LMFDB - Embedded newform 1150.4.b.o.599.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1150.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 124x^{6} + 4272x^{4} + 28129x^{2} + 4 $ x^8 + 124x^6 + 4272x^4 + 28129*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.7
Root			$0.0119250i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.o.599.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} -2.98808i q^{3} -4.00000 q^{4} +5.97615 q^{6} +2.98517i q^{7} -8.00000i q^{8} +18.0714 q^{9} +68.6262 q^{11} +11.9523i q^{12} +12.0028i q^{13} -5.97035 q^{14} +16.0000 q^{16} +106.208i q^{17} +36.1428i q^{18} +48.4891 q^{19} +8.91992 q^{21} +137.252i q^{22} +23.0000i q^{23} -23.9046 q^{24} -24.0055 q^{26} -134.677i q^{27} -11.9407i q^{28} -135.226 q^{29} -230.782 q^{31} +32.0000i q^{32} -205.060i q^{33} -212.416 q^{34} -72.2856 q^{36} -107.956i q^{37} +96.9782i q^{38} +35.8651 q^{39} +394.637 q^{41} +17.8398i q^{42} -136.063i q^{43} -274.505 q^{44} -46.0000 q^{46} -50.4512i q^{47} -47.8092i q^{48} +334.089 q^{49} +317.357 q^{51} -48.0110i q^{52} +414.707i q^{53} +269.353 q^{54} +23.8814 q^{56} -144.889i q^{57} -270.452i q^{58} +183.586 q^{59} -98.4751 q^{61} -461.565i q^{62} +53.9463i q^{63} -64.0000 q^{64} +410.121 q^{66} +136.925i q^{67} -424.831i q^{68} +68.7257 q^{69} -708.800 q^{71} -144.571i q^{72} +689.605i q^{73} +215.912 q^{74} -193.956 q^{76} +204.861i q^{77} +71.7303i q^{78} -546.583 q^{79} +85.5038 q^{81} +789.275i q^{82} +20.2622i q^{83} -35.6797 q^{84} +272.125 q^{86} +404.065i q^{87} -549.010i q^{88} +1087.31 q^{89} -35.8303 q^{91} -92.0000i q^{92} +689.595i q^{93} +100.902 q^{94} +95.6184 q^{96} -1115.90i q^{97} +668.177i q^{98} +1240.17 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) $ 8 * q - 32 * q^4 + 56 * q^6 - 128 * q^9 + 42 * q^11 - 32 * q^14 + 128 * q^16 - 346 * q^19 - 240 * q^21 - 224 * q^24 + 280 * q^26 + 236 * q^29 + 34 * q^31 - 224 * q^34 + 512 * q^36 + 442 * q^39 + 278 * q^41 - 168 * q^44 - 368 * q^46 + 248 * q^49 + 878 * q^51 - 1556 * q^54 + 128 * q^56 + 906 * q^59 - 654 * q^61 - 512 * q^64 - 356 * q^66 + 644 * q^69 + 390 * q^71 + 1372 * q^74 + 1384 * q^76 + 2280 * q^79 + 2912 * q^81 + 960 * q^84 - 200 * q^86 + 4340 * q^89 + 1114 * q^91 - 1468 * q^94 + 896 * q^96 + 5490 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	− 2.98808i	− 0.575055i	−0.957772	−	0.287528i	$-0.907167\pi$
$3$	− 2.98808i	− 0.575055i	0.957772	−	0.287528i	$-0.0928333\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	5.97615	0.406626
$7$	2.98517i	0.161184i	0.996747	+	0.0805921i	$0.0256811\pi$
$7$	2.98517i	0.161184i	−0.996747	+	0.0805921i	$0.974319\pi$
$8$	− 8.00000i	− 0.353553i
$9$	18.0714	0.669311
$10$	0	0
$11$	68.6262	1.88105	0.940527	−	0.339720i	$-0.110332\pi$
$11$	68.6262	1.88105	0.940527	+	0.339720i	$0.110332\pi$
$12$	11.9523i	0.287528i
$13$	12.0028i	0.256074i	0.991769	+	0.128037i	$0.0408677\pi$
$13$	12.0028i	0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$14$	−5.97035	−0.113974
$15$	0	0
$16$	16.0000	0.250000
$17$	106.208i	1.51525i	0.652693	+	0.757623i	$0.273639\pi$
$17$	106.208i	1.51525i	−0.652693	+	0.757623i	$0.726361\pi$
$18$	36.1428i	0.473275i
$19$	48.4891	0.585482	0.292741	−	0.956192i	$-0.405433\pi$
$19$	48.4891	0.585482	0.292741	+	0.956192i	$0.405433\pi$
$20$	0	0
$21$	8.91992	0.0926898
$22$	137.252i	1.33011i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	−23.9046	−0.203313
$25$	0	0
$26$	−24.0055	−0.181072
$27$	− 134.677i	− 0.959946i
$28$	− 11.9407i	− 0.0805921i
$29$	−135.226	−0.865890	−0.432945	−	0.901420i	$-0.642526\pi$
$29$	−135.226	−0.865890	−0.432945	+	0.901420i	$0.642526\pi$
$30$	0	0
$31$	−230.782	−1.33709	−0.668544	−	0.743673i	$-0.733082\pi$
$31$	−230.782	−1.33709	−0.668544	+	0.743673i	$0.733082\pi$
$32$	32.0000i	0.176777i
$33$	− 205.060i	− 1.08171i
$34$	−212.416	−1.07144
$35$	0	0
$36$	−72.2856	−0.334656
$37$	− 107.956i	− 0.479672i	−0.970813	−	0.239836i	$-0.922906\pi$
$37$	− 107.956i	− 0.479672i	0.970813	−	0.239836i	$-0.0770937\pi$
$38$	96.9782i	0.413998i
$39$	35.8651	0.147257
$40$	0	0
$41$	394.637	1.50322	0.751610	−	0.659608i	$-0.229278\pi$
$41$	394.637	1.50322	0.751610	+	0.659608i	$0.229278\pi$
$42$	17.8398i	0.0655416i
$43$	− 136.063i	− 0.482543i	−0.970458	−	0.241272i	$-0.922436\pi$
$43$	− 136.063i	− 0.482543i	0.970458	−	0.241272i	$-0.0775645\pi$
$44$	−274.505	−0.940527
$45$	0	0
$46$	−46.0000	−0.147442
$47$	− 50.4512i	− 0.156576i	−0.996931	−	0.0782880i	$-0.975055\pi$
$47$	− 50.4512i	− 0.156576i	0.996931	−	0.0782880i	$-0.0249454\pi$
$48$	− 47.8092i	− 0.143764i
$49$	334.089	0.974020
$50$	0	0
$51$	317.357	0.871350
$52$	− 48.0110i	− 0.128037i
$53$	414.707i	1.07480i	0.843327	+	0.537400i	$0.180594\pi$
$53$	414.707i	1.07480i	−0.843327	+	0.537400i	$0.819406\pi$
$54$	269.353	0.678785
$55$	0	0
$56$	23.8814	0.0569872
$57$	− 144.889i	− 0.336685i
$58$	− 270.452i	− 0.612277i
$59$	183.586	0.405100	0.202550	−	0.979272i	$-0.435077\pi$
$59$	183.586	0.405100	0.202550	+	0.979272i	$0.435077\pi$
$60$	0	0
$61$	−98.4751	−0.206696	−0.103348	−	0.994645i	$-0.532956\pi$
$61$	−98.4751	−0.206696	−0.103348	+	0.994645i	$0.532956\pi$
$62$	− 461.565i	− 0.945464i
$63$	53.9463i	0.107882i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	410.121	0.764884
$67$	136.925i	0.249672i	0.992177	+	0.124836i	$0.0398404\pi$
$67$	136.925i	0.249672i	−0.992177	+	0.124836i	$0.960160\pi$
$68$	− 424.831i	− 0.757623i
$69$	68.7257	0.119907
$70$	0	0
$71$	−708.800	−1.18478	−0.592388	−	0.805653i	$-0.701815\pi$
