LMFDB - Embedded newform 2205.4.a.ca.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,4,Mod(1,2205)] 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("2205.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,2,0,14,30,0,0,66,0,10,16,0,168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1163891200.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.05886$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.644648 q^{2} -7.58443 q^{4} +5.00000 q^{5} +10.0465 q^{8} -3.22324 q^{10} +47.7013 q^{11} +57.2256 q^{13} +54.1990 q^{16} +36.9686 q^{17} +30.7659 q^{19} -37.9221 q^{20} -30.7506 q^{22} -53.1282 q^{23} +25.0000 q^{25} -36.8904 q^{26} +195.663 q^{29} +257.870 q^{31} -115.311 q^{32} -23.8317 q^{34} +346.423 q^{37} -19.8332 q^{38} +50.2324 q^{40} -267.050 q^{41} -176.859 q^{43} -361.787 q^{44} +34.2490 q^{46} +311.598 q^{47} -16.1162 q^{50} -434.024 q^{52} +492.270 q^{53} +238.507 q^{55} -126.134 q^{58} -98.7653 q^{59} +82.1682 q^{61} -166.235 q^{62} -359.257 q^{64} +286.128 q^{65} +654.668 q^{67} -280.386 q^{68} -779.658 q^{71} -829.673 q^{73} -223.321 q^{74} -233.342 q^{76} -769.426 q^{79} +270.995 q^{80} +172.153 q^{82} -613.203 q^{83} +184.843 q^{85} +114.012 q^{86} +479.230 q^{88} +457.666 q^{89} +402.947 q^{92} -200.871 q^{94} +153.830 q^{95} -1412.11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31}+ \cdots + 516 q^{97}+O(q^{100}) $ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

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 $

$2$	−0.644648	−0.227917	−0.113959	−	0.993485i	$-0.536353\pi$
$2$	−0.644648	−0.227917	−0.113959	+	0.993485i	$0.536353\pi$
$3$	0	0
$3$	0	0
$4$	−7.58443	−0.948054
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	10.0465	0.443995
$9$	0	0
$10$	−3.22324	−0.101928
$11$	47.7013	1.30750	0.653750	−	0.756711i	$-0.273195\pi$
$11$	47.7013	1.30750	0.653750	+	0.756711i	$0.273195\pi$
$12$	0	0
$13$	57.2256	1.22089	0.610443	−	0.792060i	$-0.290991\pi$
$13$	57.2256	1.22089	0.610443	+	0.792060i	$0.290991\pi$
$14$	0	0
$15$	0	0
$16$	54.1990	0.846859
$17$	36.9686	0.527423	0.263712	−	0.964602i	$-0.415053\pi$
$17$	36.9686	0.527423	0.263712	+	0.964602i	$0.415053\pi$
$18$	0	0
$19$	30.7659	0.371484	0.185742	−	0.982599i	$-0.440531\pi$
$19$	30.7659	0.371484	0.185742	+	0.982599i	$0.440531\pi$
$20$	−37.9221	−0.423982
$21$	0	0
$22$	−30.7506	−0.298002
$23$	−53.1282	−0.481652	−0.240826	−	0.970568i	$-0.577418\pi$
$23$	−53.1282	−0.481652	−0.240826	+	0.970568i	$0.577418\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−36.8904	−0.278261
