LMFDB - Embedded newform 600.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.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:	$35.4011460034$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.17891$ of defining polynomial
Character	$\chi$	$=$	600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -35.0735 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -35.0735 q^{7} +9.00000 q^{9} +25.6422 q^{11} +37.6422 q^{13} -95.7891 q^{17} +50.8625 q^{19} -105.220 q^{21} +110.863 q^{23} +27.0000 q^{27} -54.5047 q^{29} +198.441 q^{31} +76.9265 q^{33} +266.945 q^{37} +112.927 q^{39} +103.853 q^{41} +108.000 q^{43} +597.009 q^{47} +887.147 q^{49} -287.367 q^{51} -305.642 q^{53} +152.588 q^{57} -223.533 q^{59} +485.450 q^{61} -315.661 q^{63} -876.166 q^{67} +332.588 q^{69} +585.597 q^{71} -1137.60 q^{73} -899.360 q^{77} +685.009 q^{79} +81.0000 q^{81} +305.725 q^{83} -163.514 q^{87} +887.175 q^{89} -1320.24 q^{91} +595.322 q^{93} -556.550 q^{97} +230.780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} + 74 q^{11} + 98 q^{13} - 78 q^{17} - 80 q^{19} - 6 q^{21} + 40 q^{23} + 54 q^{27} + 50 q^{29} - 12 q^{31} + 222 q^{33} - 34 q^{37} + 294 q^{39} + 344 q^{41} + 216 q^{43} + 876 q^{47} + 1638 q^{49} - 234 q^{51} - 634 q^{53} - 240 q^{57} + 666 q^{59} + 244 q^{61} - 18 q^{63} - 980 q^{67} + 120 q^{69} + 308 q^{71} - 1412 q^{73} + 700 q^{77} + 1052 q^{79} + 162 q^{81} + 248 q^{83} + 150 q^{87} + 684 q^{89} + 676 q^{91} - 36 q^{93} - 1840 q^{97} + 666 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 + 74 * q^11 + 98 * q^13 - 78 * q^17 - 80 * q^19 - 6 * q^21 + 40 * q^23 + 54 * q^27 + 50 * q^29 - 12 * q^31 + 222 * q^33 - 34 * q^37 + 294 * q^39 + 344 * q^41 + 216 * q^43 + 876 * q^47 + 1638 * q^49 - 234 * q^51 - 634 * q^53 - 240 * q^57 + 666 * q^59 + 244 * q^61 - 18 * q^63 - 980 * q^67 + 120 * q^69 + 308 * q^71 - 1412 * q^73 + 700 * q^77 + 1052 * q^79 + 162 * q^81 + 248 * q^83 + 150 * q^87 + 684 * q^89 + 676 * q^91 - 36 * q^93 - 1840 * q^97 + 666 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−35.0735	−1.89379	−0.946894	−	0.321545i	$-0.895798\pi$
$7$	−35.0735	−1.89379	−0.946894	+	0.321545i	$0.895798\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	25.6422	0.702855	0.351428	−	0.936215i	$-0.385696\pi$
$11$	25.6422	0.702855	0.351428	+	0.936215i	$0.385696\pi$
$12$	0	0
$13$	37.6422	0.803082	0.401541	−	0.915841i	$-0.368475\pi$
$13$	37.6422	0.803082	0.401541	+	0.915841i	$0.368475\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−95.7891	−1.36660	−0.683302	−	0.730136i	$-0.739457\pi$
$17$	−95.7891	−1.36660	−0.683302	+	0.730136i	$0.739457\pi$
$18$	0	0
$19$	50.8625	0.614140	0.307070	−	0.951687i	$-0.400651\pi$
$19$	50.8625	0.614140	0.307070	+	0.951687i	$0.400651\pi$
$20$	0	0
$21$	−105.220	−1.09338
$22$	0	0
$23$	110.863	1.00506	0.502531	−	0.864559i	$-0.332402\pi$
$23$	110.863	1.00506	0.502531	+	0.864559i	$0.332402\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−54.5047	−0.349009	−0.174505	−	0.984656i	$-0.555832\pi$
$29$	−54.5047	−0.349009	−0.174505	+	0.984656i	$0.555832\pi$
$30$	0	0
$31$	198.441	1.14971	0.574855	−	0.818255i	$-0.305059\pi$
$31$	198.441	1.14971	0.574855	+	0.818255i	$0.305059\pi$
$32$	0	0
$33$	76.9265	0.405794
$34$	0	0
$35$	0	0
$36$	0	0
$37$	266.945	1.18610	0.593048	−	0.805167i	$-0.297924\pi$
$37$	266.945	1.18610	0.593048	+	0.805167i	$0.297924\pi$
$38$	0	0
$39$	112.927	0.463659
$40$	0	0
$41$	103.853	0.395589	0.197794	−	0.980244i	$-0.436622\pi$
$41$	103.853	0.395589	0.197794	+	0.980244i	$0.436622\pi$
$42$	0	0
$43$	108.000	0.383020	0.191510	−	0.981491i	$-0.438662\pi$
$43$	108.000	0.383020	0.191510	+	0.981491i	$0.438662\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	597.009	1.85283	0.926413	−	0.376510i	$-0.122876\pi$
$47$	597.009	1.85283	0.926413	+	0.376510i	$0.122876\pi$
$48$	0	0
$49$	887.147	2.58643
$50$	0	0
$51$	−287.367	−0.789009
$52$	0	0
$53$	−305.642	−0.792136	−0.396068	−	0.918221i	$-0.629625\pi$
$53$	−305.642	−0.792136	−0.396068	+	0.918221i	$0.629625\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	152.588	0.354574
$58$	0	0
$59$	−223.533	−0.493246	−0.246623	−	0.969111i	$-0.579321\pi$
$59$	−223.533	−0.493246	−0.246623	+	0.969111i	$0.579321\pi$
$60$	0	0
$61$	485.450	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$61$	485.450	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$62$	0	0
$63$	−315.661	−0.631263
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−876.166	−1.59762	−0.798811	−	0.601582i	$-0.794537\pi$
$67$	−876.166	−1.59762	−0.798811	+	0.601582i	$0.794537\pi$
$68$	0	0
$69$	332.588	0.580273
$70$	0	0
$71$	585.597	0.978839	0.489420	−	0.872048i	$-0.337209\pi$
$71$	585.597	0.978839	0.489420	+	0.872048i	$0.337209\pi$
$72$	0	0
