LMFDB - Embedded newform 384.4.d.f.193.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,4,Mod(193,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.193"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0] 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:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1534132224.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 $ x^8 + 18x^6 + 107x^4 + 210*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.3
Root			$-2.21597i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.f.193.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +5.67763i q^{5} -33.0917 q^{7} -9.00000 q^{9} +34.6274i q^{11} -82.2421i q^{13} +17.0329 q^{15} +97.8823 q^{17} +55.8823i q^{19} +99.2750i q^{21} +130.418 q^{23} +92.7645 q^{25} +27.0000i q^{27} +147.451i q^{29} +101.223 q^{31} +103.882 q^{33} -187.882i q^{35} -184.439i q^{37} -246.726 q^{39} +237.411 q^{41} +199.882i q^{43} -51.0987i q^{45} -334.813 q^{47} +752.058 q^{49} -293.647i q^{51} -102.030i q^{53} -196.602 q^{55} +167.647 q^{57} +105.961i q^{59} +717.803i q^{61} +297.825 q^{63} +466.940 q^{65} -316.471i q^{67} -391.255i q^{69} +800.045 q^{71} +301.058 q^{73} -278.294i q^{75} -1145.88i q^{77} -42.8329 q^{79} +81.0000 q^{81} -1236.67i q^{83} +555.739i q^{85} +442.354 q^{87} +1325.29 q^{89} +2721.53i q^{91} -303.670i q^{93} -317.279 q^{95} -505.765 q^{97} -311.647i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) $ 8 * q - 72 * q^9 + 240 * q^17 - 344 * q^25 + 288 * q^33 - 816 * q^41 + 1672 * q^49 - 288 * q^57 - 1152 * q^65 - 1936 * q^73 + 648 * q^81 + 7344 * q^89 - 2960 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	5.67763i	0.507823i	0.967227	+	0.253911i	$0.0817172\pi$
$5$	5.67763i	0.507823i	−0.967227	+	0.253911i	$0.918283\pi$
$6$	0	0
$7$	−33.0917	−1.78678	−0.893391	−	0.449280i	$-0.851680\pi$
$7$	−33.0917	−1.78678	−0.893391	+	0.449280i	$0.851680\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	34.6274i	0.949142i	0.880217	+	0.474571i	$0.157397\pi$
$11$	34.6274i	0.949142i	−0.880217	+	0.474571i	$0.842603\pi$
$12$	0	0
$13$	− 82.2421i	− 1.75460i	−0.479939	−	0.877302i	$-0.659341\pi$
$13$	− 82.2421i	− 1.75460i	0.479939	−	0.877302i	$-0.340659\pi$
$14$	0	0
$15$	17.0329	0.293192
$16$	0	0
$17$	97.8823	1.39647	0.698233	−	0.715870i	$-0.253970\pi$
$17$	97.8823	1.39647	0.698233	+	0.715870i	$0.253970\pi$
$18$	0	0
$19$	55.8823i	0.674751i	0.941370	+	0.337375i	$0.109539\pi$
$19$	55.8823i	0.674751i	−0.941370	+	0.337375i	$0.890461\pi$
$20$	0	0
$21$	99.2750i	1.03160i
$22$	0	0
$23$	130.418	1.18235	0.591176	−	0.806542i	$-0.298664\pi$
$23$	130.418	1.18235	0.591176	+	0.806542i	$0.298664\pi$
$24$	0	0
$25$	92.7645	0.742116
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	147.451i	0.944173i	0.881552	+	0.472086i	$0.156499\pi$
$29$	147.451i	0.944173i	−0.881552	+	0.472086i	$0.843501\pi$
$30$	0	0
$31$	101.223	0.586459	0.293230	−	0.956042i	$-0.405270\pi$
$31$	101.223	0.586459	0.293230	+	0.956042i	$0.405270\pi$
$32$	0	0
$33$	103.882	0.547987
$34$	0	0
$35$	− 187.882i	− 0.907368i
$36$	0	0
$37$	− 184.439i	− 0.819504i	−0.912197	−	0.409752i	$-0.865615\pi$
$37$	− 184.439i	− 0.819504i	0.912197	−	0.409752i	$-0.134385\pi$
$38$	0	0
$39$	−246.726	−1.01302
$40$	0	0
$41$	237.411	0.904327	0.452164	−	0.891935i	$-0.350652\pi$
$41$	237.411	0.904327	0.452164	+	0.891935i	$0.350652\pi$
$42$	0	0
$43$	199.882i	0.708878i	0.935079	+	0.354439i	$0.115328\pi$
$43$	199.882i	0.708878i	−0.935079	+	0.354439i	$0.884672\pi$
$44$	0	0
$45$	− 51.0987i	− 0.169274i
$46$	0	0
$47$	−334.813	−1.03910	−0.519548	−	0.854441i	$-0.673900\pi$
$47$	−334.813	−1.03910	−0.519548	+	0.854441i	$0.673900\pi$
$48$	0	0
$49$	752.058	2.19259
$50$	0	0
$51$	− 293.647i	− 0.806250i
$52$	0	0
$53$	− 102.030i	− 0.264433i	−0.991221	−	0.132216i	$-0.957791\pi$
$53$	− 102.030i	− 0.264433i	0.991221	−	0.132216i	$-0.0422094\pi$
$54$	0	0
$55$	−196.602	−0.481996
$56$	0	0
$57$	167.647	0.389568
$58$	0	0
$59$	105.961i	0.233813i	0.993143	+	0.116907i	$0.0372978\pi$
$59$	105.961i	0.233813i	−0.993143	+	0.116907i	$0.962702\pi$
$60$	0	0
$61$	717.803i	1.50664i	0.657653	+	0.753321i	$0.271549\pi$
$61$	717.803i	1.50664i	−0.657653	+	0.753321i	$0.728451\pi$
$62$	0	0
$63$	297.825	0.595594
$64$	0	0
$65$	466.940	0.891028
$66$	0	0
$67$	− 316.471i	− 0.577061i	−0.957471	−	0.288530i	$-0.906833\pi$
$67$	− 316.471i	− 0.577061i	0.957471	−	0.288530i	$-0.0931667\pi$
$68$	0	0
$69$	− 391.255i	− 0.682632i
$70$	0	0
$71$	800.045	1.33729	0.668647	−	0.743580i	$-0.266874\pi$
$71$	800.045	1.33729	0.668647	+	0.743580i	$0.266874\pi$
$72$	0	0
$73$	301.058	0.482687	0.241344	−	0.970440i	$-0.422412\pi$
$73$	301.058	0.482687	0.241344	+	0.970440i	$0.422412\pi$
$74$	0	0
$75$	− 278.294i	− 0.428461i
$76$	0	0
$77$	− 1145.88i	− 1.69591i
$78$	0	0