$27$	0	0
$28$	0	0
$29$	195.663	1.25288	0.626442	−	0.779468i	$-0.284510\pi$
$29$	195.663	1.25288	0.626442	+	0.779468i	$0.284510\pi$
$30$	0	0
$31$	257.870	1.49403	0.747014	−	0.664809i	$-0.231487\pi$
$31$	257.870	1.49403	0.747014	+	0.664809i	$0.231487\pi$
$32$	−115.311	−0.637010
$33$	0	0
$34$	−23.8317	−0.120209
$35$	0	0
$36$	0	0
$37$	346.423	1.53923	0.769616	−	0.638507i	$-0.220447\pi$
$37$	346.423	1.53923	0.769616	+	0.638507i	$0.220447\pi$
$38$	−19.8332	−0.0846676
$39$	0	0
$40$	50.2324	0.198561
$41$	−267.050	−1.01722	−0.508611	−	0.860996i	$-0.669841\pi$
$41$	−267.050	−1.01722	−0.508611	+	0.860996i	$0.669841\pi$
$42$	0	0
$43$	−176.859	−0.627227	−0.313614	−	0.949551i	$-0.601540\pi$
$43$	−176.859	−0.627227	−0.313614	+	0.949551i	$0.601540\pi$
$44$	−361.787	−1.23958
$45$	0	0
$46$	34.2490	0.109777
$47$	311.598	0.967049	0.483524	−	0.875331i	$-0.339356\pi$
$47$	311.598	0.967049	0.483524	+	0.875331i	$0.339356\pi$
$48$	0	0
$49$	0	0
$50$	−16.1162	−0.0455835
$51$	0	0
$52$	−434.024	−1.15747
$53$	492.270	1.27582	0.637910	−	0.770111i	$-0.279799\pi$
$53$	492.270	1.27582	0.637910	+	0.770111i	$0.279799\pi$
$54$	0	0
$55$	238.507	0.584731
$56$	0	0
$57$	0	0
$58$	−126.134	−0.285554
$59$	−98.7653	−0.217935	−0.108967	−	0.994045i	$-0.534754\pi$
$59$	−98.7653	−0.217935	−0.108967	+	0.994045i	$0.534754\pi$
$60$	0	0
$61$	82.1682	0.172468	0.0862340	−	0.996275i	$-0.472517\pi$
$61$	82.1682	0.172468	0.0862340	+	0.996275i	$0.472517\pi$
$62$	−166.235	−0.340515
$63$	0	0
$64$	−359.257	−0.701674
$65$	286.128	0.545997
$66$	0	0
$67$	654.668	1.19374	0.596869	−	0.802339i	$-0.296411\pi$
$67$	654.668	1.19374	0.596869	+	0.802339i	$0.296411\pi$
$68$	−280.386	−0.500026
$69$	0	0
$70$	0	0
$71$	−779.658	−1.30322	−0.651608	−	0.758556i	$-0.725905\pi$
$71$	−779.658	−1.30322	−0.651608	+	0.758556i	$0.725905\pi$
$72$	0	0
$73$	−829.673	−1.33022	−0.665109	−	0.746747i	$-0.731615\pi$
$73$	−829.673	−1.33022	−0.665109	+	0.746747i	$0.731615\pi$
$74$	−223.321	−0.350818
$75$	0	0
$76$	−233.342	−0.352186
$77$	0	0
$78$	0	0
$79$	−769.426	−1.09579	−0.547894	−	0.836548i	$-0.684570\pi$
$79$	−769.426	−1.09579	−0.547894	+	0.836548i	$0.684570\pi$
$80$	270.995	0.378727
$81$	0	0
$82$	172.153	0.231843
$83$	−613.203	−0.810937	−0.405469	−	0.914109i	$-0.632892\pi$
$83$	−613.203	−0.810937	−0.405469	+	0.914109i	$0.632892\pi$
$84$	0	0
$85$	184.843	0.235871
$86$	114.012	0.142956
$87$	0	0
$88$	479.230	0.580524
$89$	457.666	0.545084	0.272542	−	0.962144i	$-0.412136\pi$
$89$	457.666	0.545084	0.272542	+	0.962144i	$0.412136\pi$
$90$	0	0
$91$	0	0
