LMFDB - Embedded newform 6128.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6128,2,Mod(1,6128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6128, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6128 = 2^{4} \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6128.a (trivial)

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:	yes
Analytic conductor:	$48.9323263586$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 383)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.08654 q^{3} -0.910566 q^{5} -2.08417 q^{7} +6.52674 q^{9} -1.45014 q^{11} -5.62225 q^{13} +2.81050 q^{15} +2.35320 q^{17} +1.05653 q^{19} +6.43287 q^{21} +5.63172 q^{23} -4.17087 q^{25} -10.8854 q^{27} +4.34646 q^{29} -8.73676 q^{31} +4.47592 q^{33} +1.89777 q^{35} -4.46866 q^{37} +17.3533 q^{39} -10.5553 q^{41} -4.24845 q^{43} -5.94303 q^{45} +8.72689 q^{47} -2.65624 q^{49} -7.26325 q^{51} +1.87073 q^{53} +1.32045 q^{55} -3.26102 q^{57} -4.30187 q^{59} -6.48215 q^{61} -13.6028 q^{63} +5.11943 q^{65} -14.6140 q^{67} -17.3825 q^{69} +5.12074 q^{71} +7.67008 q^{73} +12.8736 q^{75} +3.02234 q^{77} -11.7685 q^{79} +14.0181 q^{81} -7.99857 q^{83} -2.14274 q^{85} -13.4155 q^{87} -2.63315 q^{89} +11.7177 q^{91} +26.9664 q^{93} -0.962038 q^{95} +11.6053 q^{97} -9.46470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{3} + 3 q^{5} - 17 q^{7} + 34 q^{9} + 28 q^{13} + 2 q^{15} + 16 q^{17} - 13 q^{19} - 4 q^{21} - 7 q^{23} + 67 q^{25} + 10 q^{27} - 2 q^{29} + 24 q^{33} + 10 q^{35} + 35 q^{37} + 12 q^{39} - 34 q^{43}+ \cdots + 54 q^{99}+O(q^{100}) $ 24 * q - 2 * q^3 + 3 * q^5 - 17 * q^7 + 34 * q^9 + 28 * q^13 + 2 * q^15 + 16 * q^17 - 13 * q^19 - 4 * q^21 - 7 * q^23 + 67 * q^25 + 10 * q^27 - 2 * q^29 + 24 * q^33 + 10 * q^35 + 35 * q^37 + 12 * q^39 - 34 * q^43 - 27 * q^45 + 3 * q^47 + 49 * q^49 + 28 * q^51 - 9 * q^53 - 10 * q^55 + 47 * q^57 + 6 * q^59 + 3 * q^61 - 13 * q^63 + 8 * q^65 - 42 * q^67 - 30 * q^69 + 13 * q^71 + 83 * q^73 + 65 * q^75 - 21 * q^77 + 3 * q^79 + 28 * q^81 + 23 * q^83 + 52 * q^85 + 5 * q^87 - 38 * q^89 + 17 * q^91 + 17 * q^93 + 66 * q^95 + 63 * q^97 + 54 * 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
$3$	−3.08654	−1.78202	−0.891008	−	0.453988i	$-0.850001\pi$
$3$	−3.08654	−1.78202	−0.891008	+	0.453988i	$0.850001\pi$
$4$	0	0
$5$	−0.910566	−0.407218	−0.203609	−	0.979052i	$-0.565267\pi$
$5$	−0.910566	−0.407218	−0.203609	+	0.979052i	$0.565267\pi$
$6$	0	0
$7$	−2.08417	−0.787741	−0.393871	−	0.919166i	$-0.628864\pi$
$7$	−2.08417	−0.787741	−0.393871	+	0.919166i	$0.628864\pi$
$8$	0	0
$9$	6.52674	2.17558
$10$	0	0
$11$	−1.45014	−0.437234	−0.218617	−	0.975811i	$-0.570155\pi$
$11$	−1.45014	−0.437234	−0.218617	+	0.975811i	$0.570155\pi$
$12$	0	0
$13$	−5.62225	−1.55933	−0.779665	−	0.626197i	$-0.784611\pi$
$13$	−5.62225	−1.55933	−0.779665	+	0.626197i	$0.784611\pi$
$14$	0	0
$15$	2.81050	0.725668
$16$	0	0
$17$	2.35320	0.570735	0.285367	−	0.958418i	$-0.407884\pi$
$17$	2.35320	0.570735	0.285367	+	0.958418i	$0.407884\pi$
$18$	0	0
$19$	1.05653	0.242384	0.121192	−	0.992629i	$-0.461328\pi$
$19$	1.05653	0.242384	0.121192	+	0.992629i	$0.461328\pi$
$20$	0	0
$21$	6.43287	1.40377
$22$	0	0
$23$	5.63172	1.17430	0.587148	−	0.809480i	$-0.300251\pi$
$23$	5.63172	1.17430	0.587148	+	0.809480i	$0.300251\pi$
$24$	0	0
$25$	−4.17087	−0.834174
$26$	0	0
$27$	−10.8854	−2.09490
$28$	0	0
$29$	4.34646	0.807117	0.403559	−	0.914954i	$-0.367773\pi$
$29$	4.34646	0.807117	0.403559	+	0.914954i	$0.367773\pi$
$30$	0	0
$31$	−8.73676	−1.56917	−0.784584	−	0.620022i	$-0.787124\pi$
$31$	−8.73676	−1.56917	−0.784584	+	0.620022i	$0.787124\pi$
$32$	0	0
$33$	4.47592	0.779158
$34$	0	0
$35$	1.89777	0.320782
$36$	0	0
$37$	−4.46866	−0.734643	−0.367321	−	0.930094i	$-0.619725\pi$
$37$	−4.46866	−0.734643	−0.367321	+	0.930094i	$0.619725\pi$
$38$	0	0
$39$	17.3533	2.77875
$40$	0	0
$41$	−10.5553	−1.64847	−0.824233	−	0.566251i	$-0.808393\pi$
$41$	−10.5553	−1.64847	−0.824233	+	0.566251i	$0.808393\pi$
$42$	0	0
$43$	−4.24845	−0.647883	−0.323942	−	0.946077i	$-0.605008\pi$
$43$	−4.24845	−0.647883	−0.323942	+	0.946077i	$0.605008\pi$
$44$	0	0
$45$	−5.94303	−0.885934
$46$	0	0
$47$	8.72689	1.27295	0.636474	−	0.771298i	$-0.280392\pi$
$47$	8.72689	1.27295	0.636474	+	0.771298i	$0.280392\pi$
$48$	0	0
$49$	−2.65624	−0.379463
$50$	0	0
$51$	−7.26325	−1.01706
$52$	0	0
$53$	1.87073	0.256965	0.128483	−	0.991712i	$-0.458989\pi$
$53$	1.87073	0.256965	0.128483	+	0.991712i	$0.458989\pi$
$54$	0	0
$55$	1.32045	0.178050
$56$	0	0
$57$	−3.26102	−0.431932
$58$	0	0
$59$	−4.30187	−0.560056	−0.280028	−	0.959992i	$-0.590344\pi$
$59$	−4.30187	−0.560056	−0.280028	+	0.959992i	$0.590344\pi$
$60$	0	0