$73$	−1137.60	−1.82391	−0.911957	−	0.410287i	$-0.865429\pi$
$73$	−1137.60	−1.82391	−0.911957	+	0.410287i	$0.865429\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−899.360	−1.33106
$78$	0	0
$79$	685.009	0.975564	0.487782	−	0.872965i	$-0.337806\pi$
$79$	685.009	0.975564	0.487782	+	0.872965i	$0.337806\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	305.725	0.404309	0.202155	−	0.979354i	$-0.435206\pi$
$83$	305.725	0.404309	0.202155	+	0.979354i	$0.435206\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−163.514	−0.201501
$88$	0	0
$89$	887.175	1.05663	0.528317	−	0.849047i	$-0.322823\pi$
$89$	887.175	1.05663	0.528317	+	0.849047i	$0.322823\pi$
$90$	0	0
$91$	−1320.24	−1.52087
$92$	0	0
$93$	595.322	0.663785
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−556.550	−0.582568	−0.291284	−	0.956637i	$-0.594082\pi$
$97$	−556.550	−0.582568	−0.291284	+	0.956637i	$0.594082\pi$
$98$	0	0
$99$	230.780	0.234285
$100$	0	0
$101$	1591.09	1.56752	0.783760	−	0.621063i	$-0.213299\pi$
$101$	1591.09	1.56752	0.783760	+	0.621063i	$0.213299\pi$
$102$	0	0
$103$	1350.95	1.29236	0.646178	−	0.763187i	$-0.276366\pi$
$103$	1350.95	1.29236	0.646178	+	0.763187i	$0.276366\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1333.51	1.20481	0.602406	−	0.798190i	$-0.294209\pi$
$107$	1333.51	1.20481	0.602406	+	0.798190i	$0.294209\pi$
$108$	0	0
$109$	−609.910	−0.535952	−0.267976	−	0.963426i	$-0.586355\pi$
$109$	−609.910	−0.535952	−0.267976	+	0.963426i	$0.586355\pi$
$110$	0	0
$111$	800.836	0.684793
$112$	0	0
$113$	241.808	0.201304	0.100652	−	0.994922i	$-0.467907\pi$
$113$	241.808	0.201304	0.100652	+	0.994922i	$0.467907\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	338.780	0.267694
$118$	0	0
$119$	3359.65	2.58806
$120$	0	0
$121$	−673.478	−0.505994
$122$	0	0
$123$	311.559	0.228393
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1045.94	−0.730802	−0.365401	−	0.930850i	$-0.619068\pi$
$127$	−1045.94	−0.730802	−0.365401	+	0.930850i	$0.619068\pi$
$128$	0	0
$129$	324.000	0.221137
$130$	0	0
$131$	−886.524	−0.591267	−0.295633	−	0.955301i	$-0.595531\pi$
$131$	−886.524	−0.591267	−0.295633	+	0.955301i	$0.595531\pi$
$132$	0	0
$133$	−1783.92	−1.16305
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−160.723	−0.100230	−0.0501150	−	0.998743i	$-0.515959\pi$
$137$	−160.723	−0.100230	−0.0501150	+	0.998743i	$0.515959\pi$
$138$	0	0
$139$	−57.2655	−0.0349439	−0.0174719	−	0.999847i	$-0.505562\pi$
$139$	−57.2655	−0.0349439	−0.0174719	+	0.999847i	$0.505562\pi$
$140$	0	0
$141$	1791.03	1.06973
$142$	0	0
$143$	965.228	0.564450
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2661.44	1.49328
$148$	0	0
$149$	−1105.15	−0.607633	−0.303817	−	0.952731i	$-0.598261\pi$
$149$	−1105.15	−0.607633	−0.303817	+	0.952731i	$0.598261\pi$
$150$	0	0
$151$	2289.63	1.23396	0.616980	−	0.786979i	$-0.288356\pi$
$151$	2289.63	1.23396	0.616980	+	0.786979i	$0.288356\pi$
$152$	0	0
$153$	−862.102	−0.455535
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−161.514	−0.0821034	−0.0410517	−	0.999157i	$-0.513071\pi$
$157$	−161.514	−0.0821034	−0.0410517	+	0.999157i	$0.513071\pi$
$158$	0	0
$159$	−916.927	−0.457340
$160$	0	0
$161$	−3888.33	−1.90338
$162$	0	0
$163$	1594.15	0.766032	0.383016	−	0.923742i	$-0.374885\pi$
$163$	1594.15	0.766032	0.383016	+	0.923742i	$0.374885\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1017.39	0.471427	0.235713	−	0.971823i	$-0.424257\pi$
$167$	1017.39	0.471427	0.235713	+	0.971823i	$0.424257\pi$
$168$	0	0
$169$	−780.066	−0.355060
$170$	0	0
$171$	457.763	0.204713
$172$	0	0
$173$	−2444.49	−1.07428	−0.537142	−	0.843492i	$-0.680496\pi$
$173$	−2444.49	−1.07428	−0.537142	+	0.843492i	$0.680496\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−670.599	−0.284776
$178$	0	0
$179$	222.780	0.0930242	0.0465121	−	0.998918i	$-0.485189\pi$
$179$	222.780	0.0930242	0.0465121	+	0.998918i	$0.485189\pi$
$180$	0	0
$181$	−100.034	−0.0410798	−0.0205399	−	0.999789i	$-0.506539\pi$
$181$	−100.034	−0.0410798	−0.0205399	+	0.999789i	$0.506539\pi$
$182$	0	0
$183$	1456.35	0.588287
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2456.24	−0.960525
$188$	0	0
$189$	−946.983	−0.364460
$190$	0	0
$191$	702.403	0.266095	0.133047	−	0.991110i	$-0.457524\pi$
$191$	702.403	0.266095	0.133047	+	0.991110i	$0.457524\pi$
$192$	0	0
$193$	−4126.63	−1.53907	−0.769537	−	0.638602i	$-0.779513\pi$
$193$	−4126.63	−1.53907	−0.769537	+	0.638602i	$0.779513\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3104.45	1.12276	0.561378	−	0.827559i	$-0.310271\pi$
$197$	3104.45	1.12276	0.561378	+	0.827559i	$0.310271\pi$
$198$	0	0
$199$	−367.616	−0.130953	−0.0654764	−	0.997854i	$-0.520857\pi$
$199$	−367.616	−0.130953	−0.0654764	+	0.997854i	$0.520857\pi$
$200$	0	0
$201$	−2628.50	−0.922388
$202$	0	0
