LMFDB - Embedded newform 1984.4.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1984,4,Mod(1,1984)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1984.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1984.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-4,0,-15,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$117.059789451$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 32x^{3} + 19x^{2} + 228x + 172 $ x^5 - 2x^4 - 32x^3 + 19x^2 + 228x + 172
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.15912$ of defining polynomial
Character	$\chi$	$=$	1984.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0377000 q^{3} -13.5012 q^{5} -3.86558 q^{7} -26.9986 q^{9} +37.1306 q^{11} -48.7405 q^{13} -0.508993 q^{15} +32.0202 q^{17} -52.2455 q^{19} -0.145732 q^{21} +23.0922 q^{23} +57.2812 q^{25} -2.03574 q^{27} -221.608 q^{29} -31.0000 q^{31} +1.39982 q^{33} +52.1898 q^{35} -310.448 q^{37} -1.83752 q^{39} -432.581 q^{41} +443.095 q^{43} +364.512 q^{45} +206.668 q^{47} -328.057 q^{49} +1.20716 q^{51} -428.205 q^{53} -501.306 q^{55} -1.96965 q^{57} -621.019 q^{59} +148.908 q^{61} +104.365 q^{63} +658.053 q^{65} +648.871 q^{67} +0.870577 q^{69} -875.630 q^{71} -55.1632 q^{73} +2.15950 q^{75} -143.531 q^{77} -771.120 q^{79} +728.885 q^{81} +365.678 q^{83} -432.310 q^{85} -8.35461 q^{87} +1632.10 q^{89} +188.410 q^{91} -1.16870 q^{93} +705.374 q^{95} +27.5970 q^{97} -1002.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 15 q^{5} + 9 q^{7} + 29 q^{9} - 88 q^{11} + 28 q^{13} - 130 q^{15} + 138 q^{17} + 43 q^{19} - 170 q^{21} + 206 q^{23} + 466 q^{25} - 172 q^{27} - 474 q^{29} - 155 q^{31} - 236 q^{33} + 79 q^{35}+ \cdots - 3988 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 15 * q^5 + 9 * q^7 + 29 * q^9 - 88 * q^11 + 28 * q^13 - 130 * q^15 + 138 * q^17 + 43 * q^19 - 170 * q^21 + 206 * q^23 + 466 * q^25 - 172 * q^27 - 474 * q^29 - 155 * q^31 - 236 * q^33 + 79 * q^35 + 508 * q^37 - 792 * q^39 + 473 * q^41 + 82 * q^43 - 15 * q^45 - 644 * q^47 - 776 * q^49 + 1360 * q^51 - 374 * q^53 - 798 * q^55 + 558 * q^57 + 541 * q^59 - 440 * q^61 - 663 * q^63 + 1602 * q^65 + 1884 * q^67 - 2500 * q^69 + 491 * q^71 + 302 * q^73 + 2418 * q^75 + 1154 * q^77 - 1244 * q^79 - 127 * q^81 - 1544 * q^83 - 802 * q^85 - 220 * q^87 + 3056 * q^89 + 2042 * q^91 + 124 * q^93 + 217 * q^95 + 583 * q^97 - 3988 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	0.0377000	0.00725536	0.00362768	−	0.999993i	$-0.498845\pi$
$3$	0.0377000	0.00725536	0.00362768	+	0.999993i	$0.498845\pi$
$4$	0	0
$5$	−13.5012	−1.20758	−0.603790	−	0.797143i	$-0.706343\pi$
$5$	−13.5012	−1.20758	−0.603790	+	0.797143i	$0.706343\pi$
$6$	0	0
$7$	−3.86558	−0.208722	−0.104361	−	0.994540i	$-0.533280\pi$
$7$	−3.86558	−0.208722	−0.104361	+	0.994540i	$0.533280\pi$
$8$	0	0
$9$	−26.9986	−0.999947
$10$	0	0
$11$	37.1306	1.01776	0.508878	−	0.860839i	$-0.330061\pi$
$11$	37.1306	1.01776	0.508878	+	0.860839i	$0.330061\pi$
$12$	0	0
$13$	−48.7405	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$13$	−48.7405	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$14$	0	0
$15$	−0.508993	−0.00876143
$16$	0	0
$17$	32.0202	0.456826	0.228413	−	0.973564i	$-0.426646\pi$
$17$	32.0202	0.456826	0.228413	+	0.973564i	$0.426646\pi$
$18$	0	0
$19$	−52.2455	−0.630838	−0.315419	−	0.948952i	$-0.602145\pi$
$19$	−52.2455	−0.630838	−0.315419	+	0.948952i	$0.602145\pi$
$20$	0	0
$21$	−0.145732	−0.00151435
$22$	0	0
$23$	23.0922	0.209351	0.104675	−	0.994506i	$-0.466620\pi$
$23$	23.0922	0.209351	0.104675	+	0.994506i	$0.466620\pi$
$24$	0	0
$25$	57.2812	0.458249
$26$	0	0
$27$	−2.03574	−0.0145103
$28$	0	0
$29$	−221.608	−1.41902	−0.709510	−	0.704696i	$-0.751084\pi$
$29$	−221.608	−1.41902	−0.709510	+	0.704696i	$0.751084\pi$
$30$	0	0
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	0	0
$33$	1.39982	0.00738418
$34$	0	0
$35$	52.1898	0.252048
$36$	0	0
$37$	−310.448	−1.37939	−0.689693	−	0.724102i	$-0.742254\pi$
$37$	−310.448	−1.37939	−0.689693	+	0.724102i	$0.742254\pi$
$38$	0	0
$39$	−1.83752	−0.00754456
$40$	0	0
$41$	−432.581	−1.64775	−0.823875	−	0.566772i	$-0.808192\pi$
$41$	−432.581	−1.64775	−0.823875	+	0.566772i	$0.808192\pi$
$42$	0	0
$43$	443.095	1.57143	0.785713	−	0.618591i	$-0.212296\pi$
$43$	443.095	1.57143	0.785713	+	0.618591i	$0.212296\pi$
$44$	0	0
$45$	364.512	1.20752
$46$	0	0
$47$	206.668	0.641395	0.320698	−	0.947182i	$-0.396083\pi$
$47$	206.668	0.641395	0.320698	+	0.947182i	$0.396083\pi$
$48$	0	0
$49$	−328.057	−0.956435
$50$	0	0
$51$	1.20716	0.00331444
$52$	0	0
$53$	−428.205	−1.10978	−0.554892	−	0.831923i	$-0.687240\pi$
$53$	−428.205	−1.10978	−0.554892	+	0.831923i	$0.687240\pi$
$54$	0	0
$55$	−501.306	−1.22902
$56$	0	0
$57$	−1.96965	−0.00457696
$58$	0	0
$59$	−621.019	−1.37034	−0.685168	−	0.728385i	$-0.740271\pi$