$61$	−6.48215	−0.829954	−0.414977	−	0.909832i	$-0.636210\pi$
$61$	−6.48215	−0.829954	−0.414977	+	0.909832i	$0.636210\pi$
$62$	0	0
$63$	−13.6028	−1.71379
$64$	0	0
$65$	5.11943	0.634987
$66$	0	0
$67$	−14.6140	−1.78539	−0.892693	−	0.450665i	$-0.851187\pi$
$67$	−14.6140	−1.78539	−0.892693	+	0.450665i	$0.851187\pi$
$68$	0	0
$69$	−17.3825	−2.09261
$70$	0	0
$71$	5.12074	0.607720	0.303860	−	0.952717i	$-0.401724\pi$
$71$	5.12074	0.607720	0.303860	+	0.952717i	$0.401724\pi$
$72$	0	0
$73$	7.67008	0.897715	0.448858	−	0.893603i	$-0.351831\pi$
$73$	7.67008	0.897715	0.448858	+	0.893603i	$0.351831\pi$
$74$	0	0
$75$	12.8736	1.48651
$76$	0	0
$77$	3.02234	0.344428
$78$	0	0
$79$	−11.7685	−1.32406	−0.662029	−	0.749479i	$-0.730304\pi$
$79$	−11.7685	−1.32406	−0.662029	+	0.749479i	$0.730304\pi$
$80$	0	0
$81$	14.0181	1.55757
$82$	0	0
$83$	−7.99857	−0.877957	−0.438979	−	0.898497i	$-0.644660\pi$
$83$	−7.99857	−0.877957	−0.438979	+	0.898497i	$0.644660\pi$
$84$	0	0
$85$	−2.14274	−0.232413
$86$	0	0
$87$	−13.4155	−1.43830
$88$	0	0
$89$	−2.63315	−0.279113	−0.139557	−	0.990214i	$-0.544568\pi$
$89$	−2.63315	−0.279113	−0.139557	+	0.990214i	$0.544568\pi$
$90$	0	0
$91$	11.7177	1.22835
$92$	0	0
$93$	26.9664	2.79628
$94$	0	0
$95$	−0.962038	−0.0987030
$96$	0	0
$97$	11.6053	1.17834	0.589169	−	0.808010i	$-0.299455\pi$
$97$	11.6053	1.17834	0.589169	+	0.808010i	$0.299455\pi$
$98$	0	0
$99$	−9.46470	−0.951238
$100$	0	0
$101$	−7.93506	−0.789568	−0.394784	−	0.918774i	$-0.629181\pi$
$101$	−7.93506	−0.789568	−0.394784	+	0.918774i	$0.629181\pi$
$102$	0	0
$103$	6.37732	0.628376	0.314188	−	0.949361i	$-0.398268\pi$
$103$	6.37732	0.628376	0.314188	+	0.949361i	$0.398268\pi$
$104$	0	0
$105$	−5.85756	−0.571639
$106$	0	0
$107$	0.919710	0.0889117	0.0444559	−	0.999011i	$-0.485845\pi$
$107$	0.919710	0.0889117	0.0444559	+	0.999011i	$0.485845\pi$
$108$	0	0
$109$	−16.3419	−1.56527	−0.782634	−	0.622482i	$-0.786124\pi$
$109$	−16.3419	−1.56527	−0.782634	+	0.622482i	$0.786124\pi$
$110$	0	0
$111$	13.7927	1.30914
$112$	0	0
$113$	10.2230	0.961698	0.480849	−	0.876803i	$-0.340329\pi$
$113$	10.2230	0.961698	0.480849	+	0.876803i	$0.340329\pi$
$114$	0	0
$115$	−5.12806	−0.478194
$116$	0	0
$117$	−36.6949	−3.39245
$118$	0	0
$119$	−4.90446	−0.449591
$120$	0	0
$121$	−8.89709	−0.808826
$122$	0	0
$123$	32.5795	2.93759
$124$	0	0
$125$	8.35068	0.746908
$126$	0	0
$127$	−6.20139	−0.550284	−0.275142	−	0.961404i	$-0.588725\pi$
$127$	−6.20139	−0.550284	−0.275142	+	0.961404i	$0.588725\pi$
$128$	0	0
$129$	13.1130	1.15454
$130$	0	0
$131$	3.80228	0.332207	0.166104	−	0.986108i	$-0.446881\pi$
$131$	3.80228	0.332207	0.166104	+	0.986108i	$0.446881\pi$
$132$	0	0
$133$	−2.20198	−0.190936
$134$	0	0
$135$	9.91190	0.853081
$136$	0	0
$137$	−7.18556	−0.613904	−0.306952	−	0.951725i	$-0.599309\pi$
$137$	−7.18556	−0.613904	−0.306952	+	0.951725i	$0.599309\pi$
$138$	0	0
$139$	−22.5012	−1.90853	−0.954264	−	0.298964i	$-0.903359\pi$
$139$	−22.5012	−1.90853	−0.954264	+	0.298964i	$0.903359\pi$
$140$	0	0
$141$	−26.9359	−2.26841
$142$	0	0
$143$	8.15305	0.681793
$144$	0	0
$145$	−3.95774	−0.328672
$146$	0	0
$147$	8.19861	0.676210
$148$	0	0
$149$	10.1145	0.828608	0.414304	−	0.910139i	$-0.364025\pi$
$149$	10.1145	0.828608	0.414304	+	0.910139i	$0.364025\pi$
$150$	0	0
$151$	−19.4558	−1.58329	−0.791647	−	0.610979i	$-0.790776\pi$
$151$	−19.4558	−1.58329	−0.791647	+	0.610979i	$0.790776\pi$
$152$	0	0
$153$	15.3587	1.24168
$154$	0	0
$155$	7.95540	0.638993
$156$	0	0
$157$	4.94110	0.394343	0.197171	−	0.980369i	$-0.436824\pi$
$157$	4.94110	0.394343	0.197171	+	0.980369i	$0.436824\pi$
$158$	0	0
$159$	−5.77410	−0.457916
$160$	0	0
$161$	−11.7375	−0.925041
$162$	0	0
$163$	14.3804	1.12636	0.563182	−	0.826333i	$-0.309577\pi$
$163$	14.3804	1.12636	0.563182	+	0.826333i	$0.309577\pi$
$164$	0	0
$165$	−4.07563	−0.317287
$166$	0	0
$167$	−22.3066	−1.72613	−0.863067	−	0.505090i	$-0.831459\pi$
$167$	−22.3066	−1.72613	−0.863067	+	0.505090i	$0.831459\pi$
$168$	0	0
$169$	18.6096	1.43151
$170$	0	0
$171$	6.89568	0.527326
$172$	0	0
$173$	−13.2300	−1.00586	−0.502930	−	0.864327i	$-0.667745\pi$
$173$	−13.2300	−1.00586	−0.502930	+	0.864327i	$0.667745\pi$
$174$	0	0
$175$	8.69279	0.657113
$176$	0	0
$177$	13.2779	0.998029
$178$	0	0
$179$	−6.68349	−0.499548	−0.249774	−	0.968304i	$-0.580356\pi$
$179$	−6.68349	−0.499548	−0.249774	+	0.968304i	$0.580356\pi$
$180$	0	0
$181$	−9.78911	−0.727619	−0.363809	−	0.931473i	$-0.618524\pi$
$181$	−9.78911	−0.727619	−0.363809	+	0.931473i	$0.618524\pi$
$182$	0	0
$183$	20.0074	1.47899
$184$	0	0
$185$	4.06901	0.299159
$186$	0	0
$187$	−3.41247	−0.249545
$188$	0	0
$189$	22.6871	1.65024
$190$	0	0
$191$	21.6986	1.57005	0.785027	−	0.619462i	$-0.212649\pi$
$191$	21.6986	1.57005	0.785027	+	0.619462i	$0.212649\pi$