$203$	1911.67	0.660950
$204$	0	0
$205$	0	0
$206$	0	0
$207$	997.763	0.335021
$208$	0	0
$209$	1304.23	0.431652
$210$	0	0
$211$	2594.85	0.846619	0.423310	−	0.905985i	$-0.360868\pi$
$211$	2594.85	0.846619	0.423310	+	0.905985i	$0.360868\pi$
$212$	0	0
$213$	1756.79	0.565133
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6960.00	−2.17731
$218$	0	0
$219$	−3412.79	−1.05304
$220$	0	0
$221$	−3605.71	−1.09749
$222$	0	0
$223$	−1834.41	−0.550857	−0.275429	−	0.961321i	$-0.588820\pi$
$223$	−1834.41	−0.550857	−0.275429	+	0.961321i	$0.588820\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	175.811	0.0514053	0.0257027	−	0.999670i	$-0.491818\pi$
$227$	175.811	0.0514053	0.0257027	+	0.999670i	$0.491818\pi$
$228$	0	0
$229$	1622.35	0.468158	0.234079	−	0.972218i	$-0.424793\pi$
$229$	1622.35	0.468158	0.234079	+	0.972218i	$0.424793\pi$
$230$	0	0
$231$	−2698.08	−0.768487
$232$	0	0
$233$	3965.65	1.11501	0.557507	−	0.830172i	$-0.311758\pi$
$233$	3965.65	1.11501	0.557507	+	0.830172i	$0.311758\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2055.03	0.563242
$238$	0	0
$239$	−6323.25	−1.71137	−0.855685	−	0.517497i	$-0.826864\pi$
$239$	−6323.25	−1.71137	−0.855685	+	0.517497i	$0.826864\pi$
$240$	0	0
$241$	3407.51	0.910775	0.455388	−	0.890293i	$-0.349501\pi$
$241$	3407.51	0.910775	0.455388	+	0.890293i	$0.349501\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1914.58	0.493205
$248$	0	0
$249$	917.175	0.233428
$250$	0	0
$251$	1345.81	0.338433	0.169216	−	0.985579i	$-0.445876\pi$
$251$	1345.81	0.338433	0.169216	+	0.985579i	$0.445876\pi$
$252$	0	0
$253$	2842.76	0.706414
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4697.19	1.14009	0.570044	−	0.821614i	$-0.306926\pi$
$257$	4697.19	1.14009	0.570044	+	0.821614i	$0.306926\pi$
$258$	0	0
$259$	−9362.70	−2.24621
$260$	0	0
$261$	−490.542	−0.116336
$262$	0	0
$263$	4700.66	1.10211	0.551056	−	0.834468i	$-0.314225\pi$
$263$	4700.66	1.10211	0.551056	+	0.834468i	$0.314225\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2661.53	0.610048
$268$	0	0
$269$	−7962.67	−1.80481	−0.902403	−	0.430894i	$-0.858198\pi$
$269$	−7962.67	−1.80481	−0.902403	+	0.430894i	$0.858198\pi$
$270$	0	0
$271$	−6122.73	−1.37243	−0.686217	−	0.727397i	$-0.740730\pi$
$271$	−6122.73	−1.37243	−0.686217	+	0.727397i	$0.740730\pi$
$272$	0	0
$273$	−3960.72	−0.878073
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8417.57	−1.82586	−0.912930	−	0.408117i	$-0.866186\pi$
$277$	−8417.57	−1.82586	−0.912930	+	0.408117i	$0.866186\pi$
$278$	0	0
$279$	1785.97	0.383237
$280$	0	0
$281$	3030.99	0.643466	0.321733	−	0.946830i	$-0.395735\pi$
$281$	3030.99	0.643466	0.321733	+	0.946830i	$0.395735\pi$
$282$	0	0
$283$	2890.81	0.607211	0.303606	−	0.952798i	$-0.401809\pi$
$283$	2890.81	0.607211	0.303606	+	0.952798i	$0.401809\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3642.49	−0.749161
$288$	0	0
$289$	4262.55	0.867606
$290$	0	0
$291$	−1669.65	−0.336346
$292$	0	0
$293$	−8966.75	−1.78786	−0.893930	−	0.448206i	$-0.852063\pi$
$293$	−8966.75	−1.78786	−0.893930	+	0.448206i	$0.852063\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	692.339	0.135265
$298$	0	0
$299$	4173.11	0.807147
$300$	0	0
$301$	−3787.93	−0.725358
$302$	0	0
$303$	4773.28	0.905009
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3298.42	−0.613194	−0.306597	−	0.951839i	$-0.599190\pi$
$307$	−3298.42	−0.613194	−0.306597	+	0.951839i	$0.599190\pi$
$308$	0	0
$309$	4052.84	0.746142
$310$	0	0
$311$	−3394.92	−0.618998	−0.309499	−	0.950900i	$-0.600161\pi$
$311$	−3394.92	−0.618998	−0.309499	+	0.950900i	$0.600161\pi$
$312$	0	0
$313$	−5946.95	−1.07393	−0.536967	−	0.843603i	$-0.680430\pi$
$313$	−5946.95	−1.07393	−0.536967	+	0.843603i	$0.680430\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2520.83	0.446637	0.223318	−	0.974746i	$-0.428311\pi$
$317$	2520.83	0.446637	0.223318	+	0.974746i	$0.428311\pi$
$318$	0	0
$319$	−1397.62	−0.245303
$320$	0	0
$321$	4000.52	0.695599
$322$	0	0
$323$	−4872.08	−0.839286
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1829.73	−0.309432
$328$	0	0
$329$	−20939.2	−3.50886
$330$	0	0
$331$	−4586.86	−0.761682	−0.380841	−	0.924641i	$-0.624365\pi$
$331$	−4586.86	−0.761682	−0.380841	+	0.924641i	$0.624365\pi$
$332$	0	0
$333$	2402.51	0.395365
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8582.12	−1.38723	−0.693617	−	0.720344i	$-0.743984\pi$
$337$	−8582.12	−1.38723	−0.693617	+	0.720344i	$0.743984\pi$
$338$	0	0
$339$	725.424	0.116223
$340$	0	0
$341$	5088.45	0.808080
$342$	0	0
$343$	−19085.1	−3.00437
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2539.38	0.392857	0.196428	−	0.980518i	$-0.437066\pi$
$347$	2539.38	0.392857	0.196428	+	0.980518i	$0.437066\pi$
$348$	0	0
$349$	−9002.82	−1.38083	−0.690415	−	0.723413i	$-0.742572\pi$