$59$	−621.019	−1.37034	−0.685168	+	0.728385i	$0.740271\pi$
$60$	0	0
$61$	148.908	0.312552	0.156276	−	0.987713i	$-0.450051\pi$
$61$	148.908	0.312552	0.156276	+	0.987713i	$0.450051\pi$
$62$	0	0
$63$	104.365	0.208711
$64$	0	0
$65$	658.053	1.25571
$66$	0	0
$67$	648.871	1.18317	0.591583	−	0.806244i	$-0.298503\pi$
$67$	648.871	1.18317	0.591583	+	0.806244i	$0.298503\pi$
$68$	0	0
$69$	0.870577	0.00151891
$70$	0	0
$71$	−875.630	−1.46364	−0.731818	−	0.681500i	$-0.761328\pi$
$71$	−875.630	−1.46364	−0.731818	+	0.681500i	$0.761328\pi$
$72$	0	0
$73$	−55.1632	−0.0884434	−0.0442217	−	0.999022i	$-0.514081\pi$
$73$	−55.1632	−0.0884434	−0.0442217	+	0.999022i	$0.514081\pi$
$74$	0	0
$75$	2.15950	0.00332476
$76$	0	0
$77$	−143.531	−0.212428
$78$	0	0
$79$	−771.120	−1.09820	−0.549100	−	0.835757i	$-0.685029\pi$
$79$	−771.120	−1.09820	−0.549100	+	0.835757i	$0.685029\pi$
$80$	0	0
$81$	728.885	0.999842
$82$	0	0
$83$	365.678	0.483595	0.241797	−	0.970327i	$-0.422263\pi$
$83$	365.678	0.483595	0.241797	+	0.970327i	$0.422263\pi$
$84$	0	0
$85$	−432.310	−0.551654
$86$	0	0
$87$	−8.35461	−0.0102955
$88$	0	0
$89$	1632.10	1.94385	0.971925	−	0.235290i	$-0.0756040\pi$
$89$	1632.10	1.94385	0.971925	+	0.235290i	$0.0756040\pi$
$90$	0	0
$91$	188.410	0.217041
$92$	0	0
$93$	−1.16870	−0.00130310
$94$	0	0
$95$	705.374	0.761788
$96$	0	0
$97$	27.5970	0.0288871	0.0144435	−	0.999896i	$-0.495402\pi$
$97$	27.5970	0.0288871	0.0144435	+	0.999896i	$0.495402\pi$
$98$	0	0
$99$	−1002.47	−1.01770
$100$	0	0
$101$	311.229	0.306618	0.153309	−	0.988178i	$-0.451007\pi$
$101$	311.229	0.306618	0.153309	+	0.988178i	$0.451007\pi$
$102$	0	0
$103$	−36.1976	−0.0346277	−0.0173138	−	0.999850i	$-0.505511\pi$
$103$	−36.1976	−0.0346277	−0.0173138	+	0.999850i	$0.505511\pi$
$104$	0	0
$105$	1.96755	0.00182870
$106$	0	0
$107$	1267.17	1.14488	0.572439	−	0.819947i	$-0.305997\pi$
$107$	1267.17	1.14488	0.572439	+	0.819947i	$0.305997\pi$
$108$	0	0
$109$	−849.046	−0.746091	−0.373045	−	0.927813i	$-0.621686\pi$
$109$	−849.046	−0.746091	−0.373045	+	0.927813i	$0.621686\pi$
$110$	0	0
$111$	−11.7039	−0.0100079
$112$	0	0
$113$	465.656	0.387657	0.193828	−	0.981035i	$-0.437910\pi$
$113$	465.656	0.387657	0.193828	+	0.981035i	$0.437910\pi$
$114$	0	0
$115$	−311.772	−0.252808
$116$	0	0
$117$	1315.92	1.03981
$118$	0	0
$119$	−123.777	−0.0953495
$120$	0	0
$121$	47.6841	0.0358258
$122$	0	0
$123$	−16.3083	−0.0119550
$124$	0	0
$125$	914.282	0.654207
$126$	0	0
$127$	1350.39	0.943526	0.471763	−	0.881725i	$-0.343618\pi$
$127$	1350.39	0.943526	0.471763	+	0.881725i	$0.343618\pi$
$128$	0	0
$129$	16.7047	0.0114013
$130$	0	0
$131$	−2772.41	−1.84905	−0.924527	−	0.381116i	$-0.875540\pi$
$131$	−2772.41	−1.84905	−0.924527	+	0.381116i	$0.875540\pi$
$132$	0	0
$133$	201.959	0.131670
$134$	0	0
$135$	27.4849	0.0175224
$136$	0	0
$137$	−2135.13	−1.33151	−0.665754	−	0.746171i	$-0.731890\pi$
$137$	−2135.13	−1.33151	−0.665754	+	0.746171i	$0.731890\pi$
$138$	0	0
$139$	1942.68	1.18544	0.592720	−	0.805408i	$-0.298054\pi$
$139$	1942.68	1.18544	0.592720	+	0.805408i	$0.298054\pi$
$140$	0	0
$141$	7.79136	0.00465355
$142$	0	0
$143$	−1809.77	−1.05832
$144$	0	0
$145$	2991.96	1.71358
$146$	0	0
$147$	−12.3677	−0.00693928
$148$	0	0
$149$	−115.235	−0.0633583	−0.0316792	−	0.999498i	$-0.510085\pi$
$149$	−115.235	−0.0633583	−0.0316792	+	0.999498i	$0.510085\pi$
$150$	0	0
$151$	1445.85	0.779217	0.389608	−	0.920981i	$-0.372610\pi$
$151$	1445.85	0.779217	0.389608	+	0.920981i	$0.372610\pi$
$152$	0	0
$153$	−864.501	−0.456802
$154$	0	0
$155$	418.536	0.216888
$156$	0	0
$157$	3236.83	1.64540	0.822698	−	0.568478i	$-0.192468\pi$
$157$	3236.83	1.64540	0.822698	+	0.568478i	$0.192468\pi$
$158$	0	0
$159$	−16.1433	−0.00805188
$160$	0	0
$161$	−89.2649	−0.0436960
$162$	0	0
$163$	3104.67	1.49188	0.745940	−	0.666014i	$-0.232001\pi$
$163$	3104.67	1.49188	0.745940	+	0.666014i	$0.232001\pi$
$164$	0	0
$165$	−18.8992	−0.00891699
$166$	0	0
$167$	1992.12	0.923082	0.461541	−	0.887119i	$-0.347297\pi$
$167$	1992.12	0.923082	0.461541	+	0.887119i	$0.347297\pi$
$168$	0	0
$169$	178.638	0.0813102
$170$	0	0
$171$	1410.55	0.630805
$172$	0	0
$173$	163.951	0.0720518	0.0360259	−	0.999351i	$-0.488530\pi$
$173$	163.951	0.0720518	0.0360259	+	0.999351i	$0.488530\pi$
$174$	0	0
$175$	−221.425	−0.0956466
$176$	0	0
$177$	−23.4124	−0.00994228
$178$	0	0
$179$	−3809.18	−1.59057	−0.795284	−	0.606237i	$-0.792678\pi$
$179$	−3809.18	−1.59057	−0.795284	+	0.606237i	$0.792678\pi$
$180$	0	0
$181$	1246.33	0.511819	0.255909	−	0.966701i	$-0.417625\pi$
$181$	1246.33	0.511819	0.255909	+	0.966701i	$0.417625\pi$
$182$	0	0
$183$	5.61381	0.00226768
$184$	0	0
$185$	4191.40	1.66572
$186$	0	0
$187$	1188.93	0.464937
$188$	0	0
$189$	7.86933	0.00302862
$190$	0	0
$191$	1696.97	0.642872	0.321436	−	0.946931i	$-0.395834\pi$
$191$	1696.97	0.642872	0.321436	+	0.946931i	$0.395834\pi$