$192$	0	0
$193$	11.3873	0.819674	0.409837	−	0.912159i	$-0.365586\pi$
$193$	11.3873	0.819674	0.409837	+	0.912159i	$0.365586\pi$
$194$	0	0
$195$	−15.8013	−1.13156
$196$	0	0
$197$	−5.15003	−0.366924	−0.183462	−	0.983027i	$-0.558730\pi$
$197$	−5.15003	−0.366924	−0.183462	+	0.983027i	$0.558730\pi$
$198$	0	0
$199$	−15.9165	−1.12829	−0.564145	−	0.825676i	$-0.690794\pi$
$199$	−15.9165	−1.12829	−0.564145	+	0.825676i	$0.690794\pi$
$200$	0	0
$201$	45.1068	3.18159
$202$	0	0
$203$	−9.05875	−0.635800
$204$	0	0
$205$	9.61133	0.671284
$206$	0	0
$207$	36.7568	2.55477
$208$	0	0
$209$	−1.53211	−0.105979
$210$	0	0
$211$	4.51738	0.310989	0.155494	−	0.987837i	$-0.450303\pi$
$211$	4.51738	0.310989	0.155494	+	0.987837i	$0.450303\pi$
$212$	0	0
$213$	−15.8054	−1.08297
$214$	0	0
$215$	3.86850	0.263829
$216$	0	0
$217$	18.2089	1.23610
$218$	0	0
$219$	−23.6740	−1.59974
$220$	0	0
$221$	−13.2303	−0.889964
$222$	0	0
$223$	−13.5001	−0.904036	−0.452018	−	0.892009i	$-0.649296\pi$
$223$	−13.5001	−0.904036	−0.452018	+	0.892009i	$0.649296\pi$
$224$	0	0
$225$	−27.2222	−1.81481
$226$	0	0
$227$	18.5652	1.23222	0.616109	−	0.787661i	$-0.288708\pi$
$227$	18.5652	1.23222	0.616109	+	0.787661i	$0.288708\pi$
$228$	0	0
$229$	3.86141	0.255169	0.127584	−	0.991828i	$-0.459278\pi$
$229$	3.86141	0.255169	0.127584	+	0.991828i	$0.459278\pi$
$230$	0	0
$231$	−9.32858	−0.613775
$232$	0	0
$233$	7.30219	0.478382	0.239191	−	0.970973i	$-0.423118\pi$
$233$	7.30219	0.478382	0.239191	+	0.970973i	$0.423118\pi$
$234$	0	0
$235$	−7.94641	−0.518367
$236$	0	0
$237$	36.3239	2.35949
$238$	0	0
$239$	7.87705	0.509524	0.254762	−	0.967004i	$-0.418003\pi$
$239$	7.87705	0.509524	0.254762	+	0.967004i	$0.418003\pi$
$240$	0	0
$241$	29.1082	1.87502	0.937511	−	0.347955i	$-0.113124\pi$
$241$	29.1082	1.87502	0.937511	+	0.347955i	$0.113124\pi$
$242$	0	0
$243$	−10.6112	−0.680707
$244$	0	0
$245$	2.41869	0.154524
$246$	0	0
$247$	−5.94006	−0.377957
$248$	0	0
$249$	24.6879	1.56453
$250$	0	0
$251$	16.9063	1.06712	0.533559	−	0.845763i	$-0.320854\pi$
$251$	16.9063	1.06712	0.533559	+	0.845763i	$0.320854\pi$
$252$	0	0
$253$	−8.16680	−0.513442
$254$	0	0
$255$	6.61367	0.414164
$256$	0	0
$257$	−11.7786	−0.734726	−0.367363	−	0.930078i	$-0.619739\pi$
$257$	−11.7786	−0.734726	−0.367363	+	0.930078i	$0.619739\pi$
$258$	0	0
$259$	9.31343	0.578709
$260$	0	0
$261$	28.3682	1.75595
$262$	0	0
$263$	17.2051	1.06091	0.530456	−	0.847713i	$-0.322021\pi$
$263$	17.2051	1.06091	0.530456	+	0.847713i	$0.322021\pi$
$264$	0	0
$265$	−1.70343	−0.104641
$266$	0	0
$267$	8.12732	0.497384
$268$	0	0
$269$	−12.1597	−0.741390	−0.370695	−	0.928755i	$-0.620880\pi$
$269$	−12.1597	−0.741390	−0.370695	+	0.928755i	$0.620880\pi$
$270$	0	0
$271$	−0.144543	−0.00878034	−0.00439017	−	0.999990i	$-0.501397\pi$
$271$	−0.144543	−0.00878034	−0.00439017	+	0.999990i	$0.501397\pi$
$272$	0	0
$273$	−36.1672	−2.18894
$274$	0	0
$275$	6.04835	0.364729
$276$	0	0
$277$	4.61563	0.277327	0.138663	−	0.990340i	$-0.455719\pi$
$277$	4.61563	0.277327	0.138663	+	0.990340i	$0.455719\pi$
$278$	0	0
$279$	−57.0225	−3.41385
$280$	0	0
$281$	2.37703	0.141801	0.0709007	−	0.997483i	$-0.477413\pi$
$281$	2.37703	0.141801	0.0709007	+	0.997483i	$0.477413\pi$
$282$	0	0
$283$	−13.4835	−0.801513	−0.400756	−	0.916185i	$-0.631253\pi$
$283$	−13.4835	−0.801513	−0.400756	+	0.916185i	$0.631253\pi$
$284$	0	0
$285$	2.96937	0.175890
$286$	0	0
$287$	21.9991	1.29856
$288$	0	0
$289$	−11.4624	−0.674262
$290$	0	0
$291$	−35.8202	−2.09982
$292$	0	0
$293$	−29.2780	−1.71044	−0.855219	−	0.518268i	$-0.826577\pi$
$293$	−29.2780	−1.71044	−0.855219	+	0.518268i	$0.826577\pi$
$294$	0	0
$295$	3.91714	0.228065
$296$	0	0
$297$	15.7854	0.915963
$298$	0	0
$299$	−31.6629	−1.83111
$300$	0	0
$301$	8.85449	0.510364
$302$	0	0
$303$	24.4919	1.40702
$304$	0	0
$305$	5.90243	0.337972
$306$	0	0
$307$	27.2949	1.55780	0.778901	−	0.627147i	$-0.215778\pi$
$307$	27.2949	1.55780	0.778901	+	0.627147i	$0.215778\pi$
$308$	0	0
$309$	−19.6839	−1.11978
$310$	0	0
$311$	−14.7646	−0.837223	−0.418612	−	0.908165i	$-0.637483\pi$
$311$	−14.7646	−0.837223	−0.418612	+	0.908165i	$0.637483\pi$
$312$	0	0
$313$	4.02085	0.227272	0.113636	−	0.993522i	$-0.463750\pi$
$313$	4.02085	0.227272	0.113636	+	0.993522i	$0.463750\pi$
$314$	0	0
$315$	12.3863	0.697887
$316$	0	0
$317$	30.2041	1.69643	0.848214	−	0.529653i	$-0.177678\pi$
$317$	30.2041	1.69643	0.848214	+	0.529653i	$0.177678\pi$
$318$	0	0
$319$	−6.30299	−0.352899
$320$	0	0
$321$	−2.83872	−0.158442
$322$	0	0
$323$	2.48622	0.138337
$324$	0	0
$325$	23.4496	1.30075
$326$	0	0
$327$	50.4399	2.78933
$328$	0	0
$329$	−18.1883	−1.00275
$330$	0	0
$331$	18.7445	1.03029	0.515145	−	0.857103i	$-0.327738\pi$