$349$	−9002.82	−1.38083	−0.690415	+	0.723413i	$0.742572\pi$
$350$	0	0
$351$	1016.34	0.154553
$352$	0	0
$353$	3928.08	0.592267	0.296134	−	0.955146i	$-0.404303\pi$
$353$	3928.08	0.592267	0.296134	+	0.955146i	$0.404303\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10079.0	1.49422
$358$	0	0
$359$	10001.7	1.47039	0.735197	−	0.677854i	$-0.237090\pi$
$359$	10001.7	1.47039	0.735197	+	0.677854i	$0.237090\pi$
$360$	0	0
$361$	−4272.00	−0.622832
$362$	0	0
$363$	−2020.44	−0.292136
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5967.79	0.848818	0.424409	−	0.905471i	$-0.360482\pi$
$367$	5967.79	0.848818	0.424409	+	0.905471i	$0.360482\pi$
$368$	0	0
$369$	934.678	0.131863
$370$	0	0
$371$	10719.9	1.50014
$372$	0	0
$373$	6931.55	0.962204	0.481102	−	0.876665i	$-0.340237\pi$
$373$	6931.55	0.962204	0.481102	+	0.876665i	$0.340237\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2051.68	−0.280283
$378$	0	0
$379$	9711.70	1.31624	0.658122	−	0.752911i	$-0.271351\pi$
$379$	9711.70	1.31624	0.658122	+	0.752911i	$0.271351\pi$
$380$	0	0
$381$	−3137.81	−0.421929
$382$	0	0
$383$	5664.84	0.755769	0.377885	−	0.925853i	$-0.376652\pi$
$383$	5664.84	0.755769	0.377885	+	0.925853i	$0.376652\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	972.000	0.127673
$388$	0	0
$389$	8918.17	1.16239	0.581195	−	0.813765i	$-0.302586\pi$
$389$	8918.17	1.16239	0.581195	+	0.813765i	$0.302586\pi$
$390$	0	0
$391$	−10619.4	−1.37352
$392$	0	0
$393$	−2659.57	−0.341368
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12849.9	1.62448	0.812242	−	0.583321i	$-0.198247\pi$
$397$	12849.9	1.62448	0.812242	+	0.583321i	$0.198247\pi$
$398$	0	0
$399$	−5351.77	−0.671488
$400$	0	0
$401$	3563.08	0.443721	0.221860	−	0.975078i	$-0.428787\pi$
$401$	3563.08	0.443721	0.221860	+	0.975078i	$0.428787\pi$
$402$	0	0
$403$	7469.74	0.923311
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6845.06	0.833654
$408$	0	0
$409$	−2026.69	−0.245020	−0.122510	−	0.992467i	$-0.539094\pi$
$409$	−2026.69	−0.245020	−0.122510	+	0.992467i	$0.539094\pi$
$410$	0	0
$411$	−482.169	−0.0578678
$412$	0	0
$413$	7840.07	0.934104
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−171.796	−0.0201748
$418$	0	0
$419$	−1670.39	−0.194759	−0.0973793	−	0.995247i	$-0.531046\pi$
$419$	−1670.39	−0.194759	−0.0973793	+	0.995247i	$0.531046\pi$
$420$	0	0
$421$	6079.66	0.703812	0.351906	−	0.936035i	$-0.385534\pi$
$421$	6079.66	0.703812	0.351906	+	0.936035i	$0.385534\pi$
$422$	0	0
$423$	5373.08	0.617608
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17026.4	−1.92966
$428$	0	0
$429$	2895.68	0.325886
$430$	0	0
$431$	1719.37	0.192155	0.0960777	−	0.995374i	$-0.469370\pi$
$431$	1719.37	0.192155	0.0960777	+	0.995374i	$0.469370\pi$
$432$	0	0
$433$	12024.9	1.33459	0.667296	−	0.744792i	$-0.267451\pi$
$433$	12024.9	1.33459	0.667296	+	0.744792i	$0.267451\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5638.75	0.617249
$438$	0	0
$439$	−7542.97	−0.820060	−0.410030	−	0.912072i	$-0.634482\pi$
$439$	−7542.97	−0.820060	−0.410030	+	0.912072i	$0.634482\pi$
$440$	0	0
$441$	7984.32	0.862145
$442$	0	0
$443$	−4578.13	−0.491001	−0.245501	−	0.969396i	$-0.578952\pi$
$443$	−4578.13	−0.491001	−0.245501	+	0.969396i	$0.578952\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3315.45	−0.350817
$448$	0	0
$449$	6875.21	0.722630	0.361315	−	0.932444i	$-0.382328\pi$
$449$	6875.21	0.722630	0.361315	+	0.932444i	$0.382328\pi$
$450$	0	0
$451$	2663.02	0.278042
$452$	0	0
$453$	6868.90	0.712427
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7351.97	−0.752540	−0.376270	−	0.926510i	$-0.622793\pi$
$457$	−7351.97	−0.752540	−0.376270	+	0.926510i	$0.622793\pi$
$458$	0	0
$459$	−2586.31	−0.263003
$460$	0	0
$461$	−15614.5	−1.57752	−0.788762	−	0.614698i	$-0.789278\pi$
$461$	−15614.5	−1.57752	−0.788762	+	0.614698i	$0.789278\pi$
$462$	0	0
$463$	−1684.73	−0.169106	−0.0845530	−	0.996419i	$-0.526946\pi$
$463$	−1684.73	−0.169106	−0.0845530	+	0.996419i	$0.526946\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10235.2	1.01420	0.507099	−	0.861888i	$-0.330718\pi$
$467$	10235.2	1.01420	0.507099	+	0.861888i	$0.330718\pi$
$468$	0	0
$469$	30730.2	3.02556
$470$	0	0
$471$	−484.542	−0.0474024
$472$	0	0
$473$	2769.36	0.269207
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2750.78	−0.264045
$478$	0	0
$479$	12247.2	1.16825	0.584123	−	0.811665i	$-0.301439\pi$
$479$	12247.2	1.16825	0.584123	+	0.811665i	$0.301439\pi$
$480$	0	0
$481$	10048.4	0.952532
$482$	0	0
$483$	−11665.0	−1.09891
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7895.91	−0.734698	−0.367349	−	0.930083i	$-0.619735\pi$
$487$	−7895.91	−0.734698	−0.367349	+	0.930083i	$0.619735\pi$
$488$	0	0
$489$	4782.44	0.442269
$490$	0	0
$491$	−7625.48	−0.700882	−0.350441	−	0.936585i	$-0.613968\pi$