$192$	0	0
$193$	1757.23	0.655377	0.327689	−	0.944786i	$-0.393730\pi$
$193$	1757.23	0.655377	0.327689	+	0.944786i	$0.393730\pi$
$194$	0	0
$195$	24.8086	0.00911066
$196$	0	0
$197$	2899.42	1.04861	0.524303	−	0.851532i	$-0.324326\pi$
$197$	2899.42	1.04861	0.524303	+	0.851532i	$0.324326\pi$
$198$	0	0
$199$	3660.29	1.30387	0.651937	−	0.758273i	$-0.273957\pi$
$199$	3660.29	1.30387	0.651937	+	0.758273i	$0.273957\pi$
$200$	0	0
$201$	24.4624	0.00858430
$202$	0	0
$203$	856.643	0.296180
$204$	0	0
$205$	5840.34	1.98979
$206$	0	0
$207$	−623.458	−0.209340
$208$	0	0
$209$	−1939.91	−0.642039
$210$	0	0
$211$	−5363.21	−1.74985	−0.874925	−	0.484258i	$-0.839090\pi$
$211$	−5363.21	−1.74985	−0.874925	+	0.484258i	$0.839090\pi$
$212$	0	0
$213$	−33.0112	−0.0106192
$214$	0	0
$215$	−5982.29	−1.89762
$216$	0	0
$217$	119.833	0.0374875
$218$	0	0
$219$	−2.07965	−0.000641688 0
$220$	0	0
$221$	−1560.68	−0.475036
$222$	0	0
$223$	−4557.96	−1.36872	−0.684358	−	0.729146i	$-0.739918\pi$
$223$	−4557.96	−1.36872	−0.684358	+	0.729146i	$0.739918\pi$
$224$	0	0
$225$	−1546.51	−0.458225
$226$	0	0
$227$	−2234.01	−0.653199	−0.326600	−	0.945163i	$-0.605903\pi$
$227$	−2234.01	−0.653199	−0.326600	+	0.945163i	$0.605903\pi$
$228$	0	0
$229$	1006.17	0.290347	0.145173	−	0.989406i	$-0.453626\pi$
$229$	1006.17	0.290347	0.145173	+	0.989406i	$0.453626\pi$
$230$	0	0
$231$	−5.41113	−0.00154124
$232$	0	0
$233$	2154.84	0.605873	0.302937	−	0.953011i	$-0.402033\pi$
$233$	2154.84	0.605873	0.302937	+	0.953011i	$0.402033\pi$
$234$	0	0
$235$	−2790.25	−0.774536
$236$	0	0
$237$	−29.0712	−0.00796784
$238$	0	0
$239$	4483.80	1.21353	0.606764	−	0.794882i	$-0.292467\pi$
$239$	4483.80	1.21353	0.606764	+	0.794882i	$0.292467\pi$
$240$	0	0
$241$	−1460.11	−0.390266	−0.195133	−	0.980777i	$-0.562514\pi$
$241$	−1460.11	−0.390266	−0.195133	+	0.980777i	$0.562514\pi$
$242$	0	0
$243$	82.4440	0.0217645
$244$	0	0
$245$	4429.15	1.15497
$246$	0	0
$247$	2546.47	0.655984
$248$	0	0
$249$	13.7860	0.00350865
$250$	0	0
$251$	394.531	0.0992134	0.0496067	−	0.998769i	$-0.484203\pi$
$251$	394.531	0.0992134	0.0496067	+	0.998769i	$0.484203\pi$
$252$	0	0
$253$	857.430	0.213068
$254$	0	0
$255$	−16.2981	−0.00400245
$256$	0	0
$257$	6505.01	1.57888	0.789438	−	0.613830i	$-0.210372\pi$
$257$	6505.01	1.57888	0.789438	+	0.613830i	$0.210372\pi$
$258$	0	0
$259$	1200.06	0.287908
$260$	0	0
$261$	5983.10	1.41894
$262$	0	0
$263$	−650.590	−0.152536	−0.0762682	−	0.997087i	$-0.524301\pi$
$263$	−650.590	−0.152536	−0.0762682	+	0.997087i	$0.524301\pi$
$264$	0	0
$265$	5781.27	1.34015
$266$	0	0
$267$	61.5302	0.0141033
$268$	0	0
$269$	−2404.32	−0.544960	−0.272480	−	0.962161i	$-0.587844\pi$
$269$	−2404.32	−0.544960	−0.272480	+	0.962161i	$0.587844\pi$
$270$	0	0
$271$	−1180.82	−0.264686	−0.132343	−	0.991204i	$-0.542250\pi$
$271$	−1180.82	−0.264686	−0.132343	+	0.991204i	$0.542250\pi$
$272$	0	0
$273$	7.10306	0.00157471
$274$	0	0
$275$	2126.89	0.466386
$276$	0	0
$277$	−429.648	−0.0931952	−0.0465976	−	0.998914i	$-0.514838\pi$
$277$	−429.648	−0.0931952	−0.0465976	+	0.998914i	$0.514838\pi$
$278$	0	0
$279$	836.956	0.179596
$280$	0	0
$281$	2300.38	0.488361	0.244181	−	0.969730i	$-0.421481\pi$
$281$	2300.38	0.488361	0.244181	+	0.969730i	$0.421481\pi$
$282$	0	0
$283$	−4569.89	−0.959901	−0.479950	−	0.877296i	$-0.659345\pi$
$283$	−4569.89	−0.959901	−0.479950	+	0.877296i	$0.659345\pi$
$284$	0	0
$285$	26.5926	0.00552704
$286$	0	0
$287$	1672.17	0.343921
$288$	0	0
$289$	−3887.71	−0.791310
$290$	0	0
$291$	1.04040	0.000209586 0
$292$	0	0
$293$	2782.40	0.554776	0.277388	−	0.960758i	$-0.410531\pi$
$293$	2782.40	0.554776	0.277388	+	0.960758i	$0.410531\pi$
$294$	0	0
$295$	8384.48	1.65479
$296$	0	0
$297$	−75.5885	−0.0147680
$298$	0	0
$299$	−1125.53	−0.217696
$300$	0	0
$301$	−1712.82	−0.327991
$302$	0	0
$303$	11.7333	0.00222462
$304$	0	0
$305$	−2010.42	−0.377431
$306$	0	0
$307$	−8228.72	−1.52976	−0.764882	−	0.644170i	$-0.777203\pi$
$307$	−8228.72	−1.52976	−0.764882	+	0.644170i	$0.777203\pi$
$308$	0	0
$309$	−1.36465	−0.000251236 0
$310$	0	0
$311$	1973.40	0.359811	0.179905	−	0.983684i	$-0.442421\pi$
$311$	1973.40	0.359811	0.179905	+	0.983684i	$0.442421\pi$
$312$	0	0
$313$	−4157.98	−0.750871	−0.375436	−	0.926848i	$-0.622507\pi$
$313$	−4157.98	−0.750871	−0.375436	+	0.926848i	$0.622507\pi$
$314$	0	0
$315$	−1409.05	−0.252035
$316$	0	0
$317$	−1178.13	−0.208739	−0.104370	−	0.994539i	$-0.533283\pi$
$317$	−1178.13	−0.208739	−0.104370	+	0.994539i	$0.533283\pi$
$318$	0	0
$319$	−8228.45	−1.44421
$320$	0	0
$321$	47.7723	0.00830651
$322$	0	0
$323$	−1672.91	−0.288184
$324$	0	0
$325$	−2791.91	−0.476516
$326$	0	0
$327$	−32.0090	−0.00541316
$328$	0	0
$329$	−798.890	−0.133873
$330$	0	0
$331$	8204.49	1.36242	0.681208	−	0.732090i	$-0.261455\pi$
$331$	8204.49	1.36242	0.681208	+	0.732090i	$0.261455\pi$
$332$	0	0