$71$	−708.800	−1.18478	−0.592388	+	0.805653i	$0.701815\pi$
$72$	− 144.571i	− 0.236637i
$73$	689.605i	1.10565i	0.833299	+	0.552823i	$0.186449\pi$
$73$	689.605i	1.10565i	−0.833299	+	0.552823i	$0.813551\pi$
$74$	215.912	0.339179
$75$	0	0
$76$	−193.956	−0.292741
$77$	204.861i	0.303196i
$78$	71.7303i	0.104126i
$79$	−546.583	−0.778423	−0.389212	−	0.921148i	$-0.627252\pi$
$79$	−546.583	−0.778423	−0.389212	+	0.921148i	$0.627252\pi$
$80$	0	0
$81$	85.5038	0.117289
$82$	789.275i	1.06294i
$83$	20.2622i	0.0267959i	0.999910	+	0.0133980i	$0.00426483\pi$
$83$	20.2622i	0.0267959i	−0.999910	+	0.0133980i	$0.995735\pi$
$84$	−35.6797	−0.0463449
$85$	0	0
$86$	272.125	0.341210
$87$	404.065i	0.497935i
$88$	− 549.010i	− 0.665053i
$89$	1087.31	1.29499	0.647495	−	0.762069i	$-0.275817\pi$
$89$	1087.31	1.29499	0.647495	+	0.762069i	$0.275817\pi$
$90$	0	0
$91$	−35.8303	−0.0412751
$92$	− 92.0000i	− 0.104257i
$93$	689.595i	0.768900i
$94$	100.902	0.110716
$95$	0	0
$96$	95.6184	0.101656
$97$	− 1115.90i	− 1.16807i	−0.811728	−	0.584036i	$-0.801473\pi$
$97$	− 1115.90i	− 1.16807i	0.811728	−	0.584036i	$-0.198527\pi$
$98$	668.177i	0.688736i
$99$	1240.17	1.25901
$100$	0	0
$101$	−134.565	−0.132572	−0.0662859	−	0.997801i	$-0.521115\pi$
$101$	−134.565	−0.132572	−0.0662859	+	0.997801i	$0.521115\pi$
$102$	634.714i	0.616137i
$103$	925.953i	0.885795i	0.896572	+	0.442898i	$0.146049\pi$
$103$	925.953i	0.885795i	−0.896572	+	0.442898i	$0.853951\pi$
$104$	96.0221	0.0905360
$105$	0	0
$106$	−829.415	−0.759999
$107$	1871.13i	1.69055i	0.534334	+	0.845273i	$0.320562\pi$
$107$	1871.13i	1.69055i	−0.534334	+	0.845273i	$0.679438\pi$
$108$	538.707i	0.479973i
$109$	978.242	0.859620	0.429810	−	0.902919i	$-0.358581\pi$
$109$	978.242	0.859620	0.429810	+	0.902919i	$0.358581\pi$
$110$	0	0
$111$	−322.581	−0.275838
$112$	47.7628i	0.0402961i
$113$	1760.45i	1.46557i	0.680459	+	0.732786i	$0.261780\pi$
$113$	1760.45i	1.46557i	−0.680459	+	0.732786i	$0.738220\pi$
$114$	289.778	0.238072
$115$	0	0
$116$	540.903	0.432945
$117$	216.907i	0.171393i
$118$	367.172i	0.286449i
$119$	−317.049	−0.244234
$120$	0	0
$121$	3378.56	2.53836
$122$	− 196.950i	− 0.146156i
$123$	− 1179.21i	− 0.864434i
$124$	923.129	0.668544
$125$	0	0
$126$	−107.893	−0.0762844
$127$	− 1260.87i	− 0.880980i	−0.897758	−	0.440490i	$-0.854805\pi$
$127$	− 1260.87i	− 0.880980i	0.897758	−	0.440490i	$-0.145195\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−406.566	−0.277489
$130$	0	0
$131$	444.283	0.296314	0.148157	−	0.988964i	$-0.452666\pi$
$131$	444.283	0.296314	0.148157	+	0.988964i	$0.452666\pi$
$132$	820.241i	0.540855i
$133$	144.748i	0.0943705i
$134$	−273.849	−0.176545
$135$	0	0
$136$	849.662	0.535720
$137$	429.072i	0.267577i	0.991010	+	0.133789i	$0.0427143\pi$
$137$	429.072i	0.267577i	−0.991010	+	0.133789i	$0.957286\pi$
$138$	137.451i	0.0847873i
$139$	249.755	0.152402	0.0762012	−	0.997092i	$-0.475721\pi$
$139$	249.755	0.152402	0.0762012	+	0.997092i	$0.475721\pi$
$140$	0	0
$141$	−150.752	−0.0900398
$142$	− 1417.60i	− 0.837763i
$143$	823.704i	0.481689i
$144$	289.143	0.167328
$145$	0	0
$146$	−1379.21	−0.781809
$147$	− 998.282i	− 0.560115i
$148$	431.824i	0.239836i
$149$	1564.18	0.860020	0.430010	−	0.902824i	$-0.358510\pi$
$149$	1564.18	0.860020	0.430010	+	0.902824i	$0.358510\pi$
$150$	0	0
$151$	−1860.04	−1.00244	−0.501219	−	0.865321i	$-0.667115\pi$
$151$	−1860.04	−1.00244	−0.501219	+	0.865321i	$0.667115\pi$
$152$	− 387.913i	− 0.206999i
$153$	1919.32i	1.01417i
$154$	−409.722	−0.214392
$155$	0	0
$156$	−143.461	−0.0736285
$157$	2834.52i	1.44089i	0.693512	+	0.720445i	$0.256062\pi$
$157$	2834.52i	1.44089i	−0.693512	+	0.720445i	$0.743938\pi$
$158$	− 1093.17i	− 0.550428i
$159$	1239.18	0.618070
$160$	0	0
$161$	−68.6590	−0.0336092
$162$	171.008i	0.0829359i
$163$	− 2114.51i	− 1.01608i	−0.861333	−	0.508041i	$-0.830370\pi$
$163$	− 2114.51i	− 1.01608i	0.861333	−	0.508041i	$-0.169630\pi$
$164$	−1578.55	−0.751610
$165$	0	0
$166$	−40.5244	−0.0189476
$167$	2487.41i	1.15258i	0.817244	+	0.576291i	$0.195501\pi$
$167$	2487.41i	1.15258i	−0.817244	+	0.576291i	$0.804499\pi$
$168$	− 71.3594i	− 0.0327708i
$169$	2052.93	0.934426
$170$	0	0
$171$	876.266	0.391870
$172$	544.251i	0.241272i
$173$	− 3672.32i	− 1.61388i	−0.590634	−	0.806939i	$-0.701122\pi$
$173$	− 3672.32i	− 1.61388i	0.590634	−	0.806939i	$-0.298878\pi$
$174$	−808.130	−0.352093
$175$	0	0
$176$	1098.02	0.470263
$177$	− 548.569i	− 0.232955i
$178$	2174.61i	0.915697i
$179$	3224.63	1.34648	0.673240	−	0.739424i	$-0.264902\pi$
$179$	3224.63	1.34648	0.673240	+	0.739424i	$0.264902\pi$
$180$	0	0
$181$	−2224.23	−0.913401	−0.456700	−	0.889621i	$-0.650969\pi$
$181$	−2224.23	−0.913401	−0.456700	+	0.889621i	$0.650969\pi$
$182$	− 71.6606i	− 0.0291859i
$183$	294.251i	0.118862i
$184$	184.000	0.0737210
$185$	0	0
$186$	−1379.19	−0.543694
$187$	7288.64i	2.85026i
$188$	201.805i	0.0782880i
$189$	402.033	0.154728
$190$	0	0
$191$	3273.53	1.24013	0.620063	−	0.784552i	$-0.287107\pi$
$191$	3273.53	1.24013	0.620063	+	0.784552i	$0.287107\pi$
$192$	191.237i	0.0718819i
$193$	− 5126.86i	− 1.91212i	−0.293166	−	0.956062i	$-0.594709\pi$
$193$	− 5126.86i	− 1.91212i	0.293166	−	0.956062i	$-0.405291\pi$
$194$	2231.81	0.825951
$195$	0	0
$196$	−1336.35	−0.487010
$197$	3262.57i	1.17994i	0.807425	+	0.589971i	$0.200861\pi$
$197$	3262.57i	1.17994i	−0.807425	+	0.589971i	$0.799139\pi$
$198$	2480.34i	0.890255i
$199$	168.282	0.0599457	0.0299728	−	0.999551i	$-0.490458\pi$
$199$	168.282	0.0599457	0.0299728	+	0.999551i	$0.490458\pi$