$491$	−7625.48	−0.700882	−0.350441	+	0.936585i	$0.613968\pi$
$492$	0	0
$493$	5220.96	0.476958
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20538.9	−1.85371
$498$	0	0
$499$	−8655.23	−0.776476	−0.388238	−	0.921559i	$-0.626916\pi$
$499$	−8655.23	−0.776476	−0.388238	+	0.921559i	$0.626916\pi$
$500$	0	0
$501$	3052.18	0.272178
$502$	0	0
$503$	118.441	0.0104990	0.00524951	−	0.999986i	$-0.498329\pi$
$503$	118.441	0.0104990	0.00524951	+	0.999986i	$0.498329\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2340.20	−0.204994
$508$	0	0
$509$	5359.43	0.466704	0.233352	−	0.972392i	$-0.425030\pi$
$509$	5359.43	0.466704	0.233352	+	0.972392i	$0.425030\pi$
$510$	0	0
$511$	39899.5	3.45411
$512$	0	0
$513$	1373.29	0.118191
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15308.6	1.30227
$518$	0	0
$519$	−7333.47	−0.620238
$520$	0	0
$521$	−10862.4	−0.913414	−0.456707	−	0.889617i	$-0.650971\pi$
$521$	−10862.4	−0.913414	−0.456707	+	0.889617i	$0.650971\pi$
$522$	0	0
$523$	−9553.39	−0.798740	−0.399370	−	0.916790i	$-0.630771\pi$
$523$	−9553.39	−0.798740	−0.399370	+	0.916790i	$0.630771\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19008.5	−1.57120
$528$	0	0
$529$	123.501	0.0101505
$530$	0	0
$531$	−2011.80	−0.164415
$532$	0	0
$533$	3909.26	0.317690
$534$	0	0
$535$	0	0
$536$	0	0
$537$	668.339	0.0537076
$538$	0	0
$539$	22748.4	1.81789
$540$	0	0
$541$	−6132.47	−0.487348	−0.243674	−	0.969857i	$-0.578353\pi$
$541$	−6132.47	−0.487348	−0.243674	+	0.969857i	$0.578353\pi$
$542$	0	0
$543$	−300.101	−0.0237174
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2853.85	−0.223075	−0.111537	−	0.993760i	$-0.535578\pi$
$547$	−2853.85	−0.223075	−0.111537	+	0.993760i	$0.535578\pi$
$548$	0	0
$549$	4369.05	0.339648
$550$	0	0
$551$	−2772.25	−0.214341
$552$	0	0
$553$	−24025.6	−1.84751
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5192.60	0.395005	0.197502	−	0.980302i	$-0.436717\pi$
$557$	5192.60	0.395005	0.197502	+	0.980302i	$0.436717\pi$
$558$	0	0
$559$	4065.36	0.307596
$560$	0	0
$561$	−7368.72	−0.554559
$562$	0	0
$563$	−10907.2	−0.816492	−0.408246	−	0.912872i	$-0.633859\pi$
$563$	−10907.2	−0.816492	−0.408246	+	0.912872i	$0.633859\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2840.95	−0.210421
$568$	0	0
$569$	−155.257	−0.0114389	−0.00571945	−	0.999984i	$-0.501821\pi$
$569$	−155.257	−0.0114389	−0.00571945	+	0.999984i	$0.501821\pi$
$570$	0	0
$571$	−4925.15	−0.360965	−0.180483	−	0.983578i	$-0.557766\pi$
$571$	−4925.15	−0.360965	−0.180483	+	0.983578i	$0.557766\pi$
$572$	0	0
$573$	2107.21	0.153630
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5292.05	0.381822	0.190911	−	0.981607i	$-0.438856\pi$
$577$	5292.05	0.381822	0.190911	+	0.981607i	$0.438856\pi$
$578$	0	0
$579$	−12379.9	−0.888585
$580$	0	0
$581$	−10722.8	−0.765677
$582$	0	0
$583$	−7837.33	−0.556757
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19658.5	1.38227	0.691134	−	0.722727i	$-0.257112\pi$
$587$	19658.5	1.38227	0.691134	+	0.722727i	$0.257112\pi$
$588$	0	0
$589$	10093.2	0.706083
$590$	0	0
$591$	9313.36	0.648224
$592$	0	0
$593$	−6578.08	−0.455530	−0.227765	−	0.973716i	$-0.573142\pi$
$593$	−6578.08	−0.455530	−0.227765	+	0.973716i	$0.573142\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1102.85	−0.0756056
$598$	0	0
$599$	16915.9	1.15386	0.576931	−	0.816793i	$-0.304250\pi$
$599$	16915.9	1.15386	0.576931	+	0.816793i	$0.304250\pi$
$600$	0	0
$601$	19801.4	1.34395	0.671977	−	0.740572i	$-0.265445\pi$
$601$	19801.4	1.34395	0.671977	+	0.740572i	$0.265445\pi$
$602$	0	0
$603$	−7885.49	−0.532541
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22498.4	1.50442	0.752208	−	0.658926i	$-0.228989\pi$
$607$	22498.4	1.50442	0.752208	+	0.658926i	$0.228989\pi$
$608$	0	0
$609$	5735.01	0.381600
$610$	0	0
$611$	22472.7	1.48797
$612$	0	0
$613$	23829.3	1.57008	0.785039	−	0.619446i	$-0.212643\pi$
$613$	23829.3	1.57008	0.785039	+	0.619446i	$0.212643\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4277.52	0.279103	0.139551	−	0.990215i	$-0.455434\pi$
$617$	4277.52	0.279103	0.139551	+	0.990215i	$0.455434\pi$
$618$	0	0
$619$	4995.99	0.324404	0.162202	−	0.986758i	$-0.448140\pi$
$619$	4995.99	0.324404	0.162202	+	0.986758i	$0.448140\pi$
$620$	0	0
$621$	2993.29	0.193424
$622$	0	0
$623$	−31116.3	−2.00104
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3912.68	0.249214
$628$	0	0
$629$	−25570.5	−1.62092
$630$	0	0
$631$	−11328.0	−0.714675	−0.357337	−	0.933975i	$-0.616315\pi$
$631$	−11328.0	−0.714675	−0.357337	+	0.933975i	$0.616315\pi$
$632$	0	0
$633$	7784.54	0.488796
$634$	0	0
$635$	0	0
$636$	0	0
$637$	33394.1	2.07712
$638$	0	0
$639$	5270.37	0.326280
$640$	0	0
$641$	5955.35	0.366961	0.183481	−	0.983023i	$-0.441264\pi$