$333$	8381.64	1.37931
$334$	0	0
$335$	−8760.50	−1.42877
$336$	0	0
$337$	4623.93	0.747423	0.373711	−	0.927545i	$-0.378085\pi$
$337$	4623.93	0.747423	0.373711	+	0.927545i	$0.378085\pi$
$338$	0	0
$339$	17.5552	0.00281259
$340$	0	0
$341$	−1151.05	−0.182794
$342$	0	0
$343$	2594.02	0.408350
$344$	0	0
$345$	−11.7538	−0.00183421
$346$	0	0
$347$	7290.79	1.12793	0.563963	−	0.825800i	$-0.309276\pi$
$347$	7290.79	1.12793	0.563963	+	0.825800i	$0.309276\pi$
$348$	0	0
$349$	−758.435	−0.116327	−0.0581635	−	0.998307i	$-0.518524\pi$
$349$	−758.435	−0.116327	−0.0581635	+	0.998307i	$0.518524\pi$
$350$	0	0
$351$	99.2232	0.0150887
$352$	0	0
$353$	−6142.11	−0.926094	−0.463047	−	0.886334i	$-0.653244\pi$
$353$	−6142.11	−0.926094	−0.463047	+	0.886334i	$0.653244\pi$
$354$	0	0
$355$	11822.0	1.76746
$356$	0	0
$357$	−4.66638	−0.000691795 0
$358$	0	0
$359$	1387.74	0.204018	0.102009	−	0.994783i	$-0.467473\pi$
$359$	1387.74	0.204018	0.102009	+	0.994783i	$0.467473\pi$
$360$	0	0
$361$	−4129.41	−0.602043
$362$	0	0
$363$	1.79769	0.000259929 0
$364$	0	0
$365$	744.767	0.106802
$366$	0	0
$367$	11171.9	1.58902	0.794509	−	0.607252i	$-0.207728\pi$
$367$	11171.9	1.58902	0.794509	+	0.607252i	$0.207728\pi$
$368$	0	0
$369$	11679.1	1.64766
$370$	0	0
$371$	1655.26	0.231636
$372$	0	0
$373$	4865.95	0.675468	0.337734	−	0.941242i	$-0.390340\pi$
$373$	4865.95	0.675468	0.337734	+	0.941242i	$0.390340\pi$
$374$	0	0
$375$	34.4684	0.00474651
$376$	0	0
$377$	10801.3	1.47558
$378$	0	0
$379$	5435.85	0.736731	0.368365	−	0.929681i	$-0.379918\pi$
$379$	5435.85	0.736731	0.368365	+	0.929681i	$0.379918\pi$
$380$	0	0
$381$	50.9097	0.00684562
$382$	0	0
$383$	−11685.3	−1.55898	−0.779489	−	0.626415i	$-0.784521\pi$
$383$	−11685.3	−1.55898	−0.779489	+	0.626415i	$0.784521\pi$
$384$	0	0
$385$	1937.84	0.256523
$386$	0	0
$387$	−11962.9	−1.57134
$388$	0	0
$389$	−8798.11	−1.14674	−0.573370	−	0.819297i	$-0.694364\pi$
$389$	−8798.11	−1.14674	−0.573370	+	0.819297i	$0.694364\pi$
$390$	0	0
$391$	739.419	0.0956369
$392$	0	0
$393$	−104.520	−0.0134156
$394$	0	0
$395$	10411.0	1.32616
$396$	0	0
$397$	1169.47	0.147844	0.0739221	−	0.997264i	$-0.476448\pi$
$397$	1169.47	0.147844	0.0739221	+	0.997264i	$0.476448\pi$
$398$	0	0
$399$	7.61384	0.000955310 0
$400$	0	0
$401$	7359.83	0.916540	0.458270	−	0.888813i	$-0.348469\pi$
$401$	7359.83	0.916540	0.458270	+	0.888813i	$0.348469\pi$
$402$	0	0
$403$	1510.96	0.186764
$404$	0	0
$405$	−9840.79	−1.20739
$406$	0	0
$407$	−11527.1	−1.40388
$408$	0	0
$409$	−2808.05	−0.339484	−0.169742	−	0.985489i	$-0.554293\pi$
$409$	−2808.05	−0.339484	−0.169742	+	0.985489i	$0.554293\pi$
$410$	0	0
$411$	−80.4944	−0.00966057
$412$	0	0
$413$	2400.60	0.286019
$414$	0	0
$415$	−4937.07	−0.583979
$416$	0	0
$417$	73.2391	0.00860080
$418$	0	0
$419$	7641.52	0.890961	0.445481	−	0.895292i	$-0.353033\pi$
$419$	7641.52	0.890961	0.445481	+	0.895292i	$0.353033\pi$
$420$	0	0
$421$	9014.87	1.04361	0.521803	−	0.853066i	$-0.325260\pi$
$421$	9014.87	1.04361	0.521803	+	0.853066i	$0.325260\pi$
$422$	0	0
$423$	−5579.73	−0.641361
$424$	0	0
$425$	1834.16	0.209340
$426$	0	0
$427$	−575.614	−0.0652363
$428$	0	0
$429$	−68.2281	−0.00767852
$430$	0	0
$431$	8248.40	0.921836	0.460918	−	0.887443i	$-0.347520\pi$
$431$	8248.40	0.921836	0.460918	+	0.887443i	$0.347520\pi$
$432$	0	0
$433$	−13762.4	−1.52743	−0.763715	−	0.645553i	$-0.776627\pi$
$433$	−13762.4	−1.52743	−0.763715	+	0.645553i	$0.776627\pi$
$434$	0	0
$435$	112.797	0.0124326
$436$	0	0
$437$	−1206.46	−0.132066
$438$	0	0
$439$	−2191.21	−0.238225	−0.119113	−	0.992881i	$-0.538005\pi$
$439$	−2191.21	−0.238225	−0.119113	+	0.992881i	$0.538005\pi$
$440$	0	0
$441$	8857.08	0.956385
$442$	0	0
$443$	3397.50	0.364380	0.182190	−	0.983263i	$-0.441681\pi$
$443$	3397.50	0.364380	0.182190	+	0.983263i	$0.441681\pi$
$444$	0	0
$445$	−22035.3	−2.34735
$446$	0	0
$447$	−4.34434	−0.000459688 0
$448$	0	0
$449$	2010.24	0.211290	0.105645	−	0.994404i	$-0.466309\pi$
$449$	2010.24	0.211290	0.105645	+	0.994404i	$0.466309\pi$
$450$	0	0
$451$	−16062.0	−1.67701
$452$	0	0
$453$	54.5085	0.00565350
$454$	0	0
$455$	−2543.76	−0.262095
$456$	0	0
$457$	−5790.19	−0.592678	−0.296339	−	0.955083i	$-0.595766\pi$
$457$	−5790.19	−0.592678	−0.296339	+	0.955083i	$0.595766\pi$
$458$	0	0
$459$	−65.1850	−0.00662870
$460$	0	0
$461$	4700.84	0.474924	0.237462	−	0.971397i	$-0.423684\pi$
$461$	4700.84	0.474924	0.237462	+	0.971397i	$0.423684\pi$
$462$	0	0
$463$	12685.9	1.27335	0.636677	−	0.771131i	$-0.280308\pi$
$463$	12685.9	1.27335	0.636677	+	0.771131i	$0.280308\pi$
$464$	0	0
$465$	15.7788	0.00157360
$466$	0	0
$467$	−2483.95	−0.246131	−0.123066	−	0.992399i	$-0.539273\pi$
$467$	−2483.95	−0.246131	−0.123066	+	0.992399i	$0.539273\pi$
$468$	0	0
$469$	−2508.26	−0.246952
$470$	0	0
$471$	122.028	0.0119379
$472$	0	0
$473$	16452.4	1.59933
$474$	0	0