$200$	0	0
$201$	409.141	0.143575
$202$	− 269.131i	− 0.0937424i
$203$	− 403.673i	− 0.139568i
$204$	−1269.43	−0.435675
$205$	0	0
$206$	−1851.91	−0.626352
$207$	415.642i	0.139561i
$208$	192.044i	0.0640186i
$209$	3327.62	1.10132
$210$	0	0
$211$	4197.01	1.36936	0.684678	−	0.728846i	$-0.259943\pi$
$211$	4197.01	1.36936	0.684678	+	0.728846i	$0.259943\pi$
$212$	− 1658.83i	− 0.537400i
$213$	2117.95i	0.681312i
$214$	−3742.25	−1.19540
$215$	0	0
$216$	−1077.41	−0.339392
$217$	− 688.925i	− 0.215518i
$218$	1956.48i	0.607843i
$219$	2060.59	0.635807
$220$	0	0
$221$	−1274.79	−0.388015
$222$	− 645.162i	− 0.195047i
$223$	− 5192.66i	− 1.55931i	−0.626208	−	0.779656i	$-0.715394\pi$
$223$	− 5192.66i	− 1.55931i	0.626208	−	0.779656i	$-0.284606\pi$
$224$	−95.5256	−0.0284936
$225$	0	0
$226$	−3520.91	−1.03632
$227$	4701.24i	1.37459i	0.726378	+	0.687295i	$0.241202\pi$
$227$	4701.24i	1.37459i	−0.726378	+	0.687295i	$0.758798\pi$
$228$	579.556i	0.168342i
$229$	2125.17	0.613255	0.306628	−	0.951830i	$-0.400799\pi$
$229$	2125.17	0.613255	0.306628	+	0.951830i	$0.400799\pi$
$230$	0	0
$231$	612.141	0.174355
$232$	1081.81i	0.306138i
$233$	1646.67i	0.462992i	0.972836	+	0.231496i	$0.0743620\pi$
$233$	1646.67i	0.462992i	−0.972836	+	0.231496i	$0.925638\pi$
$234$	−433.814	−0.121193
$235$	0	0
$236$	−734.345	−0.202550
$237$	1633.23i	0.447636i
$238$	− 634.097i	− 0.172699i
$239$	−2877.88	−0.778888	−0.389444	−	0.921050i	$-0.627333\pi$
$239$	−2877.88	−0.778888	−0.389444	+	0.921050i	$0.627333\pi$
$240$	0	0
$241$	−4244.54	−1.13450	−0.567251	−	0.823545i	$-0.691993\pi$
$241$	−4244.54	−1.13450	−0.567251	+	0.823545i	$0.691993\pi$
$242$	6757.12i	1.79489i
$243$	− 3891.76i	− 1.02739i
$244$	393.900	0.103348
$245$	0	0
$246$	2358.41	0.611247
$247$	582.003i	0.149927i
$248$	1846.26i	0.472732i
$249$	60.5449	0.0154092
$250$	0	0
$251$	4401.99	1.10698	0.553488	−	0.832857i	$-0.313296\pi$
$251$	4401.99	1.10698	0.553488	+	0.832857i	$0.313296\pi$
$252$	− 215.785i	− 0.0539412i
$253$	1578.40i	0.392227i
$254$	2521.75	0.622947
$255$	0	0
$256$	256.000	0.0625000
$257$	1694.78i	0.411352i	0.978620	+	0.205676i	$0.0659393\pi$
$257$	1694.78i	0.411352i	−0.978620	+	0.205676i	$0.934061\pi$
$258$	− 813.131i	− 0.196214i
$259$	322.268	0.0773156
$260$	0	0
$261$	−2443.72	−0.579550
$262$	888.566i	0.209526i
$263$	727.462i	0.170560i	0.996357	+	0.0852799i	$0.0271784\pi$
$263$	727.462i	0.170560i	−0.996357	+	0.0852799i	$0.972822\pi$
$264$	−1640.48	−0.382442
$265$	0	0
$266$	−289.497	−0.0667300
$267$	− 3248.95i	− 0.744691i
$268$	− 547.698i	− 0.124836i
$269$	−94.9937	−0.0215311	−0.0107656	−	0.999942i	$-0.503427\pi$
$269$	−94.9937	−0.0215311	−0.0107656	+	0.999942i	$0.503427\pi$
$270$	0	0
$271$	5321.54	1.19284	0.596422	−	0.802671i	$-0.296589\pi$
$271$	5321.54	1.19284	0.596422	+	0.802671i	$0.296589\pi$
$272$	1699.32i	0.378811i
$273$	107.064i	0.0237355i
$274$	−858.143	−0.189206
$275$	0	0
$276$	−274.903	−0.0599537
$277$	2561.25i	0.555562i	0.960644	+	0.277781i	$0.0895989\pi$
$277$	2561.25i	0.555562i	−0.960644	+	0.277781i	$0.910401\pi$
$278$	499.510i	0.107765i
$279$	−4170.56	−0.894928
$280$	0	0
$281$	−8753.73	−1.85838	−0.929188	−	0.369606i	$-0.879493\pi$
$281$	−8753.73	−1.85838	−0.929188	+	0.369606i	$0.879493\pi$
$282$	− 301.504i	− 0.0636678i
$283$	2400.14i	0.504147i	0.967708	+	0.252073i	$0.0811125\pi$
$283$	2400.14i	0.504147i	−0.967708	+	0.252073i	$0.918888\pi$
$284$	2835.20	0.592388
$285$	0	0
$286$	−1647.41	−0.340606
$287$	1178.06i	0.242295i
$288$	578.285i	0.118319i
$289$	−6367.09	−1.29597
$290$	0	0
$291$	−3334.41	−0.671706
$292$	− 2758.42i	− 0.552823i
$293$	− 1045.56i	− 0.208472i	−0.994553	−	0.104236i	$-0.966760\pi$
$293$	− 1045.56i	− 0.208472i	0.994553	−	0.104236i	$-0.0332397\pi$
$294$	1996.56	0.396061
$295$	0	0
$296$	−863.649	−0.169590
$297$	− 9242.36i	− 1.80571i
$298$	3128.37i	0.608126i
$299$	−276.063	−0.0533952
$300$	0	0
$301$	406.171	0.0777784
$302$	− 3720.08i	− 0.708830i
$303$	402.091i	0.0762361i
$304$	775.826	0.146371
$305$	0	0
$306$	−3838.65	−0.717127
$307$	− 7905.84i	− 1.46974i	−0.678208	−	0.734870i	$-0.737243\pi$
$307$	− 7905.84i	− 1.46974i	0.678208	−	0.734870i	$-0.262757\pi$
$308$	− 819.445i	− 0.151598i
$309$	2766.82	0.509381
$310$	0	0
$311$	−9227.63	−1.68248	−0.841240	−	0.540662i	$-0.818174\pi$
$311$	−9227.63	−1.68248	−0.841240	+	0.540662i	$0.818174\pi$
$312$	− 286.921i	− 0.0520632i
$313$	− 2712.57i	− 0.489852i	−0.969542	−	0.244926i	$-0.921236\pi$
$313$	− 2712.57i	− 0.489852i	0.969542	−	0.244926i	$-0.0787637\pi$
$314$	−5669.05	−1.01886
$315$	0	0
$316$	2186.33	0.389212
$317$	− 3440.29i	− 0.609546i	−0.952425	−	0.304773i	$-0.901419\pi$
$317$	− 3440.29i	− 0.609546i	0.952425	−	0.304773i	$-0.0985805\pi$
$318$	2478.35i	0.437041i
$319$	−9280.04	−1.62879
$320$	0	0
$321$	5591.06	0.972158
$322$	− 137.318i	− 0.0237653i
$323$	5149.92i	0.887149i
$324$	−342.015	−0.0586446
$325$	0	0
$326$	4229.03	0.718479
$327$	− 2923.06i	− 0.494329i
$328$	− 3157.10i	− 0.531468i
$329$	150.606	0.0252376
$330$	0	0
$331$	3259.75	0.541305	0.270652	−	0.962677i	$-0.412761\pi$
$331$	3259.75	0.541305	0.270652	+	0.962677i	$0.412761\pi$
$332$	− 81.0487i	− 0.0133980i
$333$	− 1950.92i	− 0.321050i
$334$	−4974.81	−0.814999
$335$	0	0
$336$	142.719	0.0231725
$337$	7163.38i	1.15790i	0.815361	+	0.578952i	$0.196538\pi$
$337$	7163.38i	1.15790i	−0.815361	+	0.578952i	$0.803462\pi$
$338$	4105.87i	0.660739i
$339$	5260.37	0.842785
$340$	0	0
$341$	−15837.7	−2.51513
$342$	1752.53i	0.277094i
$343$	2021.23i	0.318181i
$344$	−1088.50	−0.170605