$641$	5955.35	0.366961	0.183481	+	0.983023i	$0.441264\pi$
$642$	0	0
$643$	15727.7	0.964605	0.482302	−	0.876005i	$-0.339801\pi$
$643$	15727.7	0.964605	0.482302	+	0.876005i	$0.339801\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9582.24	−0.582252	−0.291126	−	0.956685i	$-0.594030\pi$
$647$	−9582.24	−0.582252	−0.291126	+	0.956685i	$0.594030\pi$
$648$	0	0
$649$	−5731.87	−0.346681
$650$	0	0
$651$	−20880.0	−1.25707
$652$	0	0
$653$	8313.70	0.498224	0.249112	−	0.968475i	$-0.419861\pi$
$653$	8313.70	0.498224	0.249112	+	0.968475i	$0.419861\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10238.4	−0.607971
$658$	0	0
$659$	15095.5	0.892317	0.446158	−	0.894954i	$-0.352792\pi$
$659$	15095.5	0.892317	0.446158	+	0.894954i	$0.352792\pi$
$660$	0	0
$661$	8266.98	0.486457	0.243229	−	0.969969i	$-0.421793\pi$
$661$	8266.98	0.486457	0.243229	+	0.969969i	$0.421793\pi$
$662$	0	0
$663$	−10817.1	−0.633639
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6042.53	−0.350776
$668$	0	0
$669$	−5503.23	−0.318038
$670$	0	0
$671$	12448.0	0.716170
$672$	0	0
$673$	11186.7	0.640735	0.320367	−	0.947293i	$-0.396194\pi$
$673$	11186.7	0.640735	0.320367	+	0.947293i	$0.396194\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7675.84	0.435756	0.217878	−	0.975976i	$-0.430087\pi$
$677$	7675.84	0.435756	0.217878	+	0.975976i	$0.430087\pi$
$678$	0	0
$679$	19520.1	1.10326
$680$	0	0
$681$	527.434	0.0296789
$682$	0	0
$683$	10550.9	0.591099	0.295549	−	0.955327i	$-0.404497\pi$
$683$	10550.9	0.591099	0.295549	+	0.955327i	$0.404497\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4867.06	0.270291
$688$	0	0
$689$	−11505.0	−0.636150
$690$	0	0
$691$	−26950.9	−1.48373	−0.741867	−	0.670547i	$-0.766060\pi$
$691$	−26950.9	−1.48373	−0.741867	+	0.670547i	$0.766060\pi$
$692$	0	0
$693$	−8094.24	−0.443686
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9947.99	−0.540613
$698$	0	0
$699$	11897.0	0.643754
$700$	0	0
$701$	12791.6	0.689204	0.344602	−	0.938749i	$-0.388014\pi$
$701$	12791.6	0.689204	0.344602	+	0.938749i	$0.388014\pi$
$702$	0	0
$703$	13577.5	0.728429
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−55805.1	−2.96855
$708$	0	0
$709$	−16238.3	−0.860142	−0.430071	−	0.902795i	$-0.641511\pi$
$709$	−16238.3	−0.860142	−0.430071	+	0.902795i	$0.641511\pi$
$710$	0	0
$711$	6165.08	0.325188
$712$	0	0
$713$	21999.6	1.15553
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18969.8	−0.988060
$718$	0	0
$719$	−24285.8	−1.25968	−0.629839	−	0.776726i	$-0.716879\pi$
$719$	−24285.8	−1.25968	−0.629839	+	0.776726i	$0.716879\pi$
$720$	0	0
$721$	−47382.3	−2.44745
$722$	0	0
$723$	10222.5	0.525836
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4466.82	0.227875	0.113937	−	0.993488i	$-0.463654\pi$
$727$	4466.82	0.227875	0.113937	+	0.993488i	$0.463654\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−10345.2	−0.523436
$732$	0	0
$733$	−30802.1	−1.55212	−0.776059	−	0.630661i	$-0.782784\pi$
$733$	−30802.1	−1.55212	−0.776059	+	0.630661i	$0.782784\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22466.8	−1.12290
$738$	0	0
$739$	−12920.5	−0.643151	−0.321576	−	0.946884i	$-0.604212\pi$
$739$	−12920.5	−0.643151	−0.321576	+	0.946884i	$0.604212\pi$
$740$	0	0
$741$	5743.73	0.284752
$742$	0	0
$743$	2571.29	0.126960	0.0634802	−	0.997983i	$-0.479780\pi$
$743$	2571.29	0.126960	0.0634802	+	0.997983i	$0.479780\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2751.53	0.134770
$748$	0	0
$749$	−46770.7	−2.28166
$750$	0	0
$751$	−13427.4	−0.652426	−0.326213	−	0.945296i	$-0.605773\pi$
$751$	−13427.4	−0.652426	−0.326213	+	0.945296i	$0.605773\pi$
$752$	0	0
$753$	4037.42	0.195394
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7103.66	0.341066	0.170533	−	0.985352i	$-0.445451\pi$
$757$	7103.66	0.341066	0.170533	+	0.985352i	$0.445451\pi$
$758$	0	0
$759$	8528.27	0.407848
$760$	0	0
$761$	19205.2	0.914831	0.457416	−	0.889253i	$-0.348775\pi$
$761$	19205.2	0.914831	0.457416	+	0.889253i	$0.348775\pi$
$762$	0	0
$763$	21391.6	1.01498
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8414.27	−0.396117
$768$	0	0
$769$	19508.1	0.914799	0.457399	−	0.889261i	$-0.348781\pi$
$769$	19508.1	0.914799	0.457399	+	0.889261i	$0.348781\pi$
$770$	0	0
$771$	14091.6	0.658230
$772$	0	0
$773$	38852.5	1.80780	0.903899	−	0.427746i	$-0.140692\pi$
$773$	38852.5	1.80780	0.903899	+	0.427746i	$0.140692\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−28088.1	−1.29685
$778$	0	0
$779$	5282.23	0.242947
$780$	0	0
$781$	15016.0	0.687982
$782$	0	0
$783$	−1471.63	−0.0671669
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24851.1	1.12560	0.562799	−	0.826594i	$-0.309725\pi$
$787$	24851.1	1.12560	0.562799	+	0.826594i	$0.309725\pi$
$788$	0	0
$789$	14102.0	0.636304
$790$	0	0
$791$	−8481.04	−0.381228
$792$	0	0