$475$	−2992.68	−0.289081
$476$	0	0
$477$	11560.9	1.10973
$478$	0	0
$479$	4420.05	0.421622	0.210811	−	0.977527i	$-0.432389\pi$
$479$	4420.05	0.421622	0.210811	+	0.977527i	$0.432389\pi$
$480$	0	0
$481$	15131.4	1.43437
$482$	0	0
$483$	−3.36528	−0.000317030 0
$484$	0	0
$485$	−372.591	−0.0348835
$486$	0	0
$487$	3777.16	0.351457	0.175728	−	0.984439i	$-0.443772\pi$
$487$	3777.16	0.351457	0.175728	+	0.984439i	$0.443772\pi$
$488$	0	0
$489$	117.046	0.0108241
$490$	0	0
$491$	5942.03	0.546151	0.273075	−	0.961993i	$-0.411959\pi$
$491$	5942.03	0.546151	0.273075	+	0.961993i	$0.411959\pi$
$492$	0	0
$493$	−7095.94	−0.648245
$494$	0	0
$495$	13534.6	1.22896
$496$	0	0
$497$	3384.82	0.305493
$498$	0	0
$499$	−3259.79	−0.292441	−0.146221	−	0.989252i	$-0.546711\pi$
$499$	−3259.79	−0.292441	−0.146221	+	0.989252i	$0.546711\pi$
$500$	0	0
$501$	75.1028	0.00669729
$502$	0	0
$503$	−12392.2	−1.09849	−0.549246	−	0.835661i	$-0.685085\pi$
$503$	−12392.2	−1.09849	−0.549246	+	0.835661i	$0.685085\pi$
$504$	0	0
$505$	−4201.95	−0.370266
$506$	0	0
$507$	6.73466	0.000589935 0
$508$	0	0
$509$	19365.1	1.68633	0.843166	−	0.537654i	$-0.180689\pi$
$509$	19365.1	1.68633	0.843166	+	0.537654i	$0.180689\pi$
$510$	0	0
$511$	213.238	0.0184600
$512$	0	0
$513$	106.358	0.00915368
$514$	0	0
$515$	488.709	0.0418157
$516$	0	0
$517$	7673.70	0.652783
$518$	0	0
$519$	6.18095	0.000522762 0
$520$	0	0
$521$	3656.72	0.307493	0.153747	−	0.988110i	$-0.450866\pi$
$521$	3656.72	0.307493	0.153747	+	0.988110i	$0.450866\pi$
$522$	0	0
$523$	−21492.5	−1.79694	−0.898472	−	0.439031i	$-0.855322\pi$
$523$	−21492.5	−1.79694	−0.898472	+	0.439031i	$0.855322\pi$
$524$	0	0
$525$	−8.34771	−0.000693950 0
$526$	0	0
$527$	−992.627	−0.0820484
$528$	0	0
$529$	−11633.7	−0.956172
$530$	0	0
$531$	16766.6	1.37026
$532$	0	0
$533$	21084.2	1.71343
$534$	0	0
$535$	−17108.3	−1.38253
$536$	0	0
$537$	−143.606	−0.0115401
$538$	0	0
$539$	−12181.0	−0.973417
$540$	0	0
$541$	−3188.29	−0.253374	−0.126687	−	0.991943i	$-0.540434\pi$
$541$	−3188.29	−0.253374	−0.126687	+	0.991943i	$0.540434\pi$
$542$	0	0
$543$	46.9867	0.00371343
$544$	0	0
$545$	11463.1	0.900964
$546$	0	0
$547$	17131.1	1.33907	0.669536	−	0.742779i	$-0.266493\pi$
$547$	17131.1	1.33907	0.669536	+	0.742779i	$0.266493\pi$
$548$	0	0
$549$	−4020.29	−0.312535
$550$	0	0
$551$	11578.0	0.895172
$552$	0	0
$553$	2980.83	0.229218
$554$	0	0
$555$	158.016	0.0120854
$556$	0	0
$557$	−14402.4	−1.09560	−0.547800	−	0.836609i	$-0.684535\pi$
$557$	−14402.4	−1.09560	−0.547800	+	0.836609i	$0.684535\pi$
$558$	0	0
$559$	−21596.7	−1.63406
$560$	0	0
$561$	44.8227	0.00337329
$562$	0	0
$563$	8241.39	0.616933	0.308466	−	0.951235i	$-0.400184\pi$
$563$	8241.39	0.616933	0.308466	+	0.951235i	$0.400184\pi$
$564$	0	0
$565$	−6286.89	−0.468126
$566$	0	0
$567$	−2817.56	−0.208689
$568$	0	0
$569$	3903.37	0.287588	0.143794	−	0.989608i	$-0.454070\pi$
$569$	3903.37	0.287588	0.143794	+	0.989608i	$0.454070\pi$
$570$	0	0
$571$	−3136.17	−0.229851	−0.114925	−	0.993374i	$-0.536663\pi$
$571$	−3136.17	−0.229851	−0.114925	+	0.993374i	$0.536663\pi$
$572$	0	0
$573$	63.9757	0.00466427
$574$	0	0
$575$	1322.75	0.0959348
$576$	0	0
$577$	−14162.4	−1.02182	−0.510910	−	0.859634i	$-0.670691\pi$
$577$	−14162.4	−1.02182	−0.510910	+	0.859634i	$0.670691\pi$
$578$	0	0
$579$	66.2473	0.00475500
$580$	0	0
$581$	−1413.56	−0.100937
$582$	0	0
$583$	−15899.5	−1.12949
$584$	0	0
$585$	−17766.5	−1.25565
$586$	0	0
$587$	24292.8	1.70813	0.854065	−	0.520167i	$-0.174130\pi$
$587$	24292.8	1.70813	0.854065	+	0.520167i	$0.174130\pi$
$588$	0	0
$589$	1619.61	0.113302
$590$	0	0
$591$	109.308	0.00760801
$592$	0	0
$593$	−1855.30	−0.128479	−0.0642397	−	0.997935i	$-0.520462\pi$
$593$	−1855.30	−0.128479	−0.0642397	+	0.997935i	$0.520462\pi$
$594$	0	0
$595$	1671.13	0.115142
$596$	0	0
$597$	137.993	0.00946007
$598$	0	0
$599$	−6780.96	−0.462541	−0.231271	−	0.972889i	$-0.574288\pi$
$599$	−6780.96	−0.462541	−0.231271	+	0.972889i	$0.574288\pi$
$600$	0	0
$601$	13854.1	0.940298	0.470149	−	0.882587i	$-0.344200\pi$
$601$	13854.1	0.940298	0.470149	+	0.882587i	$0.344200\pi$
$602$	0	0
$603$	−17518.6	−1.18310
$604$	0	0
$605$	−643.791	−0.0432625
$606$	0	0
$607$	21406.5	1.43140	0.715701	−	0.698406i	$-0.246107\pi$
$607$	21406.5	1.43140	0.715701	+	0.698406i	$0.246107\pi$
$608$	0	0
$609$	32.2954	0.00214889
$610$	0	0
$611$	−10073.1	−0.666962
$612$	0	0
$613$	−20936.1	−1.37945	−0.689724	−	0.724073i	$-0.742268\pi$
$613$	−20936.1	−1.37945	−0.689724	+	0.724073i	$0.742268\pi$
$614$	0	0
$615$	220.180	0.0144366
$616$	0	0
$617$	−570.186	−0.0372039	−0.0186020	−	0.999827i	$-0.505922\pi$
$617$	−570.186	−0.0372039	−0.0186020	+	0.999827i	$0.505922\pi$
$618$	0	0
$619$	98.0312	0.00636544	0.00318272	−	0.999995i	$-0.498987\pi$
$619$	98.0312	0.00636544	0.00318272	+	0.999995i	$0.498987\pi$
$620$	0	0
$621$	−47.0099	−0.00303775