$345$	0	0
$346$	7344.63	1.14118
$347$	− 1833.45i	− 0.283645i	−0.989892	−	0.141822i	$-0.954704\pi$
$347$	− 1833.45i	− 0.283645i	0.989892	−	0.141822i	$-0.0452962\pi$
$348$	− 1616.26i	− 0.248967i
$349$	10400.3	1.59517	0.797583	−	0.603209i	$-0.206112\pi$
$349$	10400.3	1.59517	0.797583	+	0.603209i	$0.206112\pi$
$350$	0	0
$351$	1616.49	0.245818
$352$	2196.04i	0.332526i
$353$	3156.48i	0.475928i	0.971274	+	0.237964i	$0.0764800\pi$
$353$	3156.48i	0.475928i	−0.971274	+	0.237964i	$0.923520\pi$
$354$	1097.14	0.164724
$355$	0	0
$356$	−4349.22	−0.647495
$357$	947.365i	0.140448i
$358$	6449.25i	0.952105i
$359$	9324.16	1.37078	0.685390	−	0.728176i	$-0.259632\pi$
$359$	9324.16	1.37078	0.685390	+	0.728176i	$0.259632\pi$
$360$	0	0
$361$	−4507.81	−0.657211
$362$	− 4448.45i	− 0.645872i
$363$	− 10095.4i	− 1.45970i
$364$	143.321	0.0206376
$365$	0	0
$366$	−588.502	−0.0840478
$367$	3802.68i	0.540867i	0.962739	+	0.270434i	$0.0871671\pi$
$367$	3802.68i	0.540867i	−0.962739	+	0.270434i	$0.912833\pi$
$368$	368.000i	0.0521286i
$369$	7131.65	1.00612
$370$	0	0
$371$	−1237.97	−0.173241
$372$	− 2758.38i	− 0.384450i
$373$	8928.03i	1.23935i	0.784860	+	0.619673i	$0.212735\pi$
$373$	8928.03i	1.23935i	−0.784860	+	0.619673i	$0.787265\pi$
$374$	−14577.3	−2.01544
$375$	0	0
$376$	−403.610	−0.0553580
$377$	− 1623.08i	− 0.221732i
$378$	804.067i	0.109409i
$379$	−3798.13	−0.514768	−0.257384	−	0.966309i	$-0.582860\pi$
$379$	−3798.13	−0.514768	−0.257384	+	0.966309i	$0.582860\pi$
$380$	0	0
$381$	−3767.58	−0.506612
$382$	6547.05i	0.876901i
$383$	− 1456.41i	− 0.194306i	−0.995269	−	0.0971530i	$-0.969026\pi$
$383$	− 1456.41i	− 0.194306i	0.995269	−	0.0971530i	$-0.0309736\pi$
$384$	−382.474	−0.0508282
$385$	0	0
$386$	10253.7	1.35208
$387$	− 2458.84i	− 0.322972i
$388$	4463.62i	0.584036i
$389$	8211.34	1.07026	0.535130	−	0.844769i	$-0.320262\pi$
$389$	8211.34	1.07026	0.535130	+	0.844769i	$0.320262\pi$
$390$	0	0
$391$	−2442.78	−0.315950
$392$	− 2672.71i	− 0.344368i
$393$	− 1327.55i	− 0.170397i
$394$	−6525.14	−0.834345
$395$	0	0
$396$	−4960.69	−0.629505
$397$	914.365i	0.115594i	0.998328	+	0.0577968i	$0.0184076\pi$
$397$	914.365i	0.115594i	−0.998328	+	0.0577968i	$0.981592\pi$
$398$	336.564i	0.0423880i
$399$	432.519	0.0542683
$400$	0	0
$401$	9414.97	1.17247	0.586236	−	0.810140i	$-0.300609\pi$
$401$	9414.97	1.17247	0.586236	+	0.810140i	$0.300609\pi$
$402$	818.282i	0.101523i
$403$	− 2770.02i	− 0.342394i
$404$	538.261	0.0662859
$405$	0	0
$406$	807.345	0.0986894
$407$	− 7408.62i	− 0.902289i
$408$	− 2538.85i	− 0.308069i
$409$	−3299.41	−0.398889	−0.199444	−	0.979909i	$-0.563914\pi$
$409$	−3299.41	−0.398889	−0.199444	+	0.979909i	$0.563914\pi$
$410$	0	0
$411$	1282.10	0.153872
$412$	− 3703.81i	− 0.442898i
$413$	548.037i	0.0652957i
$414$	−831.285	−0.0986846
$415$	0	0
$416$	−384.088	−0.0452680
$417$	− 746.286i	− 0.0876398i
$418$	6655.25i	0.778753i
$419$	3554.64	0.414452	0.207226	−	0.978293i	$-0.433556\pi$
$419$	3554.64	0.414452	0.207226	+	0.978293i	$0.433556\pi$
$420$	0	0
$421$	11655.5	1.34930	0.674649	−	0.738139i	$-0.264295\pi$
$421$	11655.5	1.34930	0.674649	+	0.738139i	$0.264295\pi$
$422$	8394.02i	0.968281i
$423$	− 911.725i	− 0.104798i
$424$	3317.66	0.380000
$425$	0	0
$426$	−4235.90	−0.481760
$427$	− 293.965i	− 0.0333161i
$428$	− 7484.50i	− 0.845273i
$429$	2461.29	0.276998
$430$	0	0
$431$	4727.38	0.528330	0.264165	−	0.964478i	$-0.414904\pi$
$431$	4727.38	0.528330	0.264165	+	0.964478i	$0.414904\pi$
$432$	− 2154.83i	− 0.239987i
$433$	− 3936.24i	− 0.436868i	−0.975852	−	0.218434i	$-0.929905\pi$
$433$	− 3936.24i	− 0.436868i	0.975852	−	0.218434i	$-0.0700948\pi$
$434$	1377.85	0.152394
$435$	0	0
$436$	−3912.97	−0.429810
$437$	1115.25i	0.122081i
$438$	4121.18i	0.449584i
$439$	−6859.77	−0.745784	−0.372892	−	0.927875i	$-0.621634\pi$
$439$	−6859.77	−0.745784	−0.372892	+	0.927875i	$0.621634\pi$
$440$	0	0
$441$	6037.45	0.651922
$442$	− 2549.57i	− 0.274368i
$443$	− 16814.4i	− 1.80334i	−0.432430	−	0.901668i	$-0.642344\pi$
$443$	− 16814.4i	− 1.80334i	0.432430	−	0.901668i	$-0.357656\pi$
$444$	1290.32	0.137919
$445$	0	0
$446$	10385.3	1.10260
$447$	− 4673.90i	− 0.494559i
$448$	− 191.051i	− 0.0201480i
$449$	−10673.8	−1.12189	−0.560943	−	0.827855i	$-0.689561\pi$
$449$	−10673.8	−1.12189	−0.560943	+	0.827855i	$0.689561\pi$
$450$	0	0
$451$	27082.5	2.82764
$452$	− 7041.82i	− 0.732786i
$453$	5557.94i	0.576457i
$454$	−9402.47	−0.971982
$455$	0	0
$456$	−1159.11	−0.119036
$457$	15493.0i	1.58585i	0.609318	+	0.792926i	$0.291443\pi$
$457$	15493.0i	1.58585i	−0.609318	+	0.792926i	$0.708557\pi$
$458$	4250.35i	0.433637i
$459$	14303.7	1.45455
$460$	0	0
$461$	−10793.0	−1.09041	−0.545207	−	0.838302i	$-0.683549\pi$
$461$	−10793.0	−1.09041	−0.545207	+	0.838302i	$0.683549\pi$
$462$	1224.28i	0.123287i
$463$	6165.87i	0.618904i	0.950915	+	0.309452i	$0.100146\pi$
$463$	6165.87i	0.618904i	−0.950915	+	0.309452i	$0.899854\pi$
$464$	−2163.61	−0.216473
$465$	0	0
$466$	−3293.34	−0.327385
$467$	− 10736.4i	− 1.06386i	−0.846789	−	0.531929i	$-0.821467\pi$
$467$	− 10736.4i	− 1.06386i	0.846789	−	0.531929i	$-0.178533\pi$
$468$	− 867.627i	− 0.0856967i
$469$	−408.744	−0.0402431
$470$	0	0
$471$	8469.77	0.828591
$472$	− 1468.69i	− 0.143224i
$473$	− 9337.47i	− 0.907690i
$474$	−3266.46	−0.316527
$475$	0	0
$476$	1268.19	0.122117
$477$	7494.35i	0.719377i
$478$	− 5755.75i	− 0.550757i
$479$	−6333.30	−0.604125	−0.302062	−	0.953288i	$-0.597675\pi$
$479$	−6333.30	−0.604125	−0.302062	+	0.953288i	$0.597675\pi$
$480$	0	0