$79$	−42.8329	−0.0610010	−0.0305005	−	0.999535i	$-0.509710\pi$
$79$	−42.8329	−0.0610010	−0.0305005	+	0.999535i	$0.509710\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1236.67i	− 1.63544i	−0.575614	−	0.817721i	$-0.695237\pi$
$83$	− 1236.67i	− 1.63544i	0.575614	−	0.817721i	$-0.304763\pi$
$84$	0	0
$85$	555.739i	0.709158i
$86$	0	0
$87$	442.354	0.545119
$88$	0	0
$89$	1325.29	1.57844	0.789218	−	0.614113i	$-0.210486\pi$
$89$	1325.29	1.57844	0.789218	+	0.614113i	$0.210486\pi$
$90$	0	0
$91$	2721.53i	3.13509i
$92$	0	0
$93$	− 303.670i	− 0.338592i
$94$	0	0
$95$	−317.279	−0.342654
$96$	0	0
$97$	−505.765	−0.529408	−0.264704	−	0.964330i	$-0.585274\pi$
$97$	−505.765	−0.529408	−0.264704	+	0.964330i	$0.585274\pi$
$98$	0	0
$99$	− 311.647i	− 0.316381i
$100$	0	0
$101$	1281.97i	1.26298i	0.775383	+	0.631491i	$0.217557\pi$
$101$	1281.97i	1.26298i	−0.775383	+	0.631491i	$0.782443\pi$
$102$	0	0
$103$	161.562	0.154555	0.0772775	−	0.997010i	$-0.475377\pi$
$103$	161.562	0.154555	0.0772775	+	0.997010i	$0.475377\pi$
$104$	0	0
$105$	−563.647	−0.523869
$106$	0	0
$107$	481.726i	0.435235i	0.976034	+	0.217618i	$0.0698286\pi$
$107$	481.726i	0.435235i	−0.976034	+	0.217618i	$0.930171\pi$
$108$	0	0
$109$	− 286.637i	− 0.251879i	−0.992038	−	0.125940i	$-0.959805\pi$
$109$	− 286.637i	− 0.251879i	0.992038	−	0.125940i	$-0.0401945\pi$
$110$	0	0
$111$	−553.318	−0.473141
$112$	0	0
$113$	−1397.29	−1.16324	−0.581621	−	0.813460i	$-0.697581\pi$
$113$	−1397.29	−1.16324	−0.581621	+	0.813460i	$0.697581\pi$
$114$	0	0
$115$	740.468i	0.600426i
$116$	0	0
$117$	740.179i	0.584868i
$118$	0	0
$119$	−3239.09	−2.49518
$120$	0	0
$121$	131.942	0.0991300
$122$	0	0
$123$	− 712.234i	− 0.522113i
$124$	0	0
$125$	1236.39i	0.884686i
$126$	0	0
$127$	443.829	0.310106	0.155053	−	0.987906i	$-0.450445\pi$
$127$	443.829	0.310106	0.155053	+	0.987906i	$0.450445\pi$
$128$	0	0
$129$	599.647	0.409271
$130$	0	0
$131$	− 1365.57i	− 0.910765i	−0.890296	−	0.455382i	$-0.849503\pi$
$131$	− 1365.57i	− 0.910765i	0.890296	−	0.455382i	$-0.150497\pi$
$132$	0	0
$133$	− 1849.24i	− 1.20563i
$134$	0	0
$135$	−153.296	−0.0977305
$136$	0	0
$137$	−1223.06	−0.762721	−0.381360	−	0.924426i	$-0.624544\pi$
$137$	−1223.06	−0.762721	−0.381360	+	0.924426i	$0.624544\pi$
$138$	0	0
$139$	2176.70i	1.32824i	0.747625	+	0.664121i	$0.231194\pi$
$139$	2176.70i	1.32824i	−0.747625	+	0.664121i	$0.768806\pi$
$140$	0	0
$141$	1004.44i	0.599922i
$142$	0	0
$143$	2847.83	1.66537
$144$	0	0
$145$	−837.174	−0.479473
$146$	0	0
$147$	− 2256.17i	− 1.26589i
$148$	0	0
$149$	− 2875.88i	− 1.58122i	−0.612320	−	0.790610i	$-0.709764\pi$
$149$	− 2875.88i	− 1.58122i	0.612320	−	0.790610i	$-0.290236\pi$
$150$	0	0
$151$	2330.03	1.25573	0.627863	−	0.778323i	$-0.283930\pi$
$151$	2330.03	1.25573	0.627863	+	0.778323i	$0.283930\pi$
$152$	0	0
$153$	−880.940	−0.465489
$154$	0	0
$155$	574.708i	0.297817i
$156$	0	0
$157$	− 2447.31i	− 1.24405i	−0.782996	−	0.622027i	$-0.786310\pi$
$157$	− 2447.31i	− 1.24405i	0.782996	−	0.622027i	$-0.213690\pi$
$158$	0	0
$159$	−306.091	−0.152670
$160$	0	0
$161$	−4315.76	−2.11261
$162$	0	0
$163$	1436.82i	0.690432i	0.938523	+	0.345216i	$0.112194\pi$
$163$	1436.82i	0.690432i	−0.938523	+	0.345216i	$0.887806\pi$
$164$	0	0
$165$	589.805i	0.278280i
$166$	0	0
$167$	1769.42	0.819889	0.409945	−	0.912110i	$-0.365548\pi$
$167$	1769.42	0.819889	0.409945	+	0.912110i	$0.365548\pi$
$168$	0	0
$169$	−4566.76	−2.07863
$170$	0	0
$171$	− 502.940i	− 0.224917i
$172$	0	0
$173$	3897.86i	1.71300i	0.516148	+	0.856499i	$0.327365\pi$
$173$	3897.86i	1.71300i	−0.516148	+	0.856499i	$0.672635\pi$
$174$	0	0
$175$	−3069.73	−1.32600
$176$	0	0
$177$	317.884	0.134992
$178$	0	0
$179$	− 4146.74i	− 1.73152i	−0.500460	−	0.865760i	$-0.666836\pi$
$179$	− 4146.74i	− 1.73152i	0.500460	−	0.865760i	$-0.333164\pi$
$180$	0	0
$181$	1104.22i	0.453457i	0.973958	+	0.226728i	$0.0728030\pi$
$181$	1104.22i	0.453457i	−0.973958	+	0.226728i	$0.927197\pi$
$182$	0	0
$183$	2153.41	0.869860
$184$	0	0
$185$	1047.18	0.416163
$186$	0	0
$187$	3389.41i	1.32544i
$188$	0	0
$189$	− 893.475i	− 0.343866i
$190$	0	0
$191$	−35.0686	−0.0132852	−0.00664260	−	0.999978i	$-0.502114\pi$
$191$	−35.0686	−0.0132852	−0.00664260	+	0.999978i	$0.502114\pi$
$192$	0	0
$193$	615.884	0.229701	0.114851	−	0.993383i	$-0.463361\pi$
$193$	615.884	0.229701	0.114851	+	0.993383i	$0.463361\pi$
$194$	0	0
$195$	− 1400.82i	− 0.514435i
$196$	0	0
$197$	589.638i	0.213249i	0.994299	+	0.106624i	$0.0340042\pi$
$197$	589.638i	0.213249i	−0.994299	+	0.106624i	$0.965996\pi$
$198$	0	0
$199$	−4813.82	−1.71479	−0.857394	−	0.514661i	$-0.827918\pi$
$199$	−4813.82	−1.71479	−0.857394	+	0.514661i	$0.827918\pi$
$200$	0	0
$201$	−949.413	−0.333166
$202$	0	0
$203$	− 4879.41i	− 1.68703i
$204$	0	0
$205$	1347.93i	0.459238i
$206$	0	0
$207$	−1173.77	−0.394118