$622$	0	0
$623$	−6309.03	−0.405724
$624$	0	0
$625$	−19504.0	−1.24826
$626$	0	0
$627$	−73.1344	−0.00465822
$628$	0	0
$629$	−9940.60	−0.630140
$630$	0	0
$631$	−7723.40	−0.487264	−0.243632	−	0.969868i	$-0.578339\pi$
$631$	−7723.40	−0.487264	−0.243632	+	0.969868i	$0.578339\pi$
$632$	0	0
$633$	−202.193	−0.0126958
$634$	0	0
$635$	−18231.8	−1.13938
$636$	0	0
$637$	15989.7	0.994559
$638$	0	0
$639$	23640.8	1.46356
$640$	0	0
$641$	26310.9	1.62125	0.810624	−	0.585567i	$-0.199128\pi$
$641$	26310.9	1.62125	0.810624	+	0.585567i	$0.199128\pi$
$642$	0	0
$643$	−3462.55	−0.212363	−0.106182	−	0.994347i	$-0.533863\pi$
$643$	−3462.55	−0.212363	−0.106182	+	0.994347i	$0.533863\pi$
$644$	0	0
$645$	−225.532	−0.0137679
$646$	0	0
$647$	−11675.2	−0.709430	−0.354715	−	0.934975i	$-0.615422\pi$
$647$	−11675.2	−0.709430	−0.354715	+	0.934975i	$0.615422\pi$
$648$	0	0
$649$	−23058.8	−1.39467
$650$	0	0
$651$	4.51770	0.000271985 0
$652$	0	0
$653$	1766.63	0.105870	0.0529352	−	0.998598i	$-0.483142\pi$
$653$	1766.63	0.105870	0.0529352	+	0.998598i	$0.483142\pi$
$654$	0	0
$655$	37430.7	2.23288
$656$	0	0
$657$	1489.33	0.0884387
$658$	0	0
$659$	4676.48	0.276434	0.138217	−	0.990402i	$-0.455863\pi$
$659$	4676.48	0.276434	0.138217	+	0.990402i	$0.455863\pi$
$660$	0	0
$661$	22811.1	1.34228	0.671142	−	0.741329i	$-0.265804\pi$
$661$	22811.1	1.34228	0.671142	+	0.741329i	$0.265804\pi$
$662$	0	0
$663$	−58.8377	−0.00344655
$664$	0	0
$665$	−2726.68	−0.159002
$666$	0	0
$667$	−5117.43	−0.297073
$668$	0	0
$669$	−171.835	−0.00993053
$670$	0	0
$671$	5529.03	0.318101
$672$	0	0
$673$	−12635.5	−0.723716	−0.361858	−	0.932233i	$-0.617858\pi$
$673$	−12635.5	−0.723716	−0.361858	+	0.932233i	$0.617858\pi$
$674$	0	0
$675$	−116.610	−0.00664935
$676$	0	0
$677$	9097.82	0.516481	0.258241	−	0.966081i	$-0.416857\pi$
$677$	9097.82	0.516481	0.258241	+	0.966081i	$0.416857\pi$
$678$	0	0
$679$	−106.678	−0.00602936
$680$	0	0
$681$	−84.2220	−0.00473920
$682$	0	0
$683$	5490.48	0.307595	0.153797	−	0.988102i	$-0.450850\pi$
$683$	5490.48	0.307595	0.153797	+	0.988102i	$0.450850\pi$
$684$	0	0
$685$	28826.8	1.60790
$686$	0	0
$687$	37.9324	0.00210657
$688$	0	0
$689$	20871.0	1.15402
$690$	0	0
$691$	−21608.2	−1.18960	−0.594801	−	0.803873i	$-0.702769\pi$
$691$	−21608.2	−1.18960	−0.594801	+	0.803873i	$0.702769\pi$
$692$	0	0
$693$	3875.14	0.212416
$694$	0	0
$695$	−26228.5	−1.43151
$696$	0	0
$697$	−13851.3	−0.752735
$698$	0	0
$699$	81.2375	0.00439583
$700$	0	0
$701$	31969.4	1.72249	0.861245	−	0.508189i	$-0.169685\pi$
$701$	31969.4	1.72249	0.861245	+	0.508189i	$0.169685\pi$
$702$	0	0
$703$	16219.5	0.870170
$704$	0	0
$705$	−105.192	−0.00561954
$706$	0	0
$707$	−1203.08	−0.0639978
$708$	0	0
$709$	−10002.1	−0.529810	−0.264905	−	0.964274i	$-0.585341\pi$
$709$	−10002.1	−0.529810	−0.264905	+	0.964274i	$0.585341\pi$
$710$	0	0
$711$	20819.1	1.09814
$712$	0	0
$713$	−715.860	−0.0376005
$714$	0	0
$715$	24433.9	1.27801
$716$	0	0
$717$	169.039	0.00880458
$718$	0	0
$719$	−7201.14	−0.373515	−0.186757	−	0.982406i	$-0.559798\pi$
$719$	−7201.14	−0.373515	−0.186757	+	0.982406i	$0.559798\pi$
$720$	0	0
$721$	139.925	0.00722755
$722$	0	0
$723$	−55.0461	−0.00283152
$724$	0	0
$725$	−12694.0	−0.650265
$726$	0	0
$727$	−12589.0	−0.642226	−0.321113	−	0.947041i	$-0.604057\pi$
$727$	−12589.0	−0.642226	−0.321113	+	0.947041i	$0.604057\pi$
$728$	0	0
$729$	−19676.8	−0.999684
$730$	0	0
$731$	14188.0	0.717869
$732$	0	0
$733$	−35438.6	−1.78575	−0.892875	−	0.450304i	$-0.851316\pi$
$733$	−35438.6	−1.78575	−0.892875	+	0.450304i	$0.851316\pi$
$734$	0	0
$735$	166.979	0.00837974
$736$	0	0
$737$	24093.0	1.20417
$738$	0	0
$739$	−5883.61	−0.292872	−0.146436	−	0.989220i	$-0.546780\pi$
$739$	−5883.61	−0.292872	−0.146436	+	0.989220i	$0.546780\pi$
$740$	0	0
$741$	96.0018	0.00475940
$742$	0	0
$743$	26512.0	1.30906	0.654530	−	0.756036i	$-0.272866\pi$
$743$	26512.0	1.30906	0.654530	+	0.756036i	$0.272866\pi$
$744$	0	0
$745$	1555.80	0.0765103
$746$	0	0
$747$	−9872.78	−0.483569
$748$	0	0
$749$	−4898.35	−0.238961
$750$	0	0
$751$	27797.6	1.35066	0.675331	−	0.737515i	$-0.264001\pi$
$751$	27797.6	1.35066	0.675331	+	0.737515i	$0.264001\pi$
$752$	0	0
$753$	14.8738	0.000719829 0
$754$	0	0
$755$	−19520.7	−0.940967
$756$	0	0
$757$	−27213.3	−1.30658	−0.653291	−	0.757106i	$-0.726612\pi$
$757$	−27213.3	−1.30658	−0.653291	+	0.757106i	$0.726612\pi$
$758$	0	0
$759$	32.3251	0.00154588
$760$	0	0
$761$	−10067.1	−0.479541	−0.239771	−	0.970830i	$-0.577072\pi$
$761$	−10067.1	−0.479541	−0.239771	+	0.970830i	$0.577072\pi$
$762$	0	0
$763$	3282.06	0.155725
$764$	0	0
$765$	11671.8	0.551625
$766$	0	0
$767$	30268.8	1.42496
$768$	0	0
$769$	−15206.7	−0.713092	−0.356546	−	0.934278i	$-0.616046\pi$
$769$	−15206.7	−0.713092	−0.356546	+	0.934278i	$0.616046\pi$
$770$	0	0
$771$	245.239	0.0114553
$772$	0	0