$481$	1295.77	0.122832
$482$	− 8489.08i	− 0.802213i
$483$	205.158i	0.0193272i
$484$	−13514.2	−1.26918
$485$	0	0
$486$	7783.53	0.726477
$487$	− 16833.3i	− 1.56630i	−0.621834	−	0.783149i	$-0.713612\pi$
$487$	− 16833.3i	− 1.56630i	0.621834	−	0.783149i	$-0.286388\pi$
$488$	787.801i	0.0730780i
$489$	−6318.33	−0.584304
$490$	0	0
$491$	−4.80428	−0.000441577 0	−0.000220788	−	1.00000i	$-0.500070\pi$
$491$	−4.80428	−0.000441577 0	−0.000220788		1.00000i	$0.500070\pi$
$492$	4716.82i	0.432217i
$493$	− 14362.0i	− 1.31204i
$494$	−1164.01	−0.106014
$495$	0	0
$496$	−3692.52	−0.334272
$497$	− 2115.89i	− 0.190967i
$498$	121.090i	0.0108959i
$499$	−21206.9	−1.90251	−0.951253	−	0.308412i	$-0.900203\pi$
$499$	−21206.9	−1.90251	−0.951253	+	0.308412i	$0.900203\pi$
$500$	0	0
$501$	7432.56	0.662799
$502$	8803.98i	0.782750i
$503$	− 14513.7i	− 1.28655i	−0.765635	−	0.643275i	$-0.777575\pi$
$503$	− 14513.7i	− 1.28655i	0.765635	−	0.643275i	$-0.222425\pi$
$504$	431.570	0.0381422
$505$	0	0
$506$	−3156.81	−0.277346
$507$	− 6134.32i	− 0.537347i
$508$	5043.49i	0.440490i
$509$	2545.77	0.221689	0.110844	−	0.993838i	$-0.464645\pi$
$509$	2545.77	0.221689	0.110844	+	0.993838i	$0.464645\pi$
$510$	0	0
$511$	−2058.59	−0.178213
$512$	512.000i	0.0441942i
$513$	− 6530.35i	− 0.562032i
$514$	−3389.56	−0.290870
$515$	0	0
$516$	1626.26	0.138745
$517$	− 3462.28i	− 0.294528i
$518$	644.535i	0.0546704i
$519$	−10973.2	−0.928070
$520$	0	0
$521$	−21343.1	−1.79473	−0.897367	−	0.441285i	$-0.854523\pi$
$521$	−21343.1	−1.79473	−0.897367	+	0.441285i	$0.854523\pi$
$522$	− 4887.44i	− 0.409804i
$523$	789.191i	0.0659827i	0.999456	+	0.0329913i	$0.0105034\pi$
$523$	789.191i	0.0659827i	−0.999456	+	0.0329913i	$0.989497\pi$
$524$	−1777.13	−0.148157
$525$	0	0
$526$	−1454.92	−0.120604
$527$	− 24510.9i	− 2.02602i
$528$	− 3280.96i	− 0.270427i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	3317.66	0.271138
$532$	− 578.994i	− 0.0471852i
$533$	4736.74i	0.384936i
$534$	6497.90	0.526576
$535$	0	0
$536$	1095.40	0.0882723
$537$	− 9635.42i	− 0.774300i
$538$	− 189.987i	− 0.0152248i
$539$	22927.2	1.83218
$540$	0	0
$541$	13405.1	1.06531	0.532653	−	0.846334i	$-0.321195\pi$
$541$	13405.1	1.06531	0.532653	+	0.846334i	$0.321195\pi$
$542$	10643.1i	0.843468i
$543$	6646.16i	0.525256i
$544$	−3398.65	−0.267860
$545$	0	0
$546$	−214.127	−0.0167835
$547$	10088.8i	0.788605i	0.918981	+	0.394303i	$0.129014\pi$
$547$	10088.8i	0.788605i	−0.918981	+	0.394303i	$0.870986\pi$
$548$	− 1716.29i	− 0.133789i
$549$	−1779.58	−0.138344
$550$	0	0
$551$	−6556.98	−0.506963
$552$	− 549.806i	− 0.0423936i
$553$	− 1631.65i	− 0.125470i
$554$	−5122.50	−0.392842
$555$	0	0
$556$	−999.019	−0.0762012
$557$	− 11657.7i	− 0.886809i	−0.896322	−	0.443405i	$-0.853770\pi$
$557$	− 11657.7i	− 0.886809i	0.896322	−	0.443405i	$-0.146230\pi$
$558$	− 8341.12i	− 0.632810i
$559$	1633.13	0.123567
$560$	0	0
$561$	21779.0	1.63906
$562$	− 17507.5i	− 1.31407i
$563$	− 4839.43i	− 0.362270i	−0.983458	−	0.181135i	$-0.942023\pi$
$563$	− 4839.43i	− 0.362270i	0.983458	−	0.181135i	$-0.0579770\pi$
$564$	603.008	0.0450199
$565$	0	0
$566$	−4800.28	−0.356486
$567$	255.244i	0.0189052i
$568$	5670.40i	0.418882i
$569$	646.680	0.0476454	0.0238227	−	0.999716i	$-0.492416\pi$
$569$	646.680	0.0476454	0.0238227	+	0.999716i	$0.492416\pi$
$570$	0	0
$571$	3263.66	0.239194	0.119597	−	0.992822i	$-0.461840\pi$
$571$	3263.66	0.239194	0.119597	+	0.992822i	$0.461840\pi$
$572$	− 3294.82i	− 0.240845i
$573$	− 9781.54i	− 0.713141i
$574$	−2356.12	−0.171329
$575$	0	0
$576$	−1156.57	−0.0836639
$577$	− 8980.46i	− 0.647940i	−0.946067	−	0.323970i	$-0.894982\pi$
$577$	− 8980.46i	− 0.647940i	0.946067	−	0.323970i	$-0.105018\pi$
$578$	− 12734.2i	− 0.916388i
$579$	−15319.5	−1.09958
$580$	0	0
$581$	−60.4861	−0.00431908
$582$	− 6668.81i	− 0.474968i
$583$	28459.8i	2.02176i
$584$	5516.84	0.390905
$585$	0	0
$586$	2091.12	0.147412
$587$	− 6538.05i	− 0.459718i	−0.973224	−	0.229859i	$-0.926174\pi$
$587$	− 6538.05i	− 0.459718i	0.973224	−	0.229859i	$-0.0738265\pi$
$588$	3993.13i	0.280058i
$589$	−11190.4	−0.782841
$590$	0	0
$591$	9748.80	0.678532
$592$	− 1727.30i	− 0.119918i
$593$	− 7803.41i	− 0.540384i	−0.962806	−	0.270192i	$-0.912913\pi$
$593$	− 7803.41i	− 0.540384i	0.962806	−	0.270192i	$-0.0870872\pi$
$594$	18484.7	1.27683
$595$	0	0
$596$	−6256.73	−0.430010
$597$	− 502.839i	− 0.0344721i
$598$	− 552.127i	− 0.0377561i
$599$	27812.0	1.89711	0.948555	−	0.316613i	$-0.102546\pi$
$599$	27812.0	1.89711	0.948555	+	0.316613i	$0.102546\pi$
$600$	0	0
$601$	483.216	0.0327966	0.0163983	−	0.999866i	$-0.494780\pi$
$601$	483.216	0.0327966	0.0163983	+	0.999866i	$0.494780\pi$
$602$	812.342i	0.0549976i
$603$	2474.42i	0.167108i
$604$	7440.17	0.501219
$605$	0	0
$606$	−804.183	−0.0539071
$607$	− 18104.0i	− 1.21058i	−0.796006	−	0.605289i	$-0.793058\pi$
$607$	− 18104.0i	− 1.21058i	0.796006	−	0.605289i	$-0.206942\pi$
$608$	1551.65i	0.103500i
$609$	−1206.20	−0.0802592
$610$	0	0
$611$	605.554	0.0400951
$612$	− 7677.30i	− 0.507085i
$613$	− 2191.19i	− 0.144374i	−0.997391	−	0.0721870i	$-0.977002\pi$
$613$	− 2191.19i	− 0.144374i	0.997391	−	0.0721870i	$-0.0229978\pi$
$614$	15811.7	1.03926
$615$	0	0
$616$	1638.89	0.107196
$617$	− 8243.06i	− 0.537849i	−0.963161	−	0.268925i	$-0.913332\pi$
$617$	− 8243.06i	− 0.537849i	0.963161	−	0.268925i	$-0.0866683\pi$
$618$	5533.64i	0.360187i
$619$	−18298.0	−1.18814	−0.594069	−	0.804414i	$-0.702479\pi$
$619$	−18298.0	−1.18814	−0.594069	+	0.804414i	$0.702479\pi$
$620$	0	0
$621$	3097.57	0.200163