$793$	18273.4	0.818295
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33040.1	1.46843	0.734216	−	0.678916i	$-0.237550\pi$
$797$	33040.1	1.46843	0.734216	+	0.678916i	$0.237550\pi$
$798$	0	0
$799$	−57187.0	−2.53208
$800$	0	0
$801$	7984.58	0.352211
$802$	0	0
$803$	−29170.5	−1.28195
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23888.0	−1.04200
$808$	0	0
$809$	10756.6	0.467467	0.233734	−	0.972301i	$-0.424906\pi$
$809$	10756.6	0.467467	0.233734	+	0.972301i	$0.424906\pi$
$810$	0	0
$811$	−5985.48	−0.259160	−0.129580	−	0.991569i	$-0.541363\pi$
$811$	−5985.48	−0.259160	−0.129580	+	0.991569i	$0.541363\pi$
$812$	0	0
$813$	−18368.2	−0.792375
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5493.15	0.235228
$818$	0	0
$819$	−11882.2	−0.506956
$820$	0	0
$821$	16426.4	0.698276	0.349138	−	0.937071i	$-0.386474\pi$
$821$	16426.4	0.698276	0.349138	+	0.937071i	$0.386474\pi$
$822$	0	0
$823$	27124.3	1.14884	0.574419	−	0.818561i	$-0.305228\pi$
$823$	27124.3	1.14884	0.574419	+	0.818561i	$0.305228\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32759.9	1.37748	0.688738	−	0.725011i	$-0.258165\pi$
$827$	32759.9	1.37748	0.688738	+	0.725011i	$0.258165\pi$
$828$	0	0
$829$	−18211.8	−0.762995	−0.381498	−	0.924370i	$-0.624592\pi$
$829$	−18211.8	−0.762995	−0.381498	+	0.924370i	$0.624592\pi$
$830$	0	0
$831$	−25252.7	−1.05416
$832$	0	0
$833$	−84979.0	−3.53463
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5357.90	0.221262
$838$	0	0
$839$	−37500.6	−1.54310	−0.771552	−	0.636166i	$-0.780519\pi$
$839$	−37500.6	−1.54310	−0.771552	+	0.636166i	$0.780519\pi$
$840$	0	0
$841$	−21418.2	−0.878192
$842$	0	0
$843$	9092.98	0.371505
$844$	0	0
$845$	0	0
$846$	0	0
$847$	23621.2	0.958246
$848$	0	0
$849$	8672.43	0.350574
$850$	0	0
$851$	29594.2	1.19210
$852$	0	0
$853$	−2421.81	−0.0972112	−0.0486056	−	0.998818i	$-0.515478\pi$
$853$	−2421.81	−0.0972112	−0.0486056	+	0.998818i	$0.515478\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37700.0	−1.50269	−0.751346	−	0.659908i	$-0.770595\pi$
$857$	−37700.0	−1.50269	−0.751346	+	0.659908i	$0.770595\pi$
$858$	0	0
$859$	−14709.0	−0.584242	−0.292121	−	0.956381i	$-0.594361\pi$
$859$	−14709.0	−0.584242	−0.292121	+	0.956381i	$0.594361\pi$
$860$	0	0
$861$	−10927.5	−0.432528
$862$	0	0
$863$	28950.4	1.14193	0.570963	−	0.820975i	$-0.306570\pi$
$863$	28950.4	1.14193	0.570963	+	0.820975i	$0.306570\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12787.6	0.500913
$868$	0	0
$869$	17565.1	0.685681
$870$	0	0
$871$	−32980.8	−1.28302
$872$	0	0
$873$	−5008.95	−0.194189
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17501.3	0.673860	0.336930	−	0.941530i	$-0.390611\pi$
$877$	17501.3	0.673860	0.336930	+	0.941530i	$0.390611\pi$
$878$	0	0
$879$	−26900.2	−1.03222
$880$	0	0
$881$	−45136.0	−1.72607	−0.863037	−	0.505140i	$-0.831441\pi$
$881$	−45136.0	−1.72607	−0.863037	+	0.505140i	$0.831441\pi$
$882$	0	0
$883$	−51564.8	−1.96522	−0.982612	−	0.185673i	$-0.940554\pi$
$883$	−51564.8	−1.96522	−0.982612	+	0.185673i	$0.940554\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19226.4	0.727800	0.363900	−	0.931438i	$-0.381445\pi$
$887$	19226.4	0.727800	0.363900	+	0.931438i	$0.381445\pi$
$888$	0	0
$889$	36684.6	1.38398
$890$	0	0
$891$	2077.02	0.0780950
$892$	0	0
$893$	30365.4	1.13789
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12519.3	0.466007
$898$	0	0
$899$	−10816.0	−0.401259
$900$	0	0
$901$	29277.2	1.08254
$902$	0	0
$903$	−11363.8	−0.418786
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32050.1	1.17333	0.586663	−	0.809831i	$-0.300441\pi$
$907$	32050.1	1.17333	0.586663	+	0.809831i	$0.300441\pi$
$908$	0	0
$909$	14319.8	0.522507
$910$	0	0
$911$	−29674.6	−1.07921	−0.539606	−	0.841918i	$-0.681427\pi$
$911$	−29674.6	−1.07921	−0.539606	+	0.841918i	$0.681427\pi$
$912$	0	0
$913$	7839.46	0.284171
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31093.4	1.11973
$918$	0	0
$919$	−6029.99	−0.216443	−0.108221	−	0.994127i	$-0.534516\pi$
$919$	−6029.99	−0.216443	−0.108221	+	0.994127i	$0.534516\pi$
$920$	0	0
$921$	−9895.25	−0.354028
$922$	0	0
$923$	22043.2	0.786088
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12158.5	0.430785
$928$	0	0
$929$	−35365.2	−1.24897	−0.624485	−	0.781037i	$-0.714691\pi$
$929$	−35365.2	−1.24897	−0.624485	+	0.781037i	$0.714691\pi$
$930$	0	0
$931$	45122.5	1.58843
$932$	0	0
$933$	−10184.8	−0.357379
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35637.0	1.24249	0.621243	−	0.783618i	$-0.286628\pi$
$937$	35637.0	1.24249	0.621243	+	0.783618i	$0.286628\pi$
$938$	0	0
$939$	−17840.9	−0.620037
$940$	0	0
$941$	−6609.70	−0.228980	−0.114490	−	0.993424i	$-0.536523\pi$
$941$	−6609.70	−0.228980	−0.114490	+	0.993424i	$0.536523\pi$
$942$	0	0
$943$	11513.4	0.397591