$773$	25658.5	1.19388	0.596942	−	0.802284i	$-0.296382\pi$
$773$	25658.5	1.19388	0.596942	+	0.802284i	$0.296382\pi$
$774$	0	0
$775$	−1775.72	−0.0823040
$776$	0	0
$777$	45.2422	0.00208887
$778$	0	0
$779$	22600.4	1.03946
$780$	0	0
$781$	−32512.7	−1.48962
$782$	0	0
$783$	451.137	0.0205905
$784$	0	0
$785$	−43701.0	−1.98695
$786$	0	0
$787$	−38239.3	−1.73200	−0.865999	−	0.500046i	$-0.833317\pi$
$787$	−38239.3	−1.73200	−0.865999	+	0.500046i	$0.833317\pi$
$788$	0	0
$789$	−24.5272	−0.00110671
$790$	0	0
$791$	−1800.03	−0.0809123
$792$	0	0
$793$	−7257.84	−0.325010
$794$	0	0
$795$	217.953	0.00972329
$796$	0	0
$797$	−24256.1	−1.07804	−0.539019	−	0.842294i	$-0.681205\pi$
$797$	−24256.1	−1.07804	−0.539019	+	0.842294i	$0.681205\pi$
$798$	0	0
$799$	6617.54	0.293006
$800$	0	0
$801$	−44064.5	−1.94375
$802$	0	0
$803$	−2048.24	−0.0900137
$804$	0	0
$805$	1205.18	0.0527664
$806$	0	0
$807$	−90.6429	−0.00395388
$808$	0	0
$809$	−14967.1	−0.650450	−0.325225	−	0.945637i	$-0.605440\pi$
$809$	−14967.1	−0.650450	−0.325225	+	0.945637i	$0.605440\pi$
$810$	0	0
$811$	28851.2	1.24920	0.624601	−	0.780944i	$-0.285262\pi$
$811$	28851.2	1.24920	0.624601	+	0.780944i	$0.285262\pi$
$812$	0	0
$813$	−44.5170	−0.00192039
$814$	0	0
$815$	−41916.6	−1.80156
$816$	0	0
$817$	−23149.7	−0.991316
$818$	0	0
$819$	−5086.81	−0.217030
$820$	0	0
$821$	−35631.5	−1.51468	−0.757338	−	0.653023i	$-0.773500\pi$
$821$	−35631.5	−1.51468	−0.757338	+	0.653023i	$0.773500\pi$
$822$	0	0
$823$	−34599.6	−1.46545	−0.732725	−	0.680525i	$-0.761752\pi$
$823$	−34599.6	−1.46545	−0.732725	+	0.680525i	$0.761752\pi$
$824$	0	0
$825$	80.1835	0.00338380
$826$	0	0
$827$	−5375.51	−0.226028	−0.113014	−	0.993593i	$-0.536050\pi$
$827$	−5375.51	−0.226028	−0.113014	+	0.993593i	$0.536050\pi$
$828$	0	0
$829$	−44162.4	−1.85021	−0.925105	−	0.379710i	$-0.876024\pi$
$829$	−44162.4	−1.85021	−0.925105	+	0.379710i	$0.876024\pi$
$830$	0	0
$831$	−16.1977	−0.000676165 0
$832$	0	0
$833$	−10504.5	−0.436925
$834$	0	0
$835$	−26895.9	−1.11470
$836$	0	0
$837$	63.1081	0.00260613
$838$	0	0
$839$	8942.73	0.367983	0.183991	−	0.982928i	$-0.441098\pi$
$839$	8942.73	0.367983	0.183991	+	0.982928i	$0.441098\pi$
$840$	0	0
$841$	24721.1	1.01362
$842$	0	0
$843$	86.7244	0.00354323
$844$	0	0
$845$	−2411.83	−0.0981886
$846$	0	0
$847$	−184.327	−0.00747762
$848$	0	0
$849$	−172.285	−0.00696442
$850$	0	0
$851$	−7168.93	−0.288775
$852$	0	0
$853$	−25694.1	−1.03136	−0.515680	−	0.856781i	$-0.672461\pi$
$853$	−25694.1	−1.03136	−0.515680	+	0.856781i	$0.672461\pi$
$854$	0	0
$855$	−19044.1	−0.761748
$856$	0	0
$857$	−34742.0	−1.38479	−0.692394	−	0.721519i	$-0.743444\pi$
$857$	−34742.0	−1.38479	−0.692394	+	0.721519i	$0.743444\pi$
$858$	0	0
$859$	−1772.90	−0.0704198	−0.0352099	−	0.999380i	$-0.511210\pi$
$859$	−1772.90	−0.0704198	−0.0352099	+	0.999380i	$0.511210\pi$
$860$	0	0
$861$	63.0409	0.00249527
$862$	0	0
$863$	23432.6	0.924281	0.462140	−	0.886807i	$-0.347082\pi$
$863$	23432.6	0.924281	0.462140	+	0.886807i	$0.347082\pi$
$864$	0	0
$865$	−2213.53	−0.0870083
$866$	0	0
$867$	−146.566	−0.00574124
$868$	0	0
$869$	−28632.2	−1.11770
$870$	0	0
$871$	−31626.3	−1.23033
$872$	0	0
$873$	−745.079	−0.0288856
$874$	0	0
$875$	−3534.23	−0.136547
$876$	0	0
$877$	22630.7	0.871361	0.435681	−	0.900101i	$-0.356508\pi$
$877$	22630.7	0.871361	0.435681	+	0.900101i	$0.356508\pi$
$878$	0	0
$879$	104.896	0.00402510
$880$	0	0
$881$	−4713.81	−0.180264	−0.0901319	−	0.995930i	$-0.528729\pi$
$881$	−4713.81	−0.180264	−0.0901319	+	0.995930i	$0.528729\pi$
$882$	0	0
$883$	1947.12	0.0742080	0.0371040	−	0.999311i	$-0.488187\pi$
$883$	1947.12	0.0742080	0.0371040	+	0.999311i	$0.488187\pi$
$884$	0	0
$885$	316.094	0.0120061
$886$	0	0
$887$	14418.3	0.545794	0.272897	−	0.962043i	$-0.412018\pi$
$887$	14418.3	0.545794	0.272897	+	0.962043i	$0.412018\pi$
$888$	0	0
$889$	−5220.04	−0.196934
$890$	0	0
$891$	27064.0	1.01759
$892$	0	0
$893$	−10797.4	−0.404617
$894$	0	0
$895$	51428.4	1.92074
$896$	0	0
$897$	−42.4324	−0.00157946
$898$	0	0
$899$	6869.85	0.254863
$900$	0	0
$901$	−13711.2	−0.506978
$902$	0	0
$903$	−64.5732	−0.00237969
$904$	0	0
$905$	−16826.9	−0.618062
$906$	0	0
$907$	−634.066	−0.0232126	−0.0116063	−	0.999933i	$-0.503694\pi$
$907$	−634.066	−0.0232126	−0.0116063	+	0.999933i	$0.503694\pi$
$908$	0	0
$909$	−8402.74	−0.306602
$910$	0	0
$911$	−42359.7	−1.54055	−0.770274	−	0.637713i	$-0.779881\pi$
$911$	−42359.7	−1.54055	−0.770274	+	0.637713i	$0.779881\pi$
$912$	0	0
$913$	13577.9	0.492181
$914$	0	0
$915$	−75.7929	−0.00273840
$916$	0	0
$917$	10717.0	0.385938
$918$	0	0
$919$	−38569.5	−1.38443	−0.692215	−	0.721692i	$-0.743365\pi$
$919$	−38569.5	−1.38443	−0.692215	+	0.721692i	$0.743365\pi$
$920$	0	0
$921$	−310.222	−0.0110990
$922$	0	0
$923$	42678.7	1.52198
$924$	0	0
$925$	−17782.8	−0.632103
$926$	0	0
$927$	977.283	0.0346259
$928$	0	0