$208$	0	0
$209$	−1935.06	−0.640434
$210$	0	0
$211$	3294.35i	1.07484i	0.843313	+	0.537422i	$0.180602\pi$
$211$	3294.35i	1.07484i	−0.843313	+	0.537422i	$0.819398\pi$
$212$	0	0
$213$	− 2400.13i	− 0.772087i
$214$	0	0
$215$	−1134.86	−0.359984
$216$	0	0
$217$	−3349.65	−1.04787
$218$	0	0
$219$	− 903.174i	− 0.278680i
$220$	0	0
$221$	− 8050.04i	− 2.45025i
$222$	0	0
$223$	1572.78	0.472294	0.236147	−	0.971717i	$-0.424115\pi$
$223$	1572.78	0.472294	0.236147	+	0.971717i	$0.424115\pi$
$224$	0	0
$225$	−834.881	−0.247372
$226$	0	0
$227$	1492.51i	0.436394i	0.975905	+	0.218197i	$0.0700176\pi$
$227$	1492.51i	0.436394i	−0.975905	+	0.218197i	$0.929982\pi$
$228$	0	0
$229$	6163.31i	1.77853i	0.457393	+	0.889265i	$0.348783\pi$
$229$	6163.31i	1.77853i	−0.457393	+	0.889265i	$0.651217\pi$
$230$	0	0
$231$	−3437.64	−0.979134
$232$	0	0
$233$	2540.35	0.714266	0.357133	−	0.934054i	$-0.383754\pi$
$233$	2540.35	0.714266	0.357133	+	0.934054i	$0.383754\pi$
$234$	0	0
$235$	− 1900.95i	− 0.527677i
$236$	0	0
$237$	128.499i	0.0352190i
$238$	0	0
$239$	−1339.25	−0.362465	−0.181232	−	0.983440i	$-0.558009\pi$
$239$	−1339.25	−0.362465	−0.181232	+	0.983440i	$0.558009\pi$
$240$	0	0
$241$	2542.71	0.679627	0.339814	−	0.940493i	$-0.389636\pi$
$241$	2542.71	0.679627	0.339814	+	0.940493i	$0.389636\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	4269.91i	1.11345i
$246$	0	0
$247$	4595.87	1.18392
$248$	0	0
$249$	−3710.00	−0.944223
$250$	0	0
$251$	4701.45i	1.18228i	0.806568	+	0.591141i	$0.201322\pi$
$251$	4701.45i	1.18228i	−0.806568	+	0.591141i	$0.798678\pi$
$252$	0	0
$253$	4516.05i	1.12222i
$254$	0	0
$255$	1667.22	0.409432
$256$	0	0
$257$	−863.884	−0.209679	−0.104840	−	0.994489i	$-0.533433\pi$
$257$	−863.884	−0.209679	−0.104840	+	0.994489i	$0.533433\pi$
$258$	0	0
$259$	6103.41i	1.46428i
$260$	0	0
$261$	− 1327.06i	− 0.314724i
$262$	0	0
$263$	7091.36	1.66263	0.831315	−	0.555801i	$-0.187588\pi$
$263$	7091.36	1.66263	0.831315	+	0.555801i	$0.187588\pi$
$264$	0	0
$265$	579.290	0.134285
$266$	0	0
$267$	− 3975.88i	− 0.911311i
$268$	0	0
$269$	− 395.596i	− 0.0896650i	−0.998995	−	0.0448325i	$-0.985725\pi$
$269$	− 395.596i	− 0.0896650i	0.998995	−	0.0448325i	$-0.0142754\pi$
$270$	0	0
$271$	5532.21	1.24007	0.620033	−	0.784576i	$-0.287119\pi$
$271$	5532.21	1.24007	0.620033	+	0.784576i	$0.287119\pi$
$272$	0	0
$273$	8164.58	1.81005
$274$	0	0
$275$	3212.20i	0.704373i
$276$	0	0
$277$	− 3830.31i	− 0.830834i	−0.909631	−	0.415417i	$-0.863636\pi$
$277$	− 3830.31i	− 0.830834i	0.909631	−	0.415417i	$-0.136364\pi$
$278$	0	0
$279$	−911.009	−0.195486
$280$	0	0
$281$	5283.17	1.12159	0.560797	−	0.827954i	$-0.310495\pi$
$281$	5283.17	1.12159	0.560797	+	0.827954i	$0.310495\pi$
$282$	0	0
$283$	4639.76i	0.974577i	0.873241	+	0.487288i	$0.162014\pi$
$283$	4639.76i	0.974577i	−0.873241	+	0.487288i	$0.837986\pi$
$284$	0	0
$285$	951.836i	0.197831i
$286$	0	0
$287$	−7856.33	−1.61584
$288$	0	0
$289$	4667.94	0.950119
$290$	0	0
$291$	1517.29i	0.305654i
$292$	0	0
$293$	− 6022.12i	− 1.20074i	−0.799723	−	0.600369i	$-0.795020\pi$
$293$	− 6022.12i	− 1.20074i	0.799723	−	0.600369i	$-0.204980\pi$
$294$	0	0
$295$	−601.609	−0.118736
$296$	0	0
$297$	−934.940	−0.182662
$298$	0	0
$299$	− 10725.9i	− 2.07456i
$300$	0	0
$301$	− 6614.44i	− 1.26661i
$302$	0	0
$303$	3845.92	0.729183
$304$	0	0
$305$	−4075.42	−0.765107
$306$	0	0
$307$	− 2998.82i	− 0.557497i	−0.960364	−	0.278749i	$-0.910080\pi$
$307$	− 2998.82i	− 0.557497i	0.960364	−	0.278749i	$-0.0899196\pi$
$308$	0	0
$309$	− 484.685i	− 0.0892323i
$310$	0	0
$311$	−2403.97	−0.438318	−0.219159	−	0.975689i	$-0.570331\pi$
$311$	−2403.97	−0.438318	−0.219159	+	0.975689i	$0.570331\pi$
$312$	0	0
$313$	5845.75	1.05566	0.527830	−	0.849350i	$-0.323006\pi$
$313$	5845.75	1.05566	0.527830	+	0.849350i	$0.323006\pi$
$314$	0	0
$315$	1690.94i	0.302456i
$316$	0	0
$317$	8917.38i	1.57997i	0.613127	+	0.789984i	$0.289911\pi$
$317$	8917.38i	1.57997i	−0.613127	+	0.789984i	$0.710089\pi$
$318$	0	0
$319$	−5105.86	−0.896154
$320$	0	0
$321$	1445.18	0.251283
$322$	0	0
$323$	5469.88i	0.942267i
$324$	0	0
$325$	− 7629.15i	− 1.30212i
$326$	0	0
$327$	−859.910	−0.145422
$328$	0	0
$329$	11079.5	1.85664
$330$	0	0
$331$	− 9011.29i	− 1.49639i	−0.663478	−	0.748196i	$-0.730920\pi$
$331$	− 9011.29i	− 1.49639i	0.663478	−	0.748196i	$-0.269080\pi$
$332$	0	0
$333$	1659.96i	0.273168i
$334$	0	0
$335$	1796.81	0.293045
$336$	0	0
$337$	4516.35	0.730033	0.365017	−	0.931001i	$-0.381063\pi$
$337$	4516.35	0.730033	0.365017	+	0.931001i	$0.381063\pi$
$338$	0	0
$339$	4191.88i	0.671598i
$340$	0	0
$341$	3505.10i	0.556633i
$342$	0	0
$343$	−13536.4	−2.13090
$344$	0	0
$345$	2221.40	0.346656
$346$	0	0
$347$	− 3651.48i	− 0.564904i	−0.959281	−	0.282452i	$-0.908852\pi$
$347$	− 3651.48i	− 0.564904i	0.959281	−	0.282452i	$-0.0911478\pi$
$348$	0	0