$92$	402.947	0.456632
$93$	0	0
$94$	−200.871	−0.220407
$95$	153.830	0.166132
$96$	0	0
$97$	−1412.11	−1.47813	−0.739063	−	0.673636i	$-0.764732\pi$
$97$	−1412.11	−1.47813	−0.739063	+	0.673636i	$0.764732\pi$
$98$	0	0
$99$	0	0
$100$	−189.611	−0.189611
$101$	−1823.79	−1.79677	−0.898386	−	0.439208i	$-0.855259\pi$
$101$	−1823.79	−1.79677	−0.898386	+	0.439208i	$0.855259\pi$
$102$	0	0
$103$	407.775	0.390090	0.195045	−	0.980794i	$-0.437515\pi$
$103$	407.775	0.390090	0.195045	+	0.980794i	$0.437515\pi$
$104$	574.915	0.542068
$105$	0	0
$106$	−317.341	−0.290782
$107$	370.109	0.334390	0.167195	−	0.985924i	$-0.446529\pi$
$107$	370.109	0.334390	0.167195	+	0.985924i	$0.446529\pi$
$108$	0	0
$109$	975.570	0.857272	0.428636	−	0.903477i	$-0.358994\pi$
$109$	975.570	0.857272	0.428636	+	0.903477i	$0.358994\pi$
$110$	−153.753	−0.133270
$111$	0	0
$112$	0	0
$113$	−1978.85	−1.64739	−0.823693	−	0.567036i	$-0.808090\pi$
$113$	−1978.85	−1.64739	−0.823693	+	0.567036i	$0.808090\pi$
$114$	0	0
$115$	−265.641	−0.215401
$116$	−1483.99	−1.18780
$117$	0	0
$118$	63.6689	0.0496711
$119$	0	0
$120$	0	0
$121$	944.416	0.709554
$122$	−52.9695	−0.0393085
$123$	0	0
$124$	−1955.80	−1.41642
$125$	125.000	0.0894427
$126$	0	0
$127$	1392.38	0.972867	0.486433	−	0.873718i	$-0.338298\pi$
$127$	1392.38	0.972867	0.486433	+	0.873718i	$0.338298\pi$
$128$	1154.08	0.796933
$129$	0	0
$130$	−184.452	−0.124442
$131$	1777.02	1.18518	0.592591	−	0.805504i	$-0.298105\pi$
$131$	1777.02	1.18518	0.592591	+	0.805504i	$0.298105\pi$
$132$	0	0
$133$	0	0
$134$	−422.031	−0.272074
$135$	0	0
$136$	371.404	0.234174
$137$	1980.48	1.23506	0.617532	−	0.786545i	$-0.288132\pi$
$137$	1980.48	1.23506	0.617532	+	0.786545i	$0.288132\pi$
$138$	0	0
$139$	2182.09	1.33153	0.665763	−	0.746163i	$-0.268106\pi$
$139$	2182.09	1.33153	0.665763	+	0.746163i	$0.268106\pi$
$140$	0	0
$141$	0	0
$142$	502.605	0.297026
$143$	2729.74	1.59631
$144$	0	0
$145$	978.313	0.560307
$146$	534.847	0.303180
$147$	0	0
$148$	−2627.42	−1.45928
$149$	−670.154	−0.368464	−0.184232	−	0.982883i	$-0.558980\pi$
$149$	−670.154	−0.368464	−0.184232	+	0.982883i	$0.558980\pi$
$150$	0	0
$151$	3348.84	1.80480	0.902401	−	0.430898i	$-0.141803\pi$
$151$	3348.84	1.80480	0.902401	+	0.430898i	$0.141803\pi$
$152$	309.089	0.164937
$153$	0	0
$154$	0	0
$155$	1289.35	0.668149
$156$	0	0
$157$	2408.41	1.22428	0.612140	−	0.790749i	$-0.290309\pi$
$157$	2408.41	1.22428	0.612140	+	0.790749i	$0.290309\pi$
$158$	496.009	0.249749
$159$	0	0
$160$	−576.555	−0.284879
$161$	0	0
$162$	0	0
$163$	−3811.03	−1.83131	−0.915654	−	0.401968i	$-0.868326\pi$