$929$	2229.10	0.0787237	0.0393619	−	0.999225i	$-0.487467\pi$
$929$	2229.10	0.0787237	0.0393619	+	0.999225i	$0.487467\pi$
$930$	0	0
$931$	17139.5	0.603356
$932$	0	0
$933$	74.3970	0.00261056
$934$	0	0
$935$	−16051.9	−0.561449
$936$	0	0
$937$	−1352.28	−0.0471472	−0.0235736	−	0.999722i	$-0.507504\pi$
$937$	−1352.28	−0.0471472	−0.0235736	+	0.999722i	$0.507504\pi$
$938$	0	0
$939$	−156.756	−0.00544784
$940$	0	0
$941$	−32318.8	−1.11962	−0.559811	−	0.828621i	$-0.689126\pi$
$941$	−32318.8	−1.11962	−0.559811	+	0.828621i	$0.689126\pi$
$942$	0	0
$943$	−9989.26	−0.344958
$944$	0	0
$945$	−106.245	−0.00365730
$946$	0	0
$947$	−14722.7	−0.505198	−0.252599	−	0.967571i	$-0.581285\pi$
$947$	−14722.7	−0.505198	−0.252599	+	0.967571i	$0.581285\pi$
$948$	0	0
$949$	2688.68	0.0919688
$950$	0	0
$951$	−44.4154	−0.00151448
$952$	0	0
$953$	2361.87	0.0802817	0.0401408	−	0.999194i	$-0.487219\pi$
$953$	2361.87	0.0802817	0.0401408	+	0.999194i	$0.487219\pi$
$954$	0	0
$955$	−22911.1	−0.776319
$956$	0	0
$957$	−310.212	−0.0104783
$958$	0	0
$959$	8253.52	0.277915
$960$	0	0
$961$	961.000	0.0322581
$962$	0	0
$963$	−34211.8	−1.14482
$964$	0	0
$965$	−23724.6	−0.791421
$966$	0	0
$967$	10193.5	0.338987	0.169494	−	0.985531i	$-0.445787\pi$
$967$	10193.5	0.338987	0.169494	+	0.985531i	$0.445787\pi$
$968$	0	0
$969$	−63.0687	−0.00209088
$970$	0	0
$971$	32087.8	1.06050	0.530251	−	0.847841i	$-0.322098\pi$
$971$	32087.8	1.06050	0.530251	+	0.847841i	$0.322098\pi$
$972$	0	0
$973$	−7509.60	−0.247427
$974$	0	0
$975$	−105.255	−0.00345729
$976$	0	0
$977$	172.357	0.00564399	0.00282199	−	0.999996i	$-0.499102\pi$
$977$	172.357	0.00564399	0.00282199	+	0.999996i	$0.499102\pi$
$978$	0	0
$979$	60601.1	1.97836
$980$	0	0
$981$	22923.0	0.746052
$982$	0	0
$983$	−26441.8	−0.857946	−0.428973	−	0.903317i	$-0.641124\pi$
$983$	−26441.8	−0.857946	−0.428973	+	0.903317i	$0.641124\pi$
$984$	0	0
$985$	−39145.5	−1.26627
$986$	0	0
$987$	−30.1181	−0.000971297 0
$988$	0	0
$989$	10232.1	0.328979
$990$	0	0
$991$	−55415.8	−1.77633	−0.888164	−	0.459527i	$-0.848019\pi$
$991$	−55415.8	−1.77633	−0.888164	+	0.459527i	$0.848019\pi$
$992$	0	0
$993$	309.309	0.00988482
$994$	0	0
$995$	−49418.1	−1.57453
$996$	0	0
$997$	37593.5	1.19418	0.597090	−	0.802174i	$-0.296323\pi$
$997$	37593.5	1.19418	0.597090	+	0.802174i	$0.296323\pi$
$998$	0	0
$999$	631.992	0.0200154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.4.a.r.1.3		5
4.3	odd	2		1984.4.a.s.1.3		5
8.3	odd	2		496.4.a.i.1.3		5
8.5	even	2		31.4.a.b.1.4	✓	5
24.5	odd	2		279.4.a.h.1.2		5
40.29	even	2		775.4.a.f.1.2		5
56.13	odd	2		1519.4.a.c.1.4		5
248.61	odd	2		961.4.a.e.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.4.a.b.1.4	✓	5	8.5	even	2
279.4.a.h.1.2		5	24.5	odd	2
496.4.a.i.1.3		5	8.3	odd	2
775.4.a.f.1.2		5	40.29	even	2
961.4.a.e.1.4		5	248.61	odd	2
1519.4.a.c.1.4		5	56.13	odd	2
1984.4.a.r.1.3		5	1.1	even	1	trivial
1984.4.a.s.1.3		5	4.3	odd	2

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:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0377000 q^{3} -13.5012 q^{5} -3.86558 q^{7} -26.9986 q^{9} +37.1306 q^{11} -48.7405 q^{13} -0.508993 q^{15} +32.0202 q^{17} -52.2455 q^{19} -0.145732 q^{21} +23.0922 q^{23} +57.2812 q^{25} -2.03574 q^{27} -221.608 q^{29} -31.0000 q^{31} +1.39982 q^{33} +52.1898 q^{35} -310.448 q^{37} -1.83752 q^{39} -432.581 q^{41} +443.095 q^{43} +364.512 q^{45} +206.668 q^{47} -328.057 q^{49} +1.20716 q^{51} -428.205 q^{53} -501.306 q^{55} -1.96965 q^{57} -621.019 q^{59} +148.908 q^{61} +104.365 q^{63} +658.053 q^{65} +648.871 q^{67} +0.870577 q^{69} -875.630 q^{71} -55.1632 q^{73} +2.15950 q^{75} -143.531 q^{77} -771.120 q^{79} +728.885 q^{81} +365.678 q^{83} -432.310 q^{85} -8.35461 q^{87} +1632.10 q^{89} +188.410 q^{91} -1.16870 q^{93} +705.374 q^{95} +27.5970 q^{97} -1002.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 15 q^{5} + 9 q^{7} + 29 q^{9} - 88 q^{11} + 28 q^{13} - 130 q^{15} + 138 q^{17} + 43 q^{19} - 170 q^{21} + 206 q^{23} + 466 q^{25} - 172 q^{27} - 474 q^{29} - 155 q^{31} - 236 q^{33} + 79 q^{35}+ \cdots - 3988 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 15 * q^5 + 9 * q^7 + 29 * q^9 - 88 * q^11 + 28 * q^13 - 130 * q^15 + 138 * q^17 + 43 * q^19 - 170 * q^21 + 206 * q^23 + 466 * q^25 - 172 * q^27 - 474 * q^29 - 155 * q^31 - 236 * q^33 + 79 * q^35 + 508 * q^37 - 792 * q^39 + 473 * q^41 + 82 * q^43 - 15 * q^45 - 644 * q^47 - 776 * q^49 + 1360 * q^51 - 374 * q^53 - 798 * q^55 + 558 * q^57 + 541 * q^59 - 440 * q^61 - 663 * q^63 + 1602 * q^65 + 1884 * q^67 - 2500 * q^69 + 491 * q^71 + 302 * q^73 + 2418 * q^75 + 1154 * q^77 - 1244 * q^79 - 127 * q^81 - 1544 * q^83 - 802 * q^85 - 220 * q^87 + 3056 * q^89 + 2042 * q^91 + 124 * q^93 + 217 * q^95 + 583 * q^97 - 3988 * q^99

Introduction

L-functions

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