$331$	18.7445	1.03029	0.515145	+	0.857103i	$0.327738\pi$
$332$	0	0
$333$	−29.1658	−1.59827
$334$	0	0
$335$	13.3070	0.727041
$336$	0	0
$337$	23.4653	1.27824	0.639119	−	0.769108i	$-0.279299\pi$
$337$	23.4653	1.27824	0.639119	+	0.769108i	$0.279299\pi$
$338$	0	0
$339$	−31.5537	−1.71376
$340$	0	0
$341$	12.6695	0.686094
$342$	0	0
$343$	20.1252	1.08666
$344$	0	0
$345$	15.8280	0.852149
$346$	0	0
$347$	−8.14300	−0.437139	−0.218570	−	0.975821i	$-0.570139\pi$
$347$	−8.14300	−0.437139	−0.218570	+	0.975821i	$0.570139\pi$
$348$	0	0
$349$	−25.3501	−1.35696	−0.678480	−	0.734619i	$-0.737361\pi$
$349$	−25.3501	−1.35696	−0.678480	+	0.734619i	$0.737361\pi$
$350$	0	0
$351$	61.2005	3.26664
$352$	0	0
$353$	24.1384	1.28476	0.642378	−	0.766388i	$-0.277948\pi$
$353$	24.1384	1.28476	0.642378	+	0.766388i	$0.277948\pi$
$354$	0	0
$355$	−4.66277	−0.247474
$356$	0	0
$357$	15.1378	0.801179
$358$	0	0
$359$	36.0517	1.90274	0.951368	−	0.308055i	$-0.0996782\pi$
$359$	36.0517	1.90274	0.951368	+	0.308055i	$0.0996782\pi$
$360$	0	0
$361$	−17.8837	−0.941250
$362$	0	0
$363$	27.4612	1.44134
$364$	0	0
$365$	−6.98412	−0.365565
$366$	0	0
$367$	14.3193	0.747461	0.373730	−	0.927537i	$-0.378079\pi$
$367$	14.3193	0.747461	0.373730	+	0.927537i	$0.378079\pi$
$368$	0	0
$369$	−68.8919	−3.58637
$370$	0	0
$371$	−3.89892	−0.202422
$372$	0	0
$373$	29.7969	1.54283	0.771413	−	0.636335i	$-0.219550\pi$
$373$	29.7969	1.54283	0.771413	+	0.636335i	$0.219550\pi$
$374$	0	0
$375$	−25.7747	−1.33100
$376$	0	0
$377$	−24.4369	−1.25856
$378$	0	0
$379$	18.1874	0.934226	0.467113	−	0.884198i	$-0.345294\pi$
$379$	18.1874	0.934226	0.467113	+	0.884198i	$0.345294\pi$
$380$	0	0
$381$	19.1409	0.980615
$382$	0	0
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	−2.75204	−0.140257
$386$	0	0
$387$	−27.7285	−1.40952
$388$	0	0
$389$	−12.4224	−0.629840	−0.314920	−	0.949118i	$-0.601978\pi$
$389$	−12.4224	−0.629840	−0.314920	+	0.949118i	$0.601978\pi$
$390$	0	0
$391$	13.2526	0.670211
$392$	0	0
$393$	−11.7359	−0.591998
$394$	0	0
$395$	10.7160	0.539179
$396$	0	0
$397$	−16.8357	−0.844958	−0.422479	−	0.906373i	$-0.638840\pi$
$397$	−16.8357	−0.844958	−0.422479	+	0.906373i	$0.638840\pi$
$398$	0	0
$399$	6.79650	0.340251
$400$	0	0
$401$	−5.37256	−0.268293	−0.134146	−	0.990962i	$-0.542829\pi$
$401$	−5.37256	−0.268293	−0.134146	+	0.990962i	$0.542829\pi$
$402$	0	0
$403$	49.1202	2.44685
$404$	0	0
$405$	−12.7644	−0.634269
$406$	0	0
$407$	6.48019	0.321211
$408$	0	0
$409$	23.8792	1.18075	0.590375	−	0.807129i	$-0.298980\pi$
$409$	23.8792	1.18075	0.590375	+	0.807129i	$0.298980\pi$
$410$	0	0
$411$	22.1785	1.09399
$412$	0	0
$413$	8.96582	0.441179
$414$	0	0
$415$	7.28323	0.357520
$416$	0	0
$417$	69.4509	3.40103
$418$	0	0
$419$	8.42586	0.411630	0.205815	−	0.978591i	$-0.434015\pi$
$419$	8.42586	0.411630	0.205815	+	0.978591i	$0.434015\pi$
$420$	0	0
$421$	23.9189	1.16573	0.582867	−	0.812567i	$-0.301931\pi$
$421$	23.9189	1.16573	0.582867	+	0.812567i	$0.301931\pi$
$422$	0	0
$423$	56.9581	2.76940
$424$	0	0
$425$	−9.81489	−0.476092
$426$	0	0
$427$	13.5099	0.653789
$428$	0	0
$429$	−25.1647	−1.21497
$430$	0	0
$431$	−24.6046	−1.18516	−0.592580	−	0.805511i	$-0.701891\pi$
$431$	−24.6046	−1.18516	−0.592580	+	0.805511i	$0.701891\pi$
$432$	0	0
$433$	−21.8659	−1.05081	−0.525404	−	0.850853i	$-0.676086\pi$
$433$	−21.8659	−1.05081	−0.525404	+	0.850853i	$0.676086\pi$
$434$	0	0
$435$	12.2157	0.585699
$436$	0	0
$437$	5.95007	0.284630
$438$	0	0
$439$	2.76635	0.132031	0.0660154	−	0.997819i	$-0.478971\pi$
$439$	2.76635	0.132031	0.0660154	+	0.997819i	$0.478971\pi$
$440$	0	0
$441$	−17.3366	−0.825553
$442$	0	0
$443$	15.7990	0.750633	0.375316	−	0.926897i	$-0.377534\pi$
$443$	15.7990	0.750633	0.375316	+	0.926897i	$0.377534\pi$
$444$	0	0
$445$	2.39766	0.113660
$446$	0	0
$447$	−31.2187	−1.47659
$448$	0	0
$449$	8.09725	0.382133	0.191066	−	0.981577i	$-0.438805\pi$
$449$	8.09725	0.382133	0.191066	+	0.981577i	$0.438805\pi$
$450$	0	0
$451$	15.3067	0.720766
$452$	0	0
$453$	60.0512	2.82145
$454$	0	0
$455$	−10.6697	−0.500205
$456$	0	0
$457$	16.4155	0.767885	0.383943	−	0.923357i	$-0.374566\pi$
$457$	16.4155	0.767885	0.383943	+	0.923357i	$0.374566\pi$
$458$	0	0
$459$	−25.6156	−1.19563
$460$	0	0
$461$	−11.4672	−0.534082	−0.267041	−	0.963685i	$-0.586046\pi$
$461$	−11.4672	−0.534082	−0.267041	+	0.963685i	$0.586046\pi$
$462$	0	0
$463$	−11.8489	−0.550666	−0.275333	−	0.961349i	$-0.588788\pi$
$463$	−11.8489	−0.550666	−0.275333	+	0.961349i	$0.588788\pi$
$464$	0	0
$465$	−24.5547	−1.13870
$466$	0	0
$467$	21.6852	1.00347	0.501735	−	0.865021i	$-0.332695\pi$
$467$	21.6852	1.00347	0.501735	+	0.865021i	$0.332695\pi$
$468$	0	0
$469$	30.4581	1.40642
$470$	0	0
$471$	−15.2509	−0.702725
$472$	0	0
$473$	6.16086	0.283277