$163$	−3811.03	−1.83131	−0.915654	+	0.401968i	$0.868326\pi$
$164$	2025.42	0.964382
$165$	0	0
$166$	395.300	0.184827
$167$	−1207.15	−0.559354	−0.279677	−	0.960094i	$-0.590227\pi$
$167$	−1207.15	−0.559354	−0.279677	+	0.960094i	$0.590227\pi$
$168$	0	0
$169$	1077.77	0.490564
$170$	−119.159	−0.0537591
$171$	0	0
$172$	1341.38	0.594645
$173$	−3243.28	−1.42533	−0.712665	−	0.701505i	$-0.752512\pi$
$173$	−3243.28	−1.42533	−0.712665	+	0.701505i	$0.752512\pi$
$174$	0	0
$175$	0	0
$176$	2585.36	1.10727
$177$	0	0
$178$	−295.034	−0.124234
$179$	−859.728	−0.358989	−0.179495	−	0.983759i	$-0.557446\pi$
$179$	−859.728	−0.358989	−0.179495	+	0.983759i	$0.557446\pi$
$180$	0	0
$181$	−290.504	−0.119298	−0.0596491	−	0.998219i	$-0.518998\pi$
$181$	−290.504	−0.119298	−0.0596491	+	0.998219i	$0.518998\pi$
$182$	0	0
$183$	0	0
$184$	−533.751	−0.213851
$185$	1732.12	0.688366
$186$	0	0
$187$	1763.45	0.689605
$188$	−2363.30	−0.916814
$189$	0	0
$190$	−99.1660	−0.0378645
$191$	−4895.61	−1.85463	−0.927315	−	0.374282i	$-0.877889\pi$
$191$	−4895.61	−1.85463	−0.927315	+	0.374282i	$0.877889\pi$
$192$	0	0
$193$	−3548.73	−1.32354	−0.661770	−	0.749707i	$-0.730194\pi$
$193$	−3548.73	−1.32354	−0.661770	+	0.749707i	$0.730194\pi$
$194$	910.315	0.336891
$195$	0	0
$196$	0	0
$197$	650.107	0.235118	0.117559	−	0.993066i	$-0.462493\pi$
$197$	650.107	0.235118	0.117559	+	0.993066i	$0.462493\pi$
$198$	0	0
$199$	4055.09	1.44451	0.722256	−	0.691626i	$-0.243105\pi$
$199$	4055.09	1.44451	0.722256	+	0.691626i	$0.243105\pi$
$200$	251.162	0.0887991
$201$	0	0
$202$	1175.70	0.409516
$203$	0	0
$204$	0	0
$205$	−1335.25	−0.454916
$206$	−262.871	−0.0889083
$207$	0	0
$208$	3101.57	1.03392
$209$	1467.58	0.485714
$210$	0	0
$211$	−1569.67	−0.512134	−0.256067	−	0.966659i	$-0.582427\pi$
$211$	−1569.67	−0.512134	−0.256067	+	0.966659i	$0.582427\pi$
$212$	−3733.59	−1.20955
$213$	0	0
$214$	−238.590	−0.0762134
$215$	−884.296	−0.280504
$216$	0	0
$217$	0	0
$218$	−628.900	−0.195387
$219$	0	0
$220$	−1808.94	−0.554357
$221$	2115.55	0.643924
$222$	0	0
$223$	4723.86	1.41853	0.709267	−	0.704940i	$-0.249026\pi$
$223$	4723.86	1.41853	0.709267	+	0.704940i	$0.249026\pi$
$224$	0	0
$225$	0	0
$226$	1275.66	0.375468
$227$	−3685.02	−1.07746	−0.538730	−	0.842479i	$-0.681096\pi$
$227$	−3685.02	−1.07746	−0.538730	+	0.842479i	$0.681096\pi$
$228$	0	0
$229$	3356.78	0.968656	0.484328	−	0.874887i	$-0.339064\pi$
$229$	3356.78	0.968656	0.484328	+	0.874887i	$0.339064\pi$
$230$	171.245	0.0490937
$231$	0	0
$232$	1965.72	0.556275