$622$	− 18455.3i	− 1.18969i
$623$	3245.80i	0.208732i
$624$	573.842	0.0368142
$625$	0	0
$626$	5425.15	0.346378
$627$	− 9943.19i	− 0.633322i
$628$	− 11338.1i	− 0.720445i
$629$	11465.8	0.726821
$630$	0	0
$631$	−26472.0	−1.67010	−0.835051	−	0.550172i	$-0.814562\pi$
$631$	−26472.0	−1.67010	−0.835051	+	0.550172i	$0.814562\pi$
$632$	4372.67i	0.275214i
$633$	− 12541.0i	− 0.787455i
$634$	6880.58	0.431014
$635$	0	0
$636$	−4956.71	−0.309035
$637$	4009.99i	0.249421i
$638$	− 18560.1i	− 1.15173i
$639$	−12809.0	−0.792984
$640$	0	0
$641$	7411.93	0.456714	0.228357	−	0.973577i	$-0.426665\pi$
$641$	7411.93	0.456714	0.228357	+	0.973577i	$0.426665\pi$
$642$	11182.1i	0.687419i
$643$	− 8405.96i	− 0.515550i	−0.966205	−	0.257775i	$-0.917011\pi$
$643$	− 8405.96i	− 0.515550i	0.966205	−	0.257775i	$-0.0829894\pi$
$644$	274.636	0.0168046
$645$	0	0
$646$	−10299.8	−0.627309
$647$	27878.3i	1.69398i	0.531605	+	0.846992i	$0.321589\pi$
$647$	27878.3i	1.69398i	−0.531605	+	0.846992i	$0.678411\pi$
$648$	− 684.030i	− 0.0414680i
$649$	12598.8	0.762014
$650$	0	0
$651$	−2058.56	−0.123934
$652$	8458.06i	0.508041i
$653$	18871.1i	1.13091i	0.824779	+	0.565455i	$0.191299\pi$
$653$	18871.1i	1.13091i	−0.824779	+	0.565455i	$0.808701\pi$
$654$	5846.12	0.349543
$655$	0	0
$656$	6314.20	0.375805
$657$	12462.1i	0.740021i
$658$	301.211i	0.0178457i
$659$	12144.4	0.717872	0.358936	−	0.933362i	$-0.383140\pi$
$659$	12144.4	0.717872	0.358936	+	0.933362i	$0.383140\pi$
$660$	0	0
$661$	−32504.2	−1.91266	−0.956328	−	0.292295i	$-0.905581\pi$
$661$	−32504.2	−1.91266	−0.956328	+	0.292295i	$0.905581\pi$
$662$	6519.49i	0.382760i
$663$	3809.16i	0.223130i
$664$	162.097	0.00947380
$665$	0	0
$666$	3901.84	0.227017
$667$	− 3110.19i	− 0.180551i
$668$	− 9949.62i	− 0.576291i
$669$	−15516.1	−0.896690
$670$	0	0
$671$	−6757.98	−0.388806
$672$	285.438i	0.0163854i
$673$	21387.8i	1.22502i	0.790462	+	0.612511i	$0.209840\pi$
$673$	21387.8i	1.22502i	−0.790462	+	0.612511i	$0.790160\pi$
$674$	−14326.8	−0.818762
$675$	0	0
$676$	−8211.74	−0.467213
$677$	10494.6i	0.595776i	0.954601	+	0.297888i	$0.0962821\pi$
$677$	10494.6i	0.595776i	−0.954601	+	0.297888i	$0.903718\pi$
$678$	10520.7i	0.595939i
$679$	3331.17	0.188275
$680$	0	0
$681$	14047.6	0.790466
$682$	− 31675.4i	− 1.77847i
$683$	− 24014.2i	− 1.34535i	−0.739937	−	0.672677i	$-0.765145\pi$
$683$	− 24014.2i	− 1.34535i	0.739937	−	0.672677i	$-0.234855\pi$
$684$	−3505.07	−0.195935
$685$	0	0
$686$	−4042.45	−0.224988
$687$	− 6350.18i	− 0.352656i
$688$	− 2177.00i	− 0.120636i
$689$	−4977.63	−0.275229
$690$	0	0
$691$	−10825.8	−0.595995	−0.297997	−	0.954567i	$-0.596319\pi$
$691$	−10825.8	−0.595995	−0.297997	+	0.954567i	$0.596319\pi$
$692$	14689.3i	0.806939i
$693$	3702.13i	0.202933i
$694$	3666.90	0.200567
$695$	0	0
$696$	3232.52	0.176046
$697$	41913.6i	2.27775i
$698$	20800.5i	1.12795i
$699$	4920.38	0.266246
$700$	0	0
$701$	−171.021	−0.00921451	−0.00460726	−	0.999989i	$-0.501467\pi$
$701$	−171.021	−0.00921451	−0.00460726	+	0.999989i	$0.501467\pi$
$702$	3232.99i	0.173819i
$703$	− 5234.69i	− 0.280840i
$704$	−4392.08	−0.235132
$705$	0	0
$706$	−6312.96	−0.336532
$707$	− 401.701i	− 0.0213685i
$708$	2194.28i	0.116477i
$709$	−28732.7	−1.52197	−0.760987	−	0.648767i	$-0.775285\pi$
$709$	−28732.7	−1.52197	−0.760987	+	0.648767i	$0.775285\pi$
$710$	0	0
$711$	−9877.53	−0.521007
$712$	− 8698.45i	− 0.457848i
$713$	− 5307.99i	− 0.278802i
$714$	−1894.73	−0.0993116
$715$	0	0
$716$	−12898.5	−0.673240
$717$	8599.31i	0.447904i
$718$	18648.3i	0.969288i
$719$	−6976.22	−0.361849	−0.180924	−	0.983497i	$-0.557909\pi$
$719$	−6976.22	−0.361849	−0.180924	+	0.983497i	$0.557909\pi$
$720$	0	0
$721$	−2764.13	−0.142776
$722$	− 9015.61i	− 0.464718i
$723$	12683.0i	0.652401i
$724$	8896.91	0.456700
$725$	0	0
$726$	20190.8	1.03216
$727$	13114.6i	0.669042i	0.942388	+	0.334521i	$0.108574\pi$
$727$	13114.6i	0.669042i	−0.942388	+	0.334521i	$0.891426\pi$
$728$	286.643i	0.0145930i
$729$	−9320.28	−0.473519
$730$	0	0
$731$	14450.9	0.731172
$732$	− 1177.00i	− 0.0594308i
$733$	22681.7i	1.14293i	0.820626	+	0.571466i	$0.193625\pi$
$733$	22681.7i	1.14293i	−0.820626	+	0.571466i	$0.806375\pi$
$734$	−7605.36	−0.382451
$735$	0	0
$736$	−736.000	−0.0368605
$737$	9396.62i	0.469646i
$738$	14263.3i	0.711436i
$739$	−24237.8	−1.20650	−0.603249	−	0.797553i	$-0.706127\pi$
$739$	−24237.8	−1.20650	−0.603249	+	0.797553i	$0.706127\pi$
$740$	0	0
$741$	1739.07	0.0862163
$742$	− 2475.95i	− 0.122500i
$743$	− 27356.7i	− 1.35077i	−0.737466	−	0.675384i	$-0.763978\pi$
$743$	− 27356.7i	− 1.35077i	0.737466	−	0.675384i	$-0.236022\pi$
$744$	5516.76	0.271847
$745$	0	0
$746$	−17856.1	−0.876350
$747$	366.166i	0.0179348i
$748$	− 29154.6i	− 1.42513i
$749$	−5585.63	−0.272489
$750$	0	0
$751$	22297.8	1.08343	0.541716	−	0.840561i	$-0.317775\pi$
$751$	22297.8	1.08343	0.541716	+	0.840561i	$0.317775\pi$
$752$	− 807.220i	− 0.0391440i
$753$	− 13153.5i	− 0.636573i
$754$	3246.17	0.156788
$755$	0	0
$756$	−1608.13	−0.0773641
$757$	− 16088.4i	− 0.772447i	−0.922405	−	0.386224i	$-0.873779\pi$
$757$	− 16088.4i	− 0.772447i	0.922405	−	0.386224i	$-0.126221\pi$
$758$	− 7596.26i	− 0.363996i
$759$	4716.39	0.225552
$760$	0	0
$761$	22500.2	1.07179	0.535894	−	0.844285i	$-0.319975\pi$
$761$	22500.2	1.07179	0.535894	+	0.844285i	$0.319975\pi$
$762$	− 7535.17i	− 0.358229i
$763$	2920.22i	0.138557i
$764$	−13094.1	−0.620063
$765$	0	0
$766$	2912.82	0.137395
$767$	2203.54i	0.103736i
$768$	− 764.947i	− 0.0359410i
$769$	22677.7	1.06343	0.531716	−	0.846923i	$-0.321547\pi$