$474$	0	0
$475$	−4.40664	−0.202190
$476$	0	0
$477$	12.2098	0.559048
$478$	0	0
$479$	38.7480	1.77044	0.885220	−	0.465173i	$-0.154008\pi$
$479$	38.7480	1.77044	0.885220	+	0.465173i	$0.154008\pi$
$480$	0	0
$481$	25.1239	1.14555
$482$	0	0
$483$	36.2281	1.64844
$484$	0	0
$485$	−10.5674	−0.479840
$486$	0	0
$487$	−12.6258	−0.572129	−0.286065	−	0.958210i	$-0.592347\pi$
$487$	−12.6258	−0.572129	−0.286065	+	0.958210i	$0.592347\pi$
$488$	0	0
$489$	−44.3859	−2.00720
$490$	0	0
$491$	17.3101	0.781193	0.390597	−	0.920562i	$-0.372269\pi$
$491$	17.3101	0.781193	0.390597	+	0.920562i	$0.372269\pi$
$492$	0	0
$493$	10.2281	0.460650
$494$	0	0
$495$	8.61824	0.387361
$496$	0	0
$497$	−10.6725	−0.478726
$498$	0	0
$499$	27.8854	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$499$	27.8854	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$500$	0	0
$501$	68.8501	3.07600
$502$	0	0
$503$	12.9536	0.577572	0.288786	−	0.957394i	$-0.406748\pi$
$503$	12.9536	0.577572	0.288786	+	0.957394i	$0.406748\pi$
$504$	0	0
$505$	7.22540	0.321526
$506$	0	0
$507$	−57.4394	−2.55098
$508$	0	0
$509$	−42.3593	−1.87754	−0.938772	−	0.344540i	$-0.888035\pi$
$509$	−42.3593	−1.87754	−0.938772	+	0.344540i	$0.888035\pi$
$510$	0	0
$511$	−15.9857	−0.707167
$512$	0	0
$513$	−11.5008	−0.507771
$514$	0	0
$515$	−5.80697	−0.255886
$516$	0	0
$517$	−12.6552	−0.556577
$518$	0	0
$519$	40.8351	1.79246
$520$	0	0
$521$	−24.9719	−1.09404	−0.547019	−	0.837120i	$-0.684238\pi$
$521$	−24.9719	−1.09404	−0.547019	+	0.837120i	$0.684238\pi$
$522$	0	0
$523$	39.9277	1.74592	0.872958	−	0.487795i	$-0.162199\pi$
$523$	39.9277	1.74592	0.872958	+	0.487795i	$0.162199\pi$
$524$	0	0
$525$	−26.8307	−1.17099
$526$	0	0
$527$	−20.5593	−0.895579
$528$	0	0
$529$	8.71630	0.378970
$530$	0	0
$531$	−28.0772	−1.21845
$532$	0	0
$533$	59.3446	2.57050
$534$	0	0
$535$	−0.837457	−0.0362064
$536$	0	0
$537$	20.6289	0.890202
$538$	0	0
$539$	3.85193	0.165914
$540$	0	0
$541$	16.6446	0.715607	0.357804	−	0.933797i	$-0.383526\pi$
$541$	16.6446	0.715607	0.357804	+	0.933797i	$0.383526\pi$
$542$	0	0
$543$	30.2145	1.29663
$544$	0	0
$545$	14.8804	0.637405
$546$	0	0
$547$	−38.3956	−1.64168	−0.820839	−	0.571159i	$-0.806494\pi$
$547$	−38.3956	−1.64168	−0.820839	+	0.571159i	$0.806494\pi$
$548$	0	0
$549$	−42.3073	−1.80563
$550$	0	0
$551$	4.59216	0.195632
$552$	0	0
$553$	24.5275	1.04301
$554$	0	0
$555$	−12.5592	−0.533107
$556$	0	0
$557$	30.1962	1.27945	0.639726	−	0.768603i	$-0.279048\pi$
$557$	30.1962	1.27945	0.639726	+	0.768603i	$0.279048\pi$
$558$	0	0
$559$	23.8859	1.01026
$560$	0	0
$561$	10.5327	0.444693
$562$	0	0
$563$	36.1995	1.52563	0.762814	−	0.646618i	$-0.223817\pi$
$563$	36.1995	1.52563	0.762814	+	0.646618i	$0.223817\pi$
$564$	0	0
$565$	−9.30871	−0.391620
$566$	0	0
$567$	−29.2161	−1.22696
$568$	0	0
$569$	31.1371	1.30533	0.652667	−	0.757645i	$-0.273650\pi$
$569$	31.1371	1.30533	0.652667	+	0.757645i	$0.273650\pi$
$570$	0	0
$571$	9.41933	0.394187	0.197093	−	0.980385i	$-0.436850\pi$
$571$	9.41933	0.394187	0.197093	+	0.980385i	$0.436850\pi$
$572$	0	0
$573$	−66.9735	−2.79786
$574$	0	0
$575$	−23.4892	−0.979566
$576$	0	0
$577$	−27.0205	−1.12488	−0.562438	−	0.826839i	$-0.690137\pi$
$577$	−27.0205	−1.12488	−0.562438	+	0.826839i	$0.690137\pi$
$578$	0	0
$579$	−35.1473	−1.46067
$580$	0	0
$581$	16.6704	0.691603
$582$	0	0
$583$	−2.71283	−0.112354
$584$	0	0
$585$	33.4132	1.38146
$586$	0	0
$587$	−32.0201	−1.32161	−0.660806	−	0.750557i	$-0.729785\pi$
$587$	−32.0201	−1.32161	−0.660806	+	0.750557i	$0.729785\pi$
$588$	0	0
$589$	−9.23063	−0.380341
$590$	0	0
$591$	15.8958	0.653865
$592$	0	0
$593$	20.8329	0.855503	0.427751	−	0.903896i	$-0.359306\pi$
$593$	20.8329	0.855503	0.427751	+	0.903896i	$0.359306\pi$
$594$	0	0
$595$	4.46584	0.183082
$596$	0	0
$597$	49.1269	2.01063
$598$	0	0
$599$	−22.7908	−0.931208	−0.465604	−	0.884993i	$-0.654163\pi$
$599$	−22.7908	−0.931208	−0.465604	+	0.884993i	$0.654163\pi$
$600$	0	0
$601$	−17.4469	−0.711672	−0.355836	−	0.934548i	$-0.615804\pi$
$601$	−17.4469	−0.711672	−0.355836	+	0.934548i	$0.615804\pi$
$602$	0	0
$603$	−95.3819	−3.88425
$604$	0	0
$605$	8.10139	0.329368
$606$	0	0
$607$	−36.8843	−1.49709	−0.748543	−	0.663086i	$-0.769246\pi$
$607$	−36.8843	−1.49709	−0.748543	+	0.663086i	$0.769246\pi$
$608$	0	0
$609$	27.9602	1.13301
$610$	0	0
$611$	−49.0647	−1.98495
$612$	0	0
$613$	9.45500	0.381884	0.190942	−	0.981601i	$-0.438846\pi$
$613$	9.45500	0.381884	0.190942	+	0.981601i	$0.438846\pi$
$614$	0	0
$615$	−29.6658	−1.19624
$616$	0	0
$617$	−31.2901	−1.25969	−0.629846	−	0.776720i	$-0.716882\pi$
$617$	−31.2901	−1.25969	−0.629846	+	0.776720i	$0.716882\pi$
$618$	0	0
$619$	−27.7405	−1.11498	−0.557492	−	0.830182i	$-0.688236\pi$