$349$	3250.36i	0.498532i	0.968435	+	0.249266i	$0.0801894\pi$
$349$	3250.36i	0.498532i	−0.968435	+	0.249266i	$0.919811\pi$
$350$	0	0
$351$	2220.54	0.337674
$352$	0	0
$353$	592.345	0.0893125	0.0446563	−	0.999002i	$-0.485781\pi$
$353$	592.345	0.0893125	0.0446563	+	0.999002i	$0.485781\pi$
$354$	0	0
$355$	4542.36i	0.679108i
$356$	0	0
$357$	9717.26i	1.44059i
$358$	0	0
$359$	−3443.48	−0.506239	−0.253120	−	0.967435i	$-0.581457\pi$
$359$	−3443.48	−0.506239	−0.253120	+	0.967435i	$0.581457\pi$
$360$	0	0
$361$	3736.17	0.544711
$362$	0	0
$363$	− 395.826i	− 0.0572327i
$364$	0	0
$365$	1709.30i	0.245120i
$366$	0	0
$367$	3297.39	0.468998	0.234499	−	0.972116i	$-0.424655\pi$
$367$	3297.39	0.468998	0.234499	+	0.972116i	$0.424655\pi$
$368$	0	0
$369$	−2136.70	−0.301442
$370$	0	0
$371$	3376.35i	0.472483i
$372$	0	0
$373$	− 50.1819i	− 0.00696601i	−0.999994	−	0.00348300i	$-0.998891\pi$
$373$	− 50.1819i	− 0.00696601i	0.999994	−	0.00348300i	$-0.00110868\pi$
$374$	0	0
$375$	3709.16	0.510774
$376$	0	0
$377$	12126.7	1.65665
$378$	0	0
$379$	− 6769.28i	− 0.917453i	−0.888578	−	0.458726i	$-0.848306\pi$
$379$	− 6769.28i	− 0.917453i	0.888578	−	0.458726i	$-0.151694\pi$
$380$	0	0
$381$	− 1331.49i	− 0.179040i
$382$	0	0
$383$	4208.46	0.561468	0.280734	−	0.959786i	$-0.409422\pi$
$383$	4208.46	0.561468	0.280734	+	0.959786i	$0.409422\pi$
$384$	0	0
$385$	6505.88	0.861221
$386$	0	0
$387$	− 1798.94i	− 0.236293i
$388$	0	0
$389$	− 2490.47i	− 0.324607i	−0.986741	−	0.162303i	$-0.948108\pi$
$389$	− 2490.47i	− 0.324607i	0.986741	−	0.162303i	$-0.0518923\pi$
$390$	0	0
$391$	12765.6	1.65112
$392$	0	0
$393$	−4096.70	−0.525830
$394$	0	0
$395$	− 243.190i	− 0.0309777i
$396$	0	0
$397$	− 7905.51i	− 0.999411i	−0.866195	−	0.499706i	$-0.833442\pi$
$397$	− 7905.51i	− 0.999411i	0.866195	−	0.499706i	$-0.166558\pi$
$398$	0	0
$399$	−5547.71	−0.696072
$400$	0	0
$401$	−10389.4	−1.29382	−0.646911	−	0.762566i	$-0.723939\pi$
$401$	−10389.4	−1.29382	−0.646911	+	0.762566i	$0.723939\pi$
$402$	0	0
$403$	− 8324.81i	− 1.02900i
$404$	0	0
$405$	459.888i	0.0564248i
$406$	0	0
$407$	6386.66	0.777826
$408$	0	0
$409$	−13448.1	−1.62583	−0.812917	−	0.582379i	$-0.802122\pi$
$409$	−13448.1	−1.62583	−0.812917	+	0.582379i	$0.802122\pi$
$410$	0	0
$411$	3669.17i	0.440357i
$412$	0	0
$413$	− 3506.44i	− 0.417773i
$414$	0	0
$415$	7021.33	0.830515
$416$	0	0
$417$	6530.11	0.766860
$418$	0	0
$419$	5191.34i	0.605283i	0.953105	+	0.302641i	$0.0978684\pi$
$419$	5191.34i	0.605283i	−0.953105	+	0.302641i	$0.902132\pi$
$420$	0	0
$421$	5061.52i	0.585946i	0.956121	+	0.292973i	$0.0946447\pi$
$421$	5061.52i	0.585946i	−0.956121	+	0.292973i	$0.905355\pi$
$422$	0	0
$423$	3013.32	0.346365
$424$	0	0
$425$	9080.00	1.03634
$426$	0	0
$427$	− 23753.3i	− 2.69204i
$428$	0	0
$429$	− 8543.49i	− 0.961501i
$430$	0	0
$431$	−3005.64	−0.335909	−0.167954	−	0.985795i	$-0.553716\pi$
$431$	−3005.64	−0.335909	−0.167954	+	0.985795i	$0.553716\pi$
$432$	0	0
$433$	5895.88	0.654360	0.327180	−	0.944962i	$-0.393902\pi$
$433$	5895.88	0.654360	0.327180	+	0.944962i	$0.393902\pi$
$434$	0	0
$435$	2511.52i	0.276824i
$436$	0	0
$437$	7288.07i	0.797794i
$438$	0	0
$439$	11556.8	1.25644	0.628218	−	0.778038i	$-0.283785\pi$
$439$	11556.8	1.25644	0.628218	+	0.778038i	$0.283785\pi$
$440$	0	0
$441$	−6768.52	−0.730863
$442$	0	0
$443$	14007.8i	1.50233i	0.660117	+	0.751163i	$0.270507\pi$
$443$	14007.8i	1.50233i	−0.660117	+	0.751163i	$0.729493\pi$
$444$	0	0
$445$	7524.53i	0.801566i
$446$	0	0
$447$	−8627.65	−0.912917
$448$	0	0
$449$	3783.06	0.397625	0.198813	−	0.980038i	$-0.436291\pi$
$449$	3783.06	0.397625	0.198813	+	0.980038i	$0.436291\pi$
$450$	0	0
$451$	8220.94i	0.858335i
$452$	0	0
$453$	− 6990.08i	− 0.724994i
$454$	0	0
$455$	−15451.8	−1.59207
$456$	0	0
$457$	−1545.53	−0.158198	−0.0790992	−	0.996867i	$-0.525204\pi$
$457$	−1545.53	−0.158198	−0.0790992	+	0.996867i	$0.525204\pi$
$458$	0	0
$459$	2642.82i	0.268750i
$460$	0	0
$461$	− 12730.2i	− 1.28613i	−0.765811	−	0.643065i	$-0.777662\pi$
$461$	− 12730.2i	− 1.28613i	0.765811	−	0.643065i	$-0.222338\pi$
$462$	0	0
$463$	−19656.4	−1.97303	−0.986513	−	0.163682i	$-0.947663\pi$
$463$	−19656.4	−1.97303	−0.986513	+	0.163682i	$0.947663\pi$
$464$	0	0
$465$	1724.12	0.171945
$466$	0	0
$467$	6016.26i	0.596144i	0.954543	+	0.298072i	$0.0963436\pi$
$467$	6016.26i	0.596144i	−0.954543	+	0.298072i	$0.903656\pi$
$468$	0	0
$469$	10472.6i	1.03108i
$470$	0	0
$471$	−7341.92	−0.718255
$472$	0	0
$473$	−6921.41	−0.672826
$474$	0	0
$475$	5183.89i	0.500743i
$476$	0	0
$477$	918.272i	0.0881442i
$478$	0	0
$479$	13890.6	1.32501	0.662505	−	0.749057i	$-0.269493\pi$
$479$	13890.6	1.32501	0.662505	+	0.749057i	$0.269493\pi$
$480$	0	0
$481$	−15168.7	−1.43791
$482$	0	0
$483$	12947.3i	1.21971i
$484$	0	0
$485$	− 2871.54i	− 0.268846i
$486$	0	0