$769$	22677.7	1.06343	0.531716	+	0.846923i	$0.321547\pi$
$770$	0	0
$771$	5064.13	0.236550
$772$	20507.5i	0.956062i
$773$	− 20465.1i	− 0.952235i	−0.879382	−	0.476118i	$-0.842044\pi$
$773$	− 20465.1i	− 0.952235i	0.879382	−	0.476118i	$-0.157956\pi$
$774$	4917.69	0.228376
$775$	0	0
$776$	−8927.24	−0.412976
$777$	− 962.960i	− 0.0444607i
$778$	16422.7i	0.756789i
$779$	19135.6	0.880108
$780$	0	0
$781$	−48642.3	−2.22863
$782$	− 4885.56i	− 0.223411i
$783$	18211.8i	0.831208i
$784$	5345.42	0.243505
$785$	0	0
$786$	2655.10	0.120489
$787$	− 28850.8i	− 1.30676i	−0.757031	−	0.653379i	$-0.773351\pi$
$787$	− 28850.8i	− 1.30676i	0.757031	−	0.653379i	$-0.226649\pi$
$788$	− 13050.3i	− 0.589971i
$789$	2173.71	0.0980813
$790$	0	0
$791$	−5255.26	−0.236227
$792$	− 9921.38i	− 0.445127i
$793$	− 1181.97i	− 0.0529295i
$794$	−1828.73	−0.0817370
$795$	0	0
$796$	−673.128	−0.0299728
$797$	− 39332.9i	− 1.74811i	−0.485828	−	0.874054i	$-0.661482\pi$
$797$	− 39332.9i	− 1.74811i	0.485828	−	0.874054i	$-0.338518\pi$
$798$	865.038i	0.0383735i
$799$	5358.31	0.237251
$800$	0	0
$801$	19649.1	0.866752
$802$	18829.9i	0.829063i
$803$	47325.0i	2.07978i
$804$	−1636.56	−0.0717875
$805$	0	0
$806$	5540.05	0.242109
$807$	283.848i	0.0123816i
$808$	1076.52i	0.0468712i
$809$	−33946.5	−1.47527	−0.737637	−	0.675198i	$-0.764058\pi$
$809$	−33946.5	−1.47527	−0.737637	+	0.675198i	$0.764058\pi$
$810$	0	0
$811$	−20772.2	−0.899398	−0.449699	−	0.893180i	$-0.648469\pi$
$811$	−20772.2	−0.899398	−0.449699	+	0.893180i	$0.648469\pi$
$812$	1614.69i	0.0697839i
$813$	− 15901.2i	− 0.685951i
$814$	14817.2	0.638015
$815$	0	0
$816$	5077.71	0.217837
$817$	− 6597.56i	− 0.282521i
$818$	− 6598.83i	− 0.282057i
$819$	−647.504	−0.0276259
$820$	0	0
$821$	6466.37	0.274882	0.137441	−	0.990510i	$-0.456112\pi$
$821$	6466.37	0.274882	0.137441	+	0.990510i	$0.456112\pi$
$822$	2564.20i	0.108804i
$823$	16503.7i	0.699008i	0.936935	+	0.349504i	$0.113650\pi$
$823$	16503.7i	0.699008i	−0.936935	+	0.349504i	$0.886350\pi$
$824$	7407.63	0.313176
$825$	0	0
$826$	−1096.07	−0.0461710
$827$	36540.8i	1.53645i	0.640177	+	0.768227i	$0.278861\pi$
$827$	36540.8i	1.53645i	−0.640177	+	0.768227i	$0.721139\pi$
$828$	− 1662.57i	− 0.0697805i
$829$	24874.5	1.04213	0.521066	−	0.853516i	$-0.325535\pi$
$829$	24874.5	1.04213	0.521066	+	0.853516i	$0.325535\pi$
$830$	0	0
$831$	7653.21	0.319479
$832$	− 768.177i	− 0.0320093i
$833$	35482.8i	1.47588i
$834$	1492.57	0.0619707
$835$	0	0
$836$	−13310.5	−0.550662
$837$	31081.0i	1.28353i
$838$	7109.28i	0.293062i
$839$	−15814.3	−0.650741	−0.325370	−	0.945587i	$-0.605489\pi$
$839$	−15814.3	−0.650741	−0.325370	+	0.945587i	$0.605489\pi$
$840$	0	0
$841$	−6102.97	−0.250234
$842$	23311.0i	0.954097i
$843$	26156.8i	1.06867i
$844$	−16788.0	−0.684678
$845$	0	0
$846$	1823.45	0.0741034
$847$	10085.6i	0.409144i
$848$	6635.32i	0.268700i
$849$	7171.80	0.289912
$850$	0	0
$851$	2482.99	0.100019
$852$	− 8471.79i	− 0.340656i
$853$	31093.2i	1.24808i	0.781394	+	0.624039i	$0.214509\pi$
$853$	31093.2i	1.24808i	−0.781394	+	0.624039i	$0.785491\pi$
$854$	587.931	0.0235580
$855$	0	0
$856$	14969.0	0.597699
$857$	34212.4i	1.36368i	0.731501	+	0.681840i	$0.238820\pi$
$857$	34212.4i	1.36368i	−0.731501	+	0.681840i	$0.761180\pi$
$858$	4922.58i	0.195867i
$859$	−4821.57	−0.191513	−0.0957567	−	0.995405i	$-0.530527\pi$
$859$	−4821.57	−0.191513	−0.0957567	+	0.995405i	$0.530527\pi$
$860$	0	0
$861$	3520.13	0.139333
$862$	9454.77i	0.373585i
$863$	− 19145.6i	− 0.755185i	−0.925972	−	0.377593i	$-0.876752\pi$
$863$	− 19145.6i	− 0.755185i	0.925972	−	0.377593i	$-0.123248\pi$
$864$	4309.66	0.169696
$865$	0	0
$866$	7872.49	0.308912
$867$	19025.4i	0.745254i
$868$	2755.70i	0.107759i
$869$	−37509.9	−1.46426
$870$	0	0
$871$	−1643.47	−0.0639345
$872$	− 7825.93i	− 0.303921i
$873$	− 20166.0i	− 0.781804i
$874$	−2230.50	−0.0863246
$875$	0	0
$876$	−8242.36	−0.317904
$877$	16890.8i	0.650354i	0.945653	+	0.325177i	$0.105424\pi$
$877$	16890.8i	0.650354i	−0.945653	+	0.325177i	$0.894576\pi$
$878$	− 13719.5i	− 0.527349i
$879$	−3124.21	−0.119883
$880$	0	0
$881$	−27802.3	−1.06321	−0.531603	−	0.846994i	$-0.678410\pi$
$881$	−27802.3	−1.06321	−0.531603	+	0.846994i	$0.678410\pi$
$882$	12074.9i	0.460979i
$883$	− 9794.15i	− 0.373272i	−0.982429	−	0.186636i	$-0.940241\pi$
$883$	− 9794.15i	− 0.373272i	0.982429	−	0.186636i	$-0.0597585\pi$
$884$	5099.15	0.194008
$885$	0	0
$886$	33628.9	1.27515
$887$	− 41652.4i	− 1.57672i	−0.615214	−	0.788360i	$-0.710930\pi$
$887$	− 41652.4i	− 1.57672i	0.615214	−	0.788360i	$-0.289070\pi$
$888$	2580.65i	0.0975235i
$889$	3763.93	0.142000
$890$	0	0
$891$	5867.80	0.220627
$892$	20770.7i	0.779656i
$893$	− 2446.34i	− 0.0916724i
$894$	9347.80	0.349706
$895$	0	0
$896$	382.102	0.0142468
$897$	824.898i	0.0307052i
$898$	− 21347.5i	− 0.793293i
$899$	31207.7	1.15777
$900$	0	0
$901$	−44045.2	−1.62859
$902$	54164.9i	1.99944i
$903$	− 1213.67i	− 0.0447269i
$904$	14083.6	0.518158
$905$	0	0
$906$	−11115.9	−0.407617
$907$	5467.21i	0.200150i	0.994980	+	0.100075i	$0.0319082\pi$
$907$	5467.21i	0.200150i	−0.994980	+	0.100075i	$0.968092\pi$
$908$	− 18804.9i	− 0.687295i
$909$	−2431.79	−0.0887318
$910$	0	0
$911$	33520.9	1.21910	0.609549	−	0.792749i	$-0.291351\pi$
$911$	33520.9	1.21910	0.609549	+	0.792749i	$0.291351\pi$
$912$	− 2318.23i	− 0.0841712i
$913$	1390.52i	0.0504046i
$914$	−30986.1	−1.12137
$915$	0	0
$916$	−8500.70	−0.306628
$917$	1326.26i	0.0477612i
$918$	28607.4i	1.02853i
$919$	−21423.9	−0.768996	−0.384498	−	0.923126i	$-0.625625\pi$