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42650.9	−1.46354	−0.731768	−	0.681553i	$-0.761305\pi$
$947$	−42650.9	−1.46354	−0.731768	+	0.681553i	$0.761305\pi$
$948$	0	0
$949$	−42821.6	−1.46475
$950$	0	0
$951$	7562.48	0.257866
$952$	0	0
$953$	−38382.4	−1.30465	−0.652323	−	0.757941i	$-0.726205\pi$
$953$	−38382.4	−1.30465	−0.652323	+	0.757941i	$0.726205\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4192.86	−0.141626
$958$	0	0
$959$	5637.11	0.189814
$960$	0	0
$961$	9587.71	0.321832
$962$	0	0
$963$	12001.6	0.401604
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2611.76	0.0868549	0.0434275	−	0.999057i	$-0.486172\pi$
$967$	2611.76	0.0868549	0.0434275	+	0.999057i	$0.486172\pi$
$968$	0	0
$969$	−14616.2	−0.484562
$970$	0	0
$971$	7846.09	0.259313	0.129657	−	0.991559i	$-0.458613\pi$
$971$	7846.09	0.259313	0.129657	+	0.991559i	$0.458613\pi$
$972$	0	0
$973$	2008.50	0.0661763
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−35667.2	−1.16796	−0.583979	−	0.811769i	$-0.698505\pi$
$977$	−35667.2	−1.16796	−0.583979	+	0.811769i	$0.698505\pi$
$978$	0	0
$979$	22749.1	0.742661
$980$	0	0
$981$	−5489.19	−0.178651
$982$	0	0
$983$	−13421.6	−0.435485	−0.217742	−	0.976006i	$-0.569869\pi$
$983$	−13421.6	−0.435485	−0.217742	+	0.976006i	$0.569869\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−62817.5	−2.02584
$988$	0	0
$989$	11973.2	0.384959
$990$	0	0
$991$	40673.7	1.30378	0.651888	−	0.758315i	$-0.273977\pi$
$991$	40673.7	1.30378	0.651888	+	0.758315i	$0.273977\pi$
$992$	0	0
$993$	−13760.6	−0.439757
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29558.8	−0.938954	−0.469477	−	0.882945i	$-0.655558\pi$
$997$	−29558.8	−0.938954	−0.469477	+	0.882945i	$0.655558\pi$
$998$	0	0
$999$	7207.53	0.228264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.a.v.1.1		2
3.2	odd	2		1800.4.a.bl.1.1		2
4.3	odd	2		1200.4.a.bo.1.2		2
5.2	odd	4		120.4.f.d.49.1	✓	4
5.3	odd	4		120.4.f.d.49.3	yes	4
5.4	even	2		600.4.a.t.1.2		2
15.2	even	4		360.4.f.d.289.3		4
15.8	even	4		360.4.f.d.289.4		4
15.14	odd	2		1800.4.a.bn.1.2		2
20.3	even	4		240.4.f.g.49.1		4
20.7	even	4		240.4.f.g.49.3		4
20.19	odd	2		1200.4.a.bq.1.1		2
40.3	even	4		960.4.f.o.769.4		4
40.13	odd	4		960.4.f.n.769.2		4
40.27	even	4		960.4.f.o.769.2		4
40.37	odd	4		960.4.f.n.769.4		4
60.23	odd	4		720.4.f.i.289.4		4
60.47	odd	4		720.4.f.i.289.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.d.49.1	✓	4	5.2	odd	4
120.4.f.d.49.3	yes	4	5.3	odd	4
240.4.f.g.49.1		4	20.3	even	4
240.4.f.g.49.3		4	20.7	even	4
360.4.f.d.289.3		4	15.2	even	4
360.4.f.d.289.4		4	15.8	even	4
600.4.a.t.1.2		2	5.4	even	2
600.4.a.v.1.1		2	1.1	even	1	trivial
720.4.f.i.289.3		4	60.47	odd	4
720.4.f.i.289.4		4	60.23	odd	4
960.4.f.n.769.2		4	40.13	odd	4
960.4.f.n.769.4		4	40.37	odd	4
960.4.f.o.769.2		4	40.27	even	4
960.4.f.o.769.4		4	40.3	even	4
1200.4.a.bo.1.2		2	4.3	odd	2
1200.4.a.bq.1.1		2	20.19	odd	2
1800.4.a.bl.1.1		2	3.2	odd	2
1800.4.a.bn.1.2		2	15.14	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -35.0735 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -35.0735 q^{7} +9.00000 q^{9} +25.6422 q^{11} +37.6422 q^{13} -95.7891 q^{17} +50.8625 q^{19} -105.220 q^{21} +110.863 q^{23} +27.0000 q^{27} -54.5047 q^{29} +198.441 q^{31} +76.9265 q^{33} +266.945 q^{37} +112.927 q^{39} +103.853 q^{41} +108.000 q^{43} +597.009 q^{47} +887.147 q^{49} -287.367 q^{51} -305.642 q^{53} +152.588 q^{57} -223.533 q^{59} +485.450 q^{61} -315.661 q^{63} -876.166 q^{67} +332.588 q^{69} +585.597 q^{71} -1137.60 q^{73} -899.360 q^{77} +685.009 q^{79} +81.0000 q^{81} +305.725 q^{83} -163.514 q^{87} +887.175 q^{89} -1320.24 q^{91} +595.322 q^{93} -556.550 q^{97} +230.780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} + 74 q^{11} + 98 q^{13} - 78 q^{17} - 80 q^{19} - 6 q^{21} + 40 q^{23} + 54 q^{27} + 50 q^{29} - 12 q^{31} + 222 q^{33} - 34 q^{37} + 294 q^{39} + 344 q^{41} + 216 q^{43} + 876 q^{47} + 1638 q^{49} - 234 q^{51} - 634 q^{53} - 240 q^{57} + 666 q^{59} + 244 q^{61} - 18 q^{63} - 980 q^{67} + 120 q^{69} + 308 q^{71} - 1412 q^{73} + 700 q^{77} + 1052 q^{79} + 162 q^{81} + 248 q^{83} + 150 q^{87} + 684 q^{89} + 676 q^{91} - 36 q^{93} - 1840 q^{97} + 666 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 + 74 * q^11 + 98 * q^13 - 78 * q^17 - 80 * q^19 - 6 * q^21 + 40 * q^23 + 54 * q^27 + 50 * q^29 - 12 * q^31 + 222 * q^33 - 34 * q^37 + 294 * q^39 + 344 * q^41 + 216 * q^43 + 876 * q^47 + 1638 * q^49 - 234 * q^51 - 634 * q^53 - 240 * q^57 + 666 * q^59 + 244 * q^61 - 18 * q^63 - 980 * q^67 + 120 * q^69 + 308 * q^71 - 1412 * q^73 + 700 * q^77 + 1052 * q^79 + 162 * q^81 + 248 * q^83 + 150 * q^87 + 684 * q^89 + 676 * q^91 - 36 * q^93 - 1840 * q^97 + 666 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more