$487$	−9314.23	−0.866669	−0.433335	−	0.901233i	$-0.642663\pi$
$487$	−9314.23	−0.866669	−0.433335	+	0.901233i	$0.642663\pi$
$488$	0	0
$489$	4310.46	0.398621
$490$	0	0
$491$	− 7253.34i	− 0.666677i	−0.942807	−	0.333339i	$-0.891825\pi$
$491$	− 7253.34i	− 0.666677i	0.942807	−	0.333339i	$-0.108175\pi$
$492$	0	0
$493$	14432.9i	1.31851i
$494$	0	0
$495$	1769.42	0.160665
$496$	0	0
$497$	−26474.8	−2.38945
$498$	0	0
$499$	− 16208.2i	− 1.45407i	−0.686602	−	0.727034i	$-0.740898\pi$
$499$	− 16208.2i	− 1.45407i	0.686602	−	0.727034i	$-0.259102\pi$
$500$	0	0
$501$	− 5308.25i	− 0.473363i
$502$	0	0
$503$	2182.04	0.193425	0.0967123	−	0.995312i	$-0.469167\pi$
$503$	2182.04	0.193425	0.0967123	+	0.995312i	$0.469167\pi$
$504$	0	0
$505$	−7278.58	−0.641371
$506$	0	0
$507$	13700.3i	1.20010i
$508$	0	0
$509$	− 1795.97i	− 0.156395i	−0.996938	−	0.0781974i	$-0.975084\pi$
$509$	− 1795.97i	− 0.156395i	0.996938	−	0.0781974i	$-0.0249164\pi$
$510$	0	0
$511$	−9962.51	−0.862457
$512$	0	0
$513$	−1508.82	−0.129856
$514$	0	0
$515$	917.288i	0.0784865i
$516$	0	0
$517$	− 11593.7i	− 0.986249i
$518$	0	0
$519$	11693.6	0.989000
$520$	0	0
$521$	−20946.1	−1.76135	−0.880676	−	0.473718i	$-0.842912\pi$
$521$	−20946.1	−1.76135	−0.880676	+	0.473718i	$0.842912\pi$
$522$	0	0
$523$	− 5363.64i	− 0.448443i	−0.974538	−	0.224221i	$-0.928016\pi$
$523$	− 5363.64i	− 0.448443i	0.974538	−	0.224221i	$-0.0719838\pi$
$524$	0	0
$525$	9209.19i	0.765566i
$526$	0	0
$527$	9907.96	0.818970
$528$	0	0
$529$	4841.96	0.397958
$530$	0	0
$531$	− 953.652i	− 0.0779378i
$532$	0	0
$533$	− 19525.2i	− 1.58674i
$534$	0	0
$535$	−2735.06	−0.221022
$536$	0	0
$537$	−12440.2	−0.999693
$538$	0	0
$539$	26041.8i	2.08108i
$540$	0	0
$541$	3431.20i	0.272678i	0.990662	+	0.136339i	$0.0435337\pi$
$541$	3431.20i	0.272678i	−0.990662	+	0.136339i	$0.956466\pi$
$542$	0	0
$543$	3312.65	0.261803
$544$	0	0
$545$	1627.42	0.127910
$546$	0	0
$547$	− 19449.0i	− 1.52026i	−0.649773	−	0.760128i	$-0.725136\pi$
$547$	− 19449.0i	− 1.52026i	0.649773	−	0.760128i	$-0.274864\pi$
$548$	0	0
$549$	− 6460.22i	− 0.502214i
$550$	0	0
$551$	−8239.91	−0.637082
$552$	0	0
$553$	1417.41	0.108996
$554$	0	0
$555$	− 3141.54i	− 0.240272i
$556$	0	0
$557$	9898.44i	0.752981i	0.926420	+	0.376491i	$0.122869\pi$
$557$	9898.44i	0.752981i	−0.926420	+	0.376491i	$0.877131\pi$
$558$	0	0
$559$	16438.7	1.24380
$560$	0	0
$561$	10168.2	0.765246
$562$	0	0
$563$	− 13809.8i	− 1.03378i	−0.856053	−	0.516888i	$-0.827091\pi$
$563$	− 13809.8i	− 1.03378i	0.856053	−	0.516888i	$-0.172909\pi$
$564$	0	0
$565$	− 7933.32i	− 0.590721i
$566$	0	0
$567$	−2680.42	−0.198531
$568$	0	0
$569$	−123.053	−0.00906615	−0.00453308	−	0.999990i	$-0.501443\pi$
$569$	−123.053	−0.00906615	−0.00453308	+	0.999990i	$0.501443\pi$
$570$	0	0
$571$	2540.72i	0.186210i	0.995656	+	0.0931049i	$0.0296792\pi$
$571$	2540.72i	0.186210i	−0.995656	+	0.0931049i	$0.970321\pi$
$572$	0	0
$573$	105.206i	0.00767022i
$574$	0	0
$575$	12098.2	0.877443
$576$	0	0
$577$	−15618.2	−1.12685	−0.563427	−	0.826166i	$-0.690517\pi$
$577$	−15618.2	−1.12685	−0.563427	+	0.826166i	$0.690517\pi$
$578$	0	0
$579$	− 1847.65i	− 0.132618i
$580$	0	0
$581$	40923.3i	2.92218i
$582$	0	0
$583$	3533.04	0.250984
$584$	0	0
$585$	−4202.46	−0.297009
$586$	0	0
$587$	− 1809.56i	− 0.127238i	−0.997974	−	0.0636188i	$-0.979736\pi$
$587$	− 1809.56i	− 0.127238i	0.997974	−	0.0636188i	$-0.0202642\pi$
$588$	0	0
$589$	5656.58i	0.395714i
$590$	0	0
$591$	1768.91	0.123119
$592$	0	0
$593$	898.229	0.0622021	0.0311010	−	0.999516i	$-0.490099\pi$
$593$	898.229	0.0622021	0.0311010	+	0.999516i	$0.490099\pi$
$594$	0	0
$595$	− 18390.3i	− 1.26711i
$596$	0	0
$597$	14441.5i	0.990033i
$598$	0	0
$599$	22990.9	1.56825	0.784125	−	0.620603i	$-0.213112\pi$
$599$	22990.9	1.56825	0.784125	+	0.620603i	$0.213112\pi$
$600$	0	0
$601$	−26893.7	−1.82532	−0.912661	−	0.408717i	$-0.865976\pi$
$601$	−26893.7	−1.82532	−0.912661	+	0.408717i	$0.865976\pi$
$602$	0	0
$603$	2848.24i	0.192354i
$604$	0	0
$605$	749.118i	0.0503405i
$606$	0	0
$607$	6306.93	0.421730	0.210865	−	0.977515i	$-0.432372\pi$
$607$	6306.93	0.421730	0.210865	+	0.977515i	$0.432372\pi$
$608$	0	0
$609$	−14638.2	−0.974008
$610$	0	0
$611$	27535.7i	1.82320i
$612$	0	0
$613$	2612.97i	0.172164i	0.996288	+	0.0860822i	$0.0274348\pi$
$613$	2612.97i	0.172164i	−0.996288	+	0.0860822i	$0.972565\pi$
$614$	0	0
$615$	4043.80	0.265141
$616$	0	0
$617$	−2803.88	−0.182949	−0.0914747	−	0.995807i	$-0.529158\pi$
$617$	−2803.88	−0.182949	−0.0914747	+	0.995807i	$0.529158\pi$
$618$	0	0
$619$	− 10547.1i	− 0.684849i	−0.939545	−	0.342425i	$-0.888752\pi$
$619$	− 10547.1i	− 0.684849i	0.939545	−	0.342425i	$-0.111248\pi$
$620$	0	0
$621$	3521.30i	0.227544i
$622$	0	0
$623$	−43856.2	−2.82032
$624$	0	0
$625$	4575.82	0.292852
$626$	0	0
$627$	5805.17i	0.369755i
$628$	0	0
$629$	− 18053.3i	− 1.14441i
$630$	0	0