$919$	−21423.9	−0.768996	−0.384498	+	0.923126i	$0.625625\pi$
$920$	0	0
$921$	−23623.2	−0.845182
$922$	− 21586.0i	− 0.771039i
$923$	− 8507.56i	− 0.303391i
$924$	−2448.56	−0.0871773
$925$	0	0
$926$	−12331.7	−0.437631
$927$	16733.3i	0.592873i
$928$	− 4327.23i	− 0.153069i
$929$	−25838.9	−0.912538	−0.456269	−	0.889842i	$-0.650814\pi$
$929$	−25838.9	−0.912538	−0.456269	+	0.889842i	$0.650814\pi$
$930$	0	0
$931$	16199.7	0.570271
$932$	− 6586.69i	− 0.231496i
$933$	27572.9i	0.967519i
$934$	21472.8	0.752261
$935$	0	0
$936$	1735.25	0.0605967
$937$	48095.9i	1.67687i	0.545003	+	0.838434i	$0.316528\pi$
$937$	48095.9i	1.67687i	−0.545003	+	0.838434i	$0.683472\pi$
$938$	− 817.487i	− 0.0284562i
$939$	−8105.38	−0.281692
$940$	0	0
$941$	−32450.4	−1.12418	−0.562089	−	0.827077i	$-0.690002\pi$
$941$	−32450.4	−1.12418	−0.562089	+	0.827077i	$0.690002\pi$
$942$	16939.5i	0.585903i
$943$	9076.66i	0.313443i
$944$	2937.38	0.101275
$945$	0	0
$946$	18674.9	0.641834
$947$	39067.9i	1.34059i	0.742096	+	0.670294i	$0.233832\pi$
$947$	39067.9i	1.34059i	−0.742096	+	0.670294i	$0.766168\pi$
$948$	− 6532.93i	− 0.223818i
$949$	−8277.16	−0.283127
$950$	0	0
$951$	−10279.8	−0.350522
$952$	2536.39i	0.0863496i
$953$	− 17756.2i	− 0.603546i	−0.953380	−	0.301773i	$-0.902422\pi$
$953$	− 17756.2i	− 0.603546i	0.953380	−	0.301773i	$-0.0975784\pi$
$954$	−14988.7	−0.508676
$955$	0	0
$956$	11511.5	0.389444
$957$	27729.5i	0.936642i
$958$	− 12666.6i	− 0.427181i
$959$	−1280.85	−0.0431292
$960$	0	0
$961$	23469.5	0.787805
$962$	2591.54i	0.0868552i
$963$	33813.9i	1.13150i
$964$	16978.2	0.567251
$965$	0	0
$966$	−410.316	−0.0136664
$967$	− 10696.2i	− 0.355705i	−0.984057	−	0.177853i	$-0.943085\pi$
$967$	− 10696.2i	− 0.355705i	0.984057	−	0.177853i	$-0.0569151\pi$
$968$	− 27028.5i	− 0.897446i
$969$	15388.3	0.510160
$970$	0	0
$971$	−53635.1	−1.77264	−0.886320	−	0.463074i	$-0.846746\pi$
$971$	−53635.1	−1.77264	−0.886320	+	0.463074i	$0.846746\pi$
$972$	15567.1i	0.513697i
$973$	745.561i	0.0245649i
$974$	33666.5	1.10754
$975$	0	0
$976$	−1575.60	−0.0516740
$977$	− 21627.7i	− 0.708220i	−0.935204	−	0.354110i	$-0.884784\pi$
$977$	− 21627.7i	− 0.708220i	0.935204	−	0.354110i	$-0.115216\pi$
$978$	− 12636.7i	− 0.413165i
$979$	74617.7	2.43595
$980$	0	0
$981$	17678.2	0.575353
$982$	− 9.60856i	0 0.000312242i
$983$	− 12031.4i	− 0.390378i	−0.980766	−	0.195189i	$-0.937468\pi$
$983$	− 12031.4i	− 0.390378i	0.980766	−	0.195189i	$-0.0625321\pi$
$984$	−9433.65	−0.305624
$985$	0	0
$986$	28724.1	0.927749
$987$	− 450.021i	− 0.0145130i
$988$	− 2328.01i	− 0.0749635i
$989$	3129.44	0.100617
$990$	0	0
$991$	−24573.0	−0.787675	−0.393838	−	0.919180i	$-0.628853\pi$
$991$	−24573.0	−0.787675	−0.393838	+	0.919180i	$0.628853\pi$
$992$	− 7385.03i	− 0.236366i
$993$	− 9740.36i	− 0.311280i
$994$	4231.78	0.135034
$995$	0	0
$996$	−242.180	−0.00770458
$997$	− 14209.7i	− 0.451379i	−0.974199	−	0.225690i	$-0.927536\pi$
$997$	− 14209.7i	− 0.451379i	0.974199	−	0.225690i	$-0.0724635\pi$
$998$	− 42413.8i	− 1.34527i
$999$	−14539.2	−0.460460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.o.599.7		8
5.2	odd	4		1150.4.a.n.1.3		4
5.3	odd	4		230.4.a.j.1.2	✓	4
5.4	even	2	inner	1150.4.b.o.599.2		8
15.8	even	4		2070.4.a.bg.1.2		4
20.3	even	4		1840.4.a.k.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.2	✓	4	5.3	odd	4
1150.4.a.n.1.3		4	5.2	odd	4
1150.4.b.o.599.2		8	5.4	even	2	inner
1150.4.b.o.599.7		8	1.1	even	1	trivial
1840.4.a.k.1.3		4	20.3	even	4
2070.4.a.bg.1.2		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -2.98808i q^{3} -4.00000 q^{4} +5.97615 q^{6} +2.98517i q^{7} -8.00000i q^{8} +18.0714 q^{9} +68.6262 q^{11} +11.9523i q^{12} +12.0028i q^{13} -5.97035 q^{14} +16.0000 q^{16} +106.208i q^{17} +36.1428i q^{18} +48.4891 q^{19} +8.91992 q^{21} +137.252i q^{22} +23.0000i q^{23} -23.9046 q^{24} -24.0055 q^{26} -134.677i q^{27} -11.9407i q^{28} -135.226 q^{29} -230.782 q^{31} +32.0000i q^{32} -205.060i q^{33} -212.416 q^{34} -72.2856 q^{36} -107.956i q^{37} +96.9782i q^{38} +35.8651 q^{39} +394.637 q^{41} +17.8398i q^{42} -136.063i q^{43} -274.505 q^{44} -46.0000 q^{46} -50.4512i q^{47} -47.8092i q^{48} +334.089 q^{49} +317.357 q^{51} -48.0110i q^{52} +414.707i q^{53} +269.353 q^{54} +23.8814 q^{56} -144.889i q^{57} -270.452i q^{58} +183.586 q^{59} -98.4751 q^{61} -461.565i q^{62} +53.9463i q^{63} -64.0000 q^{64} +410.121 q^{66} +136.925i q^{67} -424.831i q^{68} +68.7257 q^{69} -708.800 q^{71} -144.571i q^{72} +689.605i q^{73} +215.912 q^{74} -193.956 q^{76} +204.861i q^{77} +71.7303i q^{78} -546.583 q^{79} +85.5038 q^{81} +789.275i q^{82} +20.2622i q^{83} -35.6797 q^{84} +272.125 q^{86} +404.065i q^{87} -549.010i q^{88} +1087.31 q^{89} -35.8303 q^{91} -92.0000i q^{92} +689.595i q^{93} +100.902 q^{94} +95.6184 q^{96} -1115.90i q^{97} +668.177i q^{98} +1240.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) \) 8 * q - 32 * q^4 + 56 * q^6 - 128 * q^9 + 42 * q^11 - 32 * q^14 + 128 * q^16 - 346 * q^19 - 240 * q^21 - 224 * q^24 + 280 * q^26 + 236 * q^29 + 34 * q^31 - 224 * q^34 + 512 * q^36 + 442 * q^39 + 278 * q^41 - 168 * q^44 - 368 * q^46 + 248 * q^49 + 878 * q^51 - 1556 * q^54 + 128 * q^56 + 906 * q^59 - 654 * q^61 - 512 * q^64 - 356 * q^66 + 644 * q^69 + 390 * q^71 + 1372 * q^74 + 1384 * q^76 + 2280 * q^79 + 2912 * q^81 + 960 * q^84 - 200 * q^86 + 4340 * q^89 + 1114 * q^91 - 1468 * q^94 + 896 * q^96 + 5490 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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