$619$	−27.7405	−1.11498	−0.557492	+	0.830182i	$0.688236\pi$
$620$	0	0
$621$	−61.3037	−2.46003
$622$	0	0
$623$	5.48792	0.219869
$624$	0	0
$625$	13.2505	0.530020
$626$	0	0
$627$	4.72894	0.188856
$628$	0	0
$629$	−10.5156	−0.419286
$630$	0	0
$631$	5.80111	0.230938	0.115469	−	0.993311i	$-0.463163\pi$
$631$	5.80111	0.230938	0.115469	+	0.993311i	$0.463163\pi$
$632$	0	0
$633$	−13.9431	−0.554187
$634$	0	0
$635$	5.64678	0.224086
$636$	0	0
$637$	14.9341	0.591709
$638$	0	0
$639$	33.4217	1.32214
$640$	0	0
$641$	−12.6431	−0.499374	−0.249687	−	0.968327i	$-0.580328\pi$
$641$	−12.6431	−0.499374	−0.249687	+	0.968327i	$0.580328\pi$
$642$	0	0
$643$	−0.739741	−0.0291725	−0.0145863	−	0.999894i	$-0.504643\pi$
$643$	−0.739741	−0.0291725	−0.0145863	+	0.999894i	$0.504643\pi$
$644$	0	0
$645$	−11.9403	−0.470148
$646$	0	0
$647$	5.76141	0.226505	0.113252	−	0.993566i	$-0.463873\pi$
$647$	5.76141	0.226505	0.113252	+	0.993566i	$0.463873\pi$
$648$	0	0
$649$	6.23833	0.244876
$650$	0	0
$651$	−56.2024	−2.20275
$652$	0	0
$653$	−23.8932	−0.935015	−0.467507	−	0.883989i	$-0.654848\pi$
$653$	−23.8932	−0.935015	−0.467507	+	0.883989i	$0.654848\pi$
$654$	0	0
$655$	−3.46223	−0.135281
$656$	0	0
$657$	50.0606	1.95305
$658$	0	0
$659$	−0.450480	−0.0175482	−0.00877410	−	0.999962i	$-0.502793\pi$
$659$	−0.450480	−0.0175482	−0.00877410	+	0.999962i	$0.502793\pi$
$660$	0	0
$661$	−27.4188	−1.06647	−0.533234	−	0.845968i	$-0.679023\pi$
$661$	−27.4188	−1.06647	−0.533234	+	0.845968i	$0.679023\pi$
$662$	0	0
$663$	40.8358	1.58593
$664$	0	0
$665$	2.00505	0.0777525
$666$	0	0
$667$	24.4781	0.947794
$668$	0	0
$669$	41.6687	1.61101
$670$	0	0
$671$	9.40004	0.362884
$672$	0	0
$673$	−39.9618	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$673$	−39.9618	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$674$	0	0
$675$	45.4017	1.74751
$676$	0	0
$677$	22.5667	0.867308	0.433654	−	0.901079i	$-0.357224\pi$
$677$	22.5667	0.867308	0.433654	+	0.901079i	$0.357224\pi$
$678$	0	0
$679$	−24.1874	−0.928226
$680$	0	0
$681$	−57.3024	−2.19583
$682$	0	0
$683$	−38.1485	−1.45971	−0.729855	−	0.683602i	$-0.760412\pi$
$683$	−38.1485	−1.45971	−0.729855	+	0.683602i	$0.760412\pi$
$684$	0	0
$685$	6.54293	0.249992
$686$	0	0
$687$	−11.9184	−0.454715
$688$	0	0
$689$	−10.5177	−0.400693
$690$	0	0
$691$	6.53892	0.248752	0.124376	−	0.992235i	$-0.460307\pi$
$691$	6.53892	0.248752	0.124376	+	0.992235i	$0.460307\pi$
$692$	0	0
$693$	19.7260	0.749330
$694$	0	0
$695$	20.4889	0.777186
$696$	0	0
$697$	−24.8388	−0.940837
$698$	0	0
$699$	−22.5385	−0.852485
$700$	0	0
$701$	10.4051	0.392996	0.196498	−	0.980504i	$-0.437043\pi$
$701$	10.4051	0.392996	0.196498	+	0.980504i	$0.437043\pi$
$702$	0	0
$703$	−4.72126	−0.178066
$704$	0	0
$705$	24.5269	0.923738
$706$	0	0
$707$	16.5380	0.621975
$708$	0	0
$709$	12.9924	0.487938	0.243969	−	0.969783i	$-0.421550\pi$
$709$	12.9924	0.487938	0.243969	+	0.969783i	$0.421550\pi$
$710$	0	0
$711$	−76.8098	−2.88059
$712$	0	0
$713$	−49.2030	−1.84267
$714$	0	0
$715$	−7.42390	−0.277638
$716$	0	0
$717$	−24.3128	−0.907980
$718$	0	0
$719$	−36.1495	−1.34815	−0.674074	−	0.738663i	$-0.735457\pi$
$719$	−36.1495	−1.34815	−0.674074	+	0.738663i	$0.735457\pi$
$720$	0	0
$721$	−13.2914	−0.494998
$722$	0	0
$723$	−89.8436	−3.34132
$724$	0	0
$725$	−18.1285	−0.673276
$726$	0	0
$727$	18.2522	0.676935	0.338468	−	0.940978i	$-0.390091\pi$
$727$	18.2522	0.676935	0.338468	+	0.940978i	$0.390091\pi$
$728$	0	0
$729$	−9.30247	−0.344536
$730$	0	0
$731$	−9.99746	−0.369769
$732$	0	0
$733$	−33.7251	−1.24567	−0.622833	−	0.782355i	$-0.714018\pi$
$733$	−33.7251	−1.24567	−0.622833	+	0.782355i	$0.714018\pi$
$734$	0	0
$735$	−7.46538	−0.275365
$736$	0	0
$737$	21.1924	0.780632
$738$	0	0
$739$	−31.2341	−1.14897	−0.574483	−	0.818516i	$-0.694797\pi$
$739$	−31.2341	−1.14897	−0.574483	+	0.818516i	$0.694797\pi$
$740$	0	0
$741$	18.3342	0.673525
$742$	0	0
$743$	−21.4211	−0.785865	−0.392932	−	0.919567i	$-0.628539\pi$
$743$	−21.4211	−0.785865	−0.392932	+	0.919567i	$0.628539\pi$
$744$	0	0
$745$	−9.20988	−0.337424
$746$	0	0
$747$	−52.2046	−1.91007
$748$	0	0
$749$	−1.91683	−0.0700394
$750$	0	0
$751$	15.0581	0.549479	0.274740	−	0.961519i	$-0.411408\pi$
$751$	15.0581	0.549479	0.274740	+	0.961519i	$0.411408\pi$
$752$	0	0
$753$	−52.1821	−1.90162
$754$	0	0
$755$	17.7158	0.644745
$756$	0	0
$757$	5.16450	0.187707	0.0938534	−	0.995586i	$-0.470082\pi$
$757$	5.16450	0.187707	0.0938534	+	0.995586i	$0.470082\pi$
$758$	0	0
$759$	25.2072	0.914962
$760$	0	0
$761$	29.8494	1.08204	0.541021	−	0.841009i	$-0.318038\pi$
$761$	29.8494	1.08204	0.541021	+	0.841009i	$0.318038\pi$
$762$	0	0
$763$	34.0592	1.23303
$764$	0	0
$765$	−13.9851	−0.505634
$766$	0	0
$767$	24.1862	0.873312
$768$	0	0