$631$	−14161.4	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$631$	−14161.4	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$632$	0	0
$633$	9883.04	0.620562
$634$	0	0
$635$	2519.90i	0.157479i
$636$	0	0
$637$	− 61850.8i	− 3.84713i
$638$	0	0
$639$	−7200.40	−0.445764
$640$	0	0
$641$	5622.61	0.346458	0.173229	−	0.984882i	$-0.444580\pi$
$641$	5622.61	0.346458	0.173229	+	0.984882i	$0.444580\pi$
$642$	0	0
$643$	− 29438.7i	− 1.80552i	−0.430146	−	0.902759i	$-0.641538\pi$
$643$	− 29438.7i	− 1.80552i	0.430146	−	0.902759i	$-0.358462\pi$
$644$	0	0
$645$	3404.57i	0.207837i
$646$	0	0
$647$	−4607.39	−0.279962	−0.139981	−	0.990154i	$-0.544704\pi$
$647$	−4607.39	−0.279962	−0.139981	+	0.990154i	$0.544704\pi$
$648$	0	0
$649$	−3669.17	−0.221922
$650$	0	0
$651$	10048.9i	0.604991i
$652$	0	0
$653$	16634.1i	0.996850i	0.866933	+	0.498425i	$0.166088\pi$
$653$	16634.1i	0.996850i	−0.866933	+	0.498425i	$0.833912\pi$
$654$	0	0
$655$	7753.19	0.462507
$656$	0	0
$657$	−2709.52	−0.160896
$658$	0	0
$659$	− 20619.9i	− 1.21887i	−0.792835	−	0.609437i	$-0.791396\pi$
$659$	− 20619.9i	− 1.21887i	0.792835	−	0.609437i	$-0.208604\pi$
$660$	0	0
$661$	− 881.112i	− 0.0518476i	−0.999664	−	0.0259238i	$-0.991747\pi$
$661$	− 881.112i	− 0.0518476i	0.999664	−	0.0259238i	$-0.00825273\pi$
$662$	0	0
$663$	−24150.1	−1.41465
$664$	0	0
$665$	10499.3	0.612248
$666$	0	0
$667$	19230.4i	1.11635i
$668$	0	0
$669$	− 4718.35i	− 0.272679i
$670$	0	0
$671$	−24855.6	−1.43002
$672$	0	0
$673$	8943.86	0.512274	0.256137	−	0.966641i	$-0.417550\pi$
$673$	8943.86	0.512274	0.256137	+	0.966641i	$0.417550\pi$
$674$	0	0
$675$	2504.64i	0.142820i
$676$	0	0
$677$	− 21748.7i	− 1.23467i	−0.786702	−	0.617333i	$-0.788213\pi$
$677$	− 21748.7i	− 1.23467i	0.786702	−	0.617333i	$-0.211787\pi$
$678$	0	0
$679$	16736.6	0.945937
$680$	0	0
$681$	4477.53	0.251952
$682$	0	0
$683$	7922.37i	0.443838i	0.975065	+	0.221919i	$0.0712320\pi$
$683$	7922.37i	0.443838i	−0.975065	+	0.221919i	$0.928768\pi$
$684$	0	0
$685$	− 6944.06i	− 0.387327i
$686$	0	0
$687$	18489.9	1.02683
$688$	0	0
$689$	−8391.18	−0.463975
$690$	0	0
$691$	177.517i	0.00977288i	0.999988	+	0.00488644i	$0.00155541\pi$
$691$	177.517i	0.00977288i	−0.999988	+	0.00488644i	$0.998445\pi$
$692$	0	0
$693$	10312.9i	0.565303i
$694$	0	0
$695$	−12358.5	−0.674511
$696$	0	0
$697$	23238.3	1.26286
$698$	0	0
$699$	− 7621.05i	− 0.412382i
$700$	0	0
$701$	− 767.980i	− 0.0413783i	−0.999786	−	0.0206891i	$-0.993414\pi$
$701$	− 767.980i	− 0.0413783i	0.999786	−	0.0206891i	$-0.00658603\pi$
$702$	0	0
$703$	10306.9	0.552961
$704$	0	0
$705$	−5702.84	−0.304654
$706$	0	0
$707$	− 42422.7i	− 2.25667i
$708$	0	0
$709$	31634.2i	1.67566i	0.545928	+	0.837832i	$0.316177\pi$
$709$	31634.2i	1.67566i	−0.545928	+	0.837832i	$0.683823\pi$
$710$	0	0
$711$	385.496	0.0203337
$712$	0	0
$713$	13201.4	0.693401
$714$	0	0
$715$	16168.9i	0.845712i
$716$	0	0
$717$	4017.76i	0.209269i
$718$	0	0
$719$	17351.7	0.900011	0.450005	−	0.893026i	$-0.351422\pi$
$719$	17351.7	0.900011	0.450005	+	0.893026i	$0.351422\pi$
$720$	0	0
$721$	−5346.35	−0.276156
$722$	0	0
$723$	− 7628.12i	− 0.392383i
$724$	0	0
$725$	13678.2i	0.700686i
$726$	0	0
$727$	−16364.6	−0.834842	−0.417421	−	0.908713i	$-0.637066\pi$
$727$	−16364.6	−0.834842	−0.417421	+	0.908713i	$0.637066\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	19564.9i	0.989925i
$732$	0	0
$733$	23552.7i	1.18682i	0.804900	+	0.593410i	$0.202219\pi$
$733$	23552.7i	1.18682i	−0.804900	+	0.593410i	$0.797781\pi$
$734$	0	0
$735$	12809.7	0.642849
$736$	0	0
$737$	10958.6	0.547713
$738$	0	0
$739$	3519.05i	0.175169i	0.996157	+	0.0875847i	$0.0279149\pi$
$739$	3519.05i	0.175169i	−0.996157	+	0.0875847i	$0.972085\pi$
$740$	0	0
$741$	− 13787.6i	− 0.683537i
$742$	0	0
$743$	20752.4	1.02467	0.512336	−	0.858785i	$-0.328780\pi$
$743$	20752.4	1.02467	0.512336	+	0.858785i	$0.328780\pi$
$744$	0	0
$745$	16328.2	0.802979
$746$	0	0
$747$	11130.0i	0.545148i
$748$	0	0
$749$	− 15941.1i	− 0.777671i
$750$	0	0
$751$	−3679.08	−0.178764	−0.0893818	−	0.995997i	$-0.528489\pi$
$751$	−3679.08	−0.178764	−0.0893818	+	0.995997i	$0.528489\pi$
$752$	0	0
$753$	14104.3	0.682591
$754$	0	0
$755$	13229.0i	0.637687i
$756$	0	0
$757$	− 527.275i	− 0.0253159i	−0.999920	−	0.0126579i	$-0.995971\pi$
$757$	− 527.275i	− 0.0253159i	0.999920	−	0.0126579i	$-0.00402926\pi$
$758$	0	0
$759$	13548.2	0.647914
$760$	0	0
$761$	3231.54	0.153933	0.0769666	−	0.997034i	$-0.475477\pi$
$761$	3231.54	0.153933	0.0769666	+	0.997034i	$0.475477\pi$
$762$	0	0
$763$	9485.29i	0.450053i
$764$	0	0
$765$	− 5001.65i	− 0.236386i
$766$	0	0
$767$	8714.48	0.410250
$768$	0	0
$769$	13681.7	0.641581	0.320791	−	0.947150i	$-0.396051\pi$
$769$	13681.7	0.641581	0.320791	+	0.947150i	$0.396051\pi$
$770$	0	0
$771$	2591.65i	0.121058i
$772$	0	0
