757722i \(0.726312\pi\)
\(98\) −240.616 − 38.3981i −0.248020 − 0.0395795i
\(99\) − 255.342i − 0.259221i
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 200.4.o.a.109.3 ✓ 352
8.5 even 2 inner 200.4.o.a.109.56 yes 352
25.14 even 10 inner 200.4.o.a.189.56 yes 352
200.189 even 10 inner 200.4.o.a.189.3 yes 352
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.o.a.109.3 ✓ 352 1.1 even 1 trivial
200.4.o.a.109.56 yes 352 8.5 even 2 inner
200.4.o.a.189.3 yes 352 200.189 even 10 inner
200.4.o.a.189.56 yes 352 25.14 even 10 inner