$769$	15.6136	0.563042	0.281521	−	0.959555i	$-0.409161\pi$
$769$	15.6136	0.563042	0.281521	+	0.959555i	$0.409161\pi$
$770$	0	0
$771$	36.3550	1.30929
$772$	0	0
$773$	33.2962	1.19758	0.598790	−	0.800906i	$-0.295648\pi$
$773$	33.2962	1.19758	0.598790	+	0.800906i	$0.295648\pi$
$774$	0	0
$775$	36.4399	1.30896
$776$	0	0
$777$	−28.7463	−1.03127
$778$	0	0
$779$	−11.1520	−0.399562
$780$	0	0
$781$	−7.42580	−0.265716
$782$	0	0
$783$	−47.3131	−1.69083
$784$	0	0
$785$	−4.49920	−0.160583
$786$	0	0
$787$	−25.0491	−0.892905	−0.446453	−	0.894807i	$-0.647313\pi$
$787$	−25.0491	−0.892905	−0.446453	+	0.894807i	$0.647313\pi$
$788$	0	0
$789$	−53.1042	−1.89056
$790$	0	0
$791$	−21.3064	−0.757569
$792$	0	0
$793$	36.4442	1.29417
$794$	0	0
$795$	5.25770	0.186471
$796$	0	0
$797$	−51.0298	−1.80757	−0.903785	−	0.427987i	$-0.859223\pi$
$797$	−51.0298	−1.80757	−0.903785	+	0.427987i	$0.859223\pi$
$798$	0	0
$799$	20.5361	0.726516
$800$	0	0
$801$	−17.1859	−0.607233
$802$	0	0
$803$	−11.1227	−0.392512
$804$	0	0
$805$	10.6877	0.376693
$806$	0	0
$807$	37.5314	1.32117
$808$	0	0
$809$	25.1705	0.884947	0.442473	−	0.896782i	$-0.354101\pi$
$809$	25.1705	0.884947	0.442473	+	0.896782i	$0.354101\pi$
$810$	0	0
$811$	5.64533	0.198235	0.0991173	−	0.995076i	$-0.468398\pi$
$811$	5.64533	0.198235	0.0991173	+	0.995076i	$0.468398\pi$
$812$	0	0
$813$	0.446137	0.0156467
$814$	0	0
$815$	−13.0944	−0.458675
$816$	0	0
$817$	−4.48861	−0.157037
$818$	0	0
$819$	76.4784	2.67237
$820$	0	0
$821$	10.6903	0.373096	0.186548	−	0.982446i	$-0.440270\pi$
$821$	10.6903	0.373096	0.186548	+	0.982446i	$0.440270\pi$
$822$	0	0
$823$	−12.4143	−0.432734	−0.216367	−	0.976312i	$-0.569421\pi$
$823$	−12.4143	−0.432734	−0.216367	+	0.976312i	$0.569421\pi$
$824$	0	0
$825$	−18.6685	−0.649953
$826$	0	0
$827$	44.5662	1.54972	0.774859	−	0.632134i	$-0.217821\pi$
$827$	44.5662	1.54972	0.774859	+	0.632134i	$0.217821\pi$
$828$	0	0
$829$	−18.5663	−0.644834	−0.322417	−	0.946598i	$-0.604495\pi$
$829$	−18.5663	−0.644834	−0.322417	+	0.946598i	$0.604495\pi$
$830$	0	0
$831$	−14.2463	−0.494200
$832$	0	0
$833$	−6.25067	−0.216573
$834$	0	0
$835$	20.3116	0.702912
$836$	0	0
$837$	95.1033	3.28725
$838$	0	0
$839$	−43.3676	−1.49722	−0.748608	−	0.663012i	$-0.769278\pi$
$839$	−43.3676	−1.49722	−0.748608	+	0.663012i	$0.769278\pi$
$840$	0	0
$841$	−10.1083	−0.348561
$842$	0	0
$843$	−7.33679	−0.252692
$844$	0	0
$845$	−16.9453	−0.582937
$846$	0	0
$847$	18.5430	0.637146
$848$	0	0
$849$	41.6175	1.42831
$850$	0	0
$851$	−25.1662	−0.862688
$852$	0	0
$853$	19.3360	0.662050	0.331025	−	0.943622i	$-0.392605\pi$
$853$	19.3360	0.662050	0.331025	+	0.943622i	$0.392605\pi$
$854$	0	0
$855$	−6.27897	−0.214736
$856$	0	0
$857$	−33.8462	−1.15616	−0.578082	−	0.815979i	$-0.696199\pi$
$857$	−33.8462	−1.15616	−0.578082	+	0.815979i	$0.696199\pi$
$858$	0	0
$859$	12.4242	0.423908	0.211954	−	0.977280i	$-0.432017\pi$
$859$	12.4242	0.423908	0.211954	+	0.977280i	$0.432017\pi$
$860$	0	0
$861$	−67.9011	−2.31406
$862$	0	0
$863$	46.9463	1.59807	0.799035	−	0.601284i	$-0.205344\pi$
$863$	46.9463	1.59807	0.799035	+	0.601284i	$0.205344\pi$
$864$	0	0
$865$	12.0468	0.409604
$866$	0	0
$867$	35.3793	1.20154
$868$	0	0
$869$	17.0660	0.578923
$870$	0	0
$871$	82.1636	2.78401
$872$	0	0
$873$	75.7447	2.56357
$874$	0	0
$875$	−17.4042	−0.588370
$876$	0	0
$877$	32.6960	1.10407	0.552033	−	0.833822i	$-0.313852\pi$
$877$	32.6960	1.10407	0.552033	+	0.833822i	$0.313852\pi$
$878$	0	0
$879$	90.3677	3.04803
$880$	0	0
$881$	−12.2877	−0.413982	−0.206991	−	0.978343i	$-0.566367\pi$
$881$	−12.2877	−0.413982	−0.206991	+	0.978343i	$0.566367\pi$
$882$	0	0
$883$	2.03500	0.0684831	0.0342415	−	0.999414i	$-0.489098\pi$
$883$	2.03500	0.0684831	0.0342415	+	0.999414i	$0.489098\pi$
$884$	0	0
$885$	−12.0904	−0.406415
$886$	0	0
$887$	−47.0813	−1.58083	−0.790417	−	0.612569i	$-0.790136\pi$
$887$	−47.0813	−1.58083	−0.790417	+	0.612569i	$0.790136\pi$
$888$	0	0
$889$	12.9247	0.433482
$890$	0	0
$891$	−20.3282	−0.681022
$892$	0	0
$893$	9.22020	0.308542
$894$	0	0
$895$	6.08576	0.203425
$896$	0	0
$897$	97.7289	3.26307
$898$	0	0
$899$	−37.9740	−1.26650
$900$	0	0
$901$	4.40221	0.146659
$902$	0	0
$903$	−27.3298	−0.909477
$904$	0	0
$905$	8.91363	0.296299
$906$	0	0
$907$	21.3028	0.707349	0.353674	−	0.935369i	$-0.384932\pi$
$907$	21.3028	0.707349	0.353674	+	0.935369i	$0.384932\pi$
$908$	0	0
$909$	−51.7901	−1.71777
$910$	0	0
$911$	14.2825	0.473199	0.236600	−	0.971607i	$-0.423967\pi$
$911$	14.2825	0.473199	0.236600	+	0.971607i	$0.423967\pi$
$912$	0	0
$913$	11.5991	0.383873
$914$	0	0
$915$	−18.2181	−0.602271
$916$	0	0
$917$	−7.92460	−0.261693
$918$	0	0
$919$	−10.2972	−0.339673	−0.169837	−	0.985472i	$-0.554324\pi$
$919$	−10.2972	−0.339673	−0.169837	+	0.985472i	$0.554324\pi$