$773$	− 9216.63i	− 0.428847i	−0.976741	−	0.214424i	$-0.931213\pi$
$773$	− 9216.63i	− 0.428847i	0.976741	−	0.214424i	$-0.0687873\pi$
$774$	0	0
$775$	9389.92	0.435221
$776$	0	0
$777$	18310.2	0.845400
$778$	0	0
$779$	13267.1i	0.610196i
$780$	0	0
$781$	27703.5i	1.26928i
$782$	0	0
$783$	−3981.18	−0.181706
$784$	0	0
$785$	13894.9	0.631759
$786$	0	0
$787$	33169.0i	1.50235i	0.660104	+	0.751174i	$0.270512\pi$
$787$	33169.0i	1.50235i	−0.660104	+	0.751174i	$0.729488\pi$
$788$	0	0
$789$	− 21274.1i	− 0.959920i
$790$	0	0
$791$	46238.8	2.07846
$792$	0	0
$793$	59033.6	2.64356
$794$	0	0
$795$	− 1737.87i	− 0.0775294i
$796$	0	0
$797$	23781.9i	1.05696i	0.848945	+	0.528481i	$0.177238\pi$
$797$	23781.9i	1.05696i	−0.848945	+	0.528481i	$0.822762\pi$
$798$	0	0
$799$	−32772.3	−1.45106
$800$	0	0
$801$	−11927.6	−0.526145
$802$	0	0
$803$	10424.9i	0.458139i
$804$	0	0
$805$	− 24503.3i	− 1.07283i
$806$	0	0
$807$	−1186.79	−0.0517681
$808$	0	0
$809$	−29842.4	−1.29691	−0.648456	−	0.761252i	$-0.724585\pi$
$809$	−29842.4	−1.29691	−0.648456	+	0.761252i	$0.724585\pi$
$810$	0	0
$811$	− 19074.5i	− 0.825888i	−0.910756	−	0.412944i	$-0.864500\pi$
$811$	− 19074.5i	− 0.825888i	0.910756	−	0.412944i	$-0.135500\pi$
$812$	0	0
$813$	− 16596.6i	− 0.715952i
$814$	0	0
$815$	−8157.74	−0.350617
$816$	0	0
$817$	−11169.9	−0.478316
$818$	0	0
$819$	− 24493.7i	− 1.04503i
$820$	0	0
$821$	− 7493.97i	− 0.318564i	−0.987233	−	0.159282i	$-0.949082\pi$
$821$	− 7493.97i	− 0.318564i	0.987233	−	0.159282i	$-0.0509180\pi$
$822$	0	0
$823$	−1780.90	−0.0754294	−0.0377147	−	0.999289i	$-0.512008\pi$
$823$	−1780.90	−0.0754294	−0.0377147	+	0.999289i	$0.512008\pi$
$824$	0	0
$825$	9636.59	0.406670
$826$	0	0
$827$	10860.1i	0.456640i	0.973586	+	0.228320i	$0.0733232\pi$
$827$	10860.1i	0.456640i	−0.973586	+	0.228320i	$0.926677\pi$
$828$	0	0
$829$	34105.1i	1.42885i	0.699711	+	0.714426i	$0.253312\pi$
$829$	34105.1i	1.42885i	−0.699711	+	0.714426i	$0.746688\pi$
$830$	0	0
$831$	−11490.9	−0.479682
$832$	0	0
$833$	73613.1	3.06188
$834$	0	0
$835$	10046.1i	0.416358i
$836$	0	0
$837$	2733.03i	0.112864i
$838$	0	0
$839$	−32688.9	−1.34511	−0.672555	−	0.740047i	$-0.734803\pi$
$839$	−32688.9	−1.34511	−0.672555	+	0.740047i	$0.734803\pi$
$840$	0	0
$841$	2647.12	0.108537
$842$	0	0
$843$	− 15849.5i	− 0.647552i
$844$	0	0
$845$	− 25928.4i	− 1.05558i
$846$	0	0
$847$	−4366.18	−0.177124
$848$	0	0
$849$	13919.3	0.562672
$850$	0	0
$851$	− 24054.3i	− 0.968943i
$852$	0	0
$853$	17391.2i	0.698080i	0.937108	+	0.349040i	$0.113492\pi$
$853$	17391.2i	0.698080i	−0.937108	+	0.349040i	$0.886508\pi$
$854$	0	0
$855$	2855.51	0.114218
$856$	0	0
$857$	−4182.34	−0.166705	−0.0833525	−	0.996520i	$-0.526563\pi$
$857$	−4182.34	−0.166705	−0.0833525	+	0.996520i	$0.526563\pi$
$858$	0	0
$859$	20585.0i	0.817639i	0.912615	+	0.408820i	$0.134059\pi$
$859$	20585.0i	0.817639i	−0.912615	+	0.408820i	$0.865941\pi$
$860$	0	0
$861$	23569.0i	0.932903i
$862$	0	0
$863$	−14159.7	−0.558518	−0.279259	−	0.960216i	$-0.590089\pi$
$863$	−14159.7	−0.558518	−0.279259	+	0.960216i	$0.590089\pi$
$864$	0	0
$865$	−22130.6	−0.869900
$866$	0	0
$867$	− 14003.8i	− 0.548552i
$868$	0	0
$869$	− 1483.19i	− 0.0578986i
$870$	0	0
$871$	−26027.2	−1.01251
$872$	0	0
$873$	4551.88	0.176469
$874$	0	0
$875$	− 40914.1i	− 1.58074i
$876$	0	0
$877$	5156.38i	0.198539i	0.995061	+	0.0992695i	$0.0316506\pi$
$877$	5156.38i	0.198539i	−0.995061	+	0.0992695i	$0.968349\pi$
$878$	0	0
$879$	−18066.4	−0.693246
$880$	0	0
$881$	841.065	0.0321637	0.0160818	−	0.999871i	$-0.494881\pi$
$881$	841.065	0.0321637	0.0160818	+	0.999871i	$0.494881\pi$
$882$	0	0
$883$	− 7845.99i	− 0.299024i	−0.988760	−	0.149512i	$-0.952230\pi$
$883$	− 7845.99i	− 0.299024i	0.988760	−	0.149512i	$-0.0477703\pi$
$884$	0	0
$885$	1804.83i	0.0685521i
$886$	0	0
$887$	−38781.9	−1.46806	−0.734029	−	0.679118i	$-0.762363\pi$
$887$	−38781.9	−1.46806	−0.734029	+	0.679118i	$0.762363\pi$
$888$	0	0
$889$	−14687.1	−0.554092
$890$	0	0
$891$	2804.82i	0.105460i
$892$	0	0
$893$	− 18710.1i	− 0.701131i
$894$	0	0
$895$	23543.7	0.879305
$896$	0	0
$897$	−32177.6	−1.19775
$898$	0	0
$899$	14925.5i	0.553719i
$900$	0	0
$901$	− 9986.95i	− 0.369271i
$902$	0	0
$903$	−19843.3	−0.731278
$904$	0	0
$905$	−6269.33	−0.230276
$906$	0	0
$907$	36431.2i	1.33371i	0.745187	+	0.666856i	$0.232360\pi$
$907$	36431.2i	1.33371i	−0.745187	+	0.666856i	$0.767640\pi$
$908$	0	0
$909$	− 11537.8i	− 0.420994i
$910$	0	0
$911$	51077.9	1.85762	0.928808	−	0.370562i	$-0.120835\pi$
$911$	51077.9	1.85762	0.928808	+	0.370562i	$0.120835\pi$
$912$	0	0
$913$	42822.6	1.55227
$914$	0	0
$915$	12226.3i	0.441735i
$916$	0	0
$917$	45188.9i	1.62734i
$918$	0	0
$919$	37080.1	1.33097	0.665484	−	0.746412i	$-0.268225\pi$
$919$	37080.1	1.33097	0.665484	+	0.746412i	$0.268225\pi$
$920$	0	0
$921$	−8996.46	−0.321871
$922$	0	0
$923$	− 65797.3i	− 2.34642i