$920$	0	0
$921$	−84.2468	−2.77603
$922$	0	0
$923$	−28.7901	−0.947636
$924$	0	0
$925$	18.6382	0.612820
$926$	0	0
$927$	41.6231	1.36708
$928$	0	0
$929$	3.77218	0.123761	0.0618806	−	0.998084i	$-0.480290\pi$
$929$	3.77218	0.123761	0.0618806	+	0.998084i	$0.480290\pi$
$930$	0	0
$931$	−2.80640	−0.0919759
$932$	0	0
$933$	45.5715	1.49194
$934$	0	0
$935$	3.10728	0.101619
$936$	0	0
$937$	55.6015	1.81642	0.908211	−	0.418513i	$-0.137449\pi$
$937$	55.6015	1.81642	0.908211	+	0.418513i	$0.137449\pi$
$938$	0	0
$939$	−12.4105	−0.405002
$940$	0	0
$941$	1.44384	0.0470680	0.0235340	−	0.999723i	$-0.492508\pi$
$941$	1.44384	0.0470680	0.0235340	+	0.999723i	$0.492508\pi$
$942$	0	0
$943$	−59.4447	−1.93579
$944$	0	0
$945$	−20.6581	−0.672007
$946$	0	0
$947$	−41.0784	−1.33487	−0.667434	−	0.744669i	$-0.732607\pi$
$947$	−41.0784	−1.33487	−0.667434	+	0.744669i	$0.732607\pi$
$948$	0	0
$949$	−43.1231	−1.39983
$950$	0	0
$951$	−93.2261	−3.02306
$952$	0	0
$953$	26.5179	0.859000	0.429500	−	0.903067i	$-0.358690\pi$
$953$	26.5179	0.859000	0.429500	+	0.903067i	$0.358690\pi$
$954$	0	0
$955$	−19.7580	−0.639354
$956$	0	0
$957$	19.4544	0.628872
$958$	0	0
$959$	14.9759	0.483597
$960$	0	0
$961$	45.3310	1.46229
$962$	0	0
$963$	6.00271	0.193434
$964$	0	0
$965$	−10.3689	−0.333786
$966$	0	0
$967$	−30.1124	−0.968349	−0.484175	−	0.874971i	$-0.660880\pi$
$967$	−30.1124	−0.968349	−0.484175	+	0.874971i	$0.660880\pi$
$968$	0	0
$969$	−7.67382	−0.246519
$970$	0	0
$971$	−3.49524	−0.112168	−0.0560838	−	0.998426i	$-0.517861\pi$
$971$	−3.49524	−0.112168	−0.0560838	+	0.998426i	$0.517861\pi$
$972$	0	0
$973$	46.8963	1.50343
$974$	0	0
$975$	−72.3783	−2.31796
$976$	0	0
$977$	55.8789	1.78772	0.893862	−	0.448341i	$-0.147985\pi$
$977$	55.8789	1.78772	0.893862	+	0.448341i	$0.147985\pi$
$978$	0	0
$979$	3.81844	0.122038
$980$	0	0
$981$	−106.659	−3.40537
$982$	0	0
$983$	−52.8623	−1.68604	−0.843022	−	0.537879i	$-0.819226\pi$
$983$	−52.8623	−1.68604	−0.843022	+	0.537879i	$0.819226\pi$
$984$	0	0
$985$	4.68944	0.149418
$986$	0	0
$987$	56.1390	1.78692
$988$	0	0
$989$	−23.9261	−0.760806
$990$	0	0
$991$	−8.06158	−0.256084	−0.128042	−	0.991769i	$-0.540869\pi$
$991$	−8.06158	−0.256084	−0.128042	+	0.991769i	$0.540869\pi$
$992$	0	0
$993$	−57.8556	−1.83599
$994$	0	0
$995$	14.4930	0.459460
$996$	0	0
$997$	−18.7766	−0.594662	−0.297331	−	0.954775i	$-0.596096\pi$
$997$	−18.7766	−0.594662	−0.297331	+	0.954775i	$0.596096\pi$
$998$	0	0
$999$	48.6432	1.53900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6128.2.a.p.1.1		24
4.3	odd	2		383.2.a.c.1.16	✓	24
12.11	even	2		3447.2.a.j.1.9		24
20.19	odd	2		9575.2.a.e.1.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
383.2.a.c.1.16	✓	24	4.3	odd	2
3447.2.a.j.1.9		24	12.11	even	2
6128.2.a.p.1.1		24	1.1	even	1	trivial
9575.2.a.e.1.9		24	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6128 = 2^{4} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6128.a (trivial)

\(f(q)\)	\(=\)	\(q-3.08654 q^{3} -0.910566 q^{5} -2.08417 q^{7} +6.52674 q^{9} -1.45014 q^{11} -5.62225 q^{13} +2.81050 q^{15} +2.35320 q^{17} +1.05653 q^{19} +6.43287 q^{21} +5.63172 q^{23} -4.17087 q^{25} -10.8854 q^{27} +4.34646 q^{29} -8.73676 q^{31} +4.47592 q^{33} +1.89777 q^{35} -4.46866 q^{37} +17.3533 q^{39} -10.5553 q^{41} -4.24845 q^{43} -5.94303 q^{45} +8.72689 q^{47} -2.65624 q^{49} -7.26325 q^{51} +1.87073 q^{53} +1.32045 q^{55} -3.26102 q^{57} -4.30187 q^{59} -6.48215 q^{61} -13.6028 q^{63} +5.11943 q^{65} -14.6140 q^{67} -17.3825 q^{69} +5.12074 q^{71} +7.67008 q^{73} +12.8736 q^{75} +3.02234 q^{77} -11.7685 q^{79} +14.0181 q^{81} -7.99857 q^{83} -2.14274 q^{85} -13.4155 q^{87} -2.63315 q^{89} +11.7177 q^{91} +26.9664 q^{93} -0.962038 q^{95} +11.6053 q^{97} -9.46470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{3} + 3 q^{5} - 17 q^{7} + 34 q^{9} + 28 q^{13} + 2 q^{15} + 16 q^{17} - 13 q^{19} - 4 q^{21} - 7 q^{23} + 67 q^{25} + 10 q^{27} - 2 q^{29} + 24 q^{33} + 10 q^{35} + 35 q^{37} + 12 q^{39} - 34 q^{43}+ \cdots + 54 q^{99}+O(q^{100}) \) 24 * q - 2 * q^3 + 3 * q^5 - 17 * q^7 + 34 * q^9 + 28 * q^13 + 2 * q^15 + 16 * q^17 - 13 * q^19 - 4 * q^21 - 7 * q^23 + 67 * q^25 + 10 * q^27 - 2 * q^29 + 24 * q^33 + 10 * q^35 + 35 * q^37 + 12 * q^39 - 34 * q^43 - 27 * q^45 + 3 * q^47 + 49 * q^49 + 28 * q^51 - 9 * q^53 - 10 * q^55 + 47 * q^57 + 6 * q^59 + 3 * q^61 - 13 * q^63 + 8 * q^65 - 42 * q^67 - 30 * q^69 + 13 * q^71 + 83 * q^73 + 65 * q^75 - 21 * q^77 + 3 * q^79 + 28 * q^81 + 23 * q^83 + 52 * q^85 + 5 * q^87 - 38 * q^89 + 17 * q^91 + 17 * q^93 + 66 * q^95 + 63 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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