$924$	0	0
$925$	− 17109.4i	− 0.608167i
$926$	0	0
$927$	−1454.06	−0.0515183
$928$	0	0
$929$	38259.5	1.35119	0.675595	−	0.737273i	$-0.263887\pi$
$929$	38259.5	1.35119	0.675595	+	0.737273i	$0.263887\pi$
$930$	0	0
$931$	42026.7i	1.47945i
$932$	0	0
$933$	7211.92i	0.253063i
$934$	0	0
$935$	−19243.8	−0.673091
$936$	0	0
$937$	4413.55	0.153879	0.0769394	−	0.997036i	$-0.475485\pi$
$937$	4413.55	0.153879	0.0769394	+	0.997036i	$0.475485\pi$
$938$	0	0
$939$	− 17537.3i	− 0.609486i
$940$	0	0
$941$	− 31511.2i	− 1.09164i	−0.837902	−	0.545821i	$-0.816218\pi$
$941$	− 31511.2i	− 1.09164i	0.837902	−	0.545821i	$-0.183782\pi$
$942$	0	0
$943$	30962.8	1.06923
$944$	0	0
$945$	5072.82	0.174623
$946$	0	0
$947$	− 23561.3i	− 0.808490i	−0.914651	−	0.404245i	$-0.867534\pi$
$947$	− 23561.3i	− 0.808490i	0.914651	−	0.404245i	$-0.132466\pi$
$948$	0	0
$949$	− 24759.6i	− 0.846925i
$950$	0	0
$951$	26752.1	0.912195
$952$	0	0
$953$	35039.7	1.19103	0.595513	−	0.803346i	$-0.296949\pi$
$953$	35039.7	1.19103	0.595513	+	0.803346i	$0.296949\pi$
$954$	0	0
$955$	− 199.107i	− 0.00674653i
$956$	0	0
$957$	15317.6i	0.517395i
$958$	0	0
$959$	40473.0	1.36282
$960$	0	0
$961$	−19544.9	−0.656066
$962$	0	0
$963$	− 4335.53i	− 0.145078i
$964$	0	0
$965$	3496.76i	0.116647i
$966$	0	0
$967$	−7975.38	−0.265223	−0.132612	−	0.991168i	$-0.542336\pi$
$967$	−7975.38	−0.265223	−0.132612	+	0.991168i	$0.542336\pi$
$968$	0	0
$969$	16409.6	0.544018
$970$	0	0
$971$	21386.8i	0.706834i	0.935466	+	0.353417i	$0.114980\pi$
$971$	21386.8i	0.706834i	−0.935466	+	0.353417i	$0.885020\pi$
$972$	0	0
$973$	− 72030.7i	− 2.37328i
$974$	0	0
$975$	−22887.4	−0.751779
$976$	0	0
$977$	−40142.3	−1.31450	−0.657250	−	0.753673i	$-0.728280\pi$
$977$	−40142.3	−1.31450	−0.657250	+	0.753673i	$0.728280\pi$
$978$	0	0
$979$	45891.5i	1.49816i
$980$	0	0
$981$	2579.73i	0.0839597i
$982$	0	0
$983$	9205.44	0.298686	0.149343	−	0.988785i	$-0.452284\pi$
$983$	9205.44	0.298686	0.149343	+	0.988785i	$0.452284\pi$
$984$	0	0
$985$	−3347.75	−0.108292
$986$	0	0
$987$	− 33238.6i	− 1.07193i
$988$	0	0
$989$	26068.3i	0.838144i
$990$	0	0
$991$	−22082.2	−0.707834	−0.353917	−	0.935277i	$-0.615150\pi$
$991$	−22082.2	−0.707834	−0.353917	+	0.935277i	$0.615150\pi$
$992$	0	0
$993$	−27033.9	−0.863942
$994$	0	0
$995$	− 27331.1i	− 0.870808i
$996$	0	0
$997$	− 7207.66i	− 0.228956i	−0.993426	−	0.114478i	$-0.963480\pi$
$997$	− 7207.66i	− 0.228956i	0.993426	−	0.114478i	$-0.0365195\pi$
$998$	0	0
$999$	4979.87	0.157714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.d.f.193.3	yes	8
3.2	odd	2		1152.4.d.p.577.3		8
4.3	odd	2	inner	384.4.d.f.193.7	yes	8
8.3	odd	2	inner	384.4.d.f.193.2	✓	8
8.5	even	2	inner	384.4.d.f.193.6	yes	8
12.11	even	2		1152.4.d.p.577.4		8
16.3	odd	4		768.4.a.u.1.3		4
16.5	even	4		768.4.a.u.1.2		4
16.11	odd	4		768.4.a.v.1.2		4
16.13	even	4		768.4.a.v.1.3		4
24.5	odd	2		1152.4.d.p.577.5		8
24.11	even	2		1152.4.d.p.577.6		8
48.5	odd	4		2304.4.a.cb.1.3		4
48.11	even	4		2304.4.a.by.1.3		4
48.29	odd	4		2304.4.a.by.1.2		4
48.35	even	4		2304.4.a.cb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.f.193.2	✓	8	8.3	odd	2	inner
384.4.d.f.193.3	yes	8	1.1	even	1	trivial
384.4.d.f.193.6	yes	8	8.5	even	2	inner
384.4.d.f.193.7	yes	8	4.3	odd	2	inner
768.4.a.u.1.2		4	16.5	even	4
768.4.a.u.1.3		4	16.3	odd	4
768.4.a.v.1.2		4	16.11	odd	4
768.4.a.v.1.3		4	16.13	even	4
1152.4.d.p.577.3		8	3.2	odd	2
1152.4.d.p.577.4		8	12.11	even	2
1152.4.d.p.577.5		8	24.5	odd	2
1152.4.d.p.577.6		8	24.11	even	2
2304.4.a.by.1.2		4	48.29	odd	4
2304.4.a.by.1.3		4	48.11	even	4
2304.4.a.cb.1.2		4	48.35	even	4
2304.4.a.cb.1.3		4	48.5	odd	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +5.67763i q^{5} -33.0917 q^{7} -9.00000 q^{9} +34.6274i q^{11} -82.2421i q^{13} +17.0329 q^{15} +97.8823 q^{17} +55.8823i q^{19} +99.2750i q^{21} +130.418 q^{23} +92.7645 q^{25} +27.0000i q^{27} +147.451i q^{29} +101.223 q^{31} +103.882 q^{33} -187.882i q^{35} -184.439i q^{37} -246.726 q^{39} +237.411 q^{41} +199.882i q^{43} -51.0987i q^{45} -334.813 q^{47} +752.058 q^{49} -293.647i q^{51} -102.030i q^{53} -196.602 q^{55} +167.647 q^{57} +105.961i q^{59} +717.803i q^{61} +297.825 q^{63} +466.940 q^{65} -316.471i q^{67} -391.255i q^{69} +800.045 q^{71} +301.058 q^{73} -278.294i q^{75} -1145.88i q^{77} -42.8329 q^{79} +81.0000 q^{81} -1236.67i q^{83} +555.739i q^{85} +442.354 q^{87} +1325.29 q^{89} +2721.53i q^{91} -303.670i q^{93} -317.279 q^{95} -505.765 q^{97} -311.647i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) \) 8 * q - 72 * q^9 + 240 * q^17 - 344 * q^25 + 288 * q^33 - 816 * q^41 + 1672 * q^49 - 288 * q^57 - 1152 * q^65 - 1936 * q^73 + 648 * q^81 + 7344 * q^89 - 2960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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