$233$	−2314.01	−0.650627	−0.325314	−	0.945606i	$-0.605470\pi$
$233$	−2314.01	−0.650627	−0.325314	+	0.945606i	$0.605470\pi$
$234$	0	0
$235$	1557.99	0.432477
$236$	749.079	0.206614
$237$	0	0
$238$	0	0
$239$	−941.179	−0.254727	−0.127364	−	0.991856i	$-0.540652\pi$
$239$	−941.179	−0.254727	−0.127364	+	0.991856i	$0.540652\pi$
$240$	0	0
$241$	5638.24	1.50702	0.753509	−	0.657438i	$-0.228360\pi$
$241$	5638.24	1.50702	0.753509	+	0.657438i	$0.228360\pi$
$242$	−608.816	−0.161720
$243$	0	0
$244$	−623.199	−0.163509
$245$	0	0
$246$	0	0
$247$	1760.60	0.453539
$248$	2590.69	0.663341
$249$	0	0
$250$	−80.5810	−0.0203856
$251$	−365.822	−0.0919940	−0.0459970	−	0.998942i	$-0.514646\pi$
$251$	−365.822	−0.0919940	−0.0459970	+	0.998942i	$0.514646\pi$
$252$	0	0
$253$	−2534.28	−0.629759
$254$	−897.598	−0.221733
$255$	0	0
$256$	2130.08	0.520039
$257$	6276.46	1.52340	0.761702	−	0.647928i	$-0.224364\pi$
$257$	6276.46	1.52340	0.761702	+	0.647928i	$0.224364\pi$
$258$	0	0
$259$	0	0
$260$	−2170.12	−0.517635
$261$	0	0
$262$	−1145.55	−0.270124
$263$	4225.97	0.990817	0.495408	−	0.868660i	$-0.335018\pi$
$263$	4225.97	0.990817	0.495408	+	0.868660i	$0.335018\pi$
$264$	0	0
$265$	2461.35	0.570564
$266$	0	0
$267$	0	0
$268$	−4965.28	−1.13173
$269$	−981.591	−0.222486	−0.111243	−	0.993793i	$-0.535483\pi$
$269$	−981.591	−0.222486	−0.111243	+	0.993793i	$0.535483\pi$
$270$	0	0
$271$	3884.42	0.870708	0.435354	−	0.900259i	$-0.356623\pi$
$271$	3884.42	0.870708	0.435354	+	0.900259i	$0.356623\pi$
$272$	2003.66	0.446653
$273$	0	0
$274$	−1276.71	−0.281493
$275$	1192.53	0.261500
$276$	0	0
$277$	3614.93	0.784116	0.392058	−	0.919941i	$-0.371763\pi$
$277$	3614.93	0.784116	0.392058	+	0.919941i	$0.371763\pi$
$278$	−1406.68	−0.303478
$279$	0	0
$280$	0	0
$281$	−72.6835	−0.0154304	−0.00771518	−	0.999970i	$-0.502456\pi$
$281$	−72.6835	−0.0154304	−0.00771518	+	0.999970i	$0.502456\pi$
$282$	0	0
$283$	1743.04	0.366125	0.183062	−	0.983101i	$-0.441399\pi$
$283$	1743.04	0.366125	0.183062	+	0.983101i	$0.441399\pi$
$284$	5913.26	1.23552
$285$	0	0
$286$	−1759.72	−0.363827
$287$	0	0
$288$	0	0
$289$	−3546.32	−0.721825
$290$	−630.668	−0.127704
$291$	0	0
$292$	6292.59	1.26112
$293$	4989.29	0.994804	0.497402	−	0.867520i	$-0.334287\pi$
$293$	4989.29	0.994804	0.497402	+	0.867520i	$0.334287\pi$
$294$	0	0
$295$	−493.827	−0.0974634
$296$	3480.33	0.683412
$297$	0	0
$298$	432.013	0.0839794
$299$	−3040.29	−0.588042
$300$	0	0
$301$	0	0
$302$	−2158.82	−0.411346
$303$	0	0
$304$	1667.48	0.314594
$305$	410.841	0.0771301
$306$	0	0
$307$	1664.61	0.309461	0.154731	−	0.987957i	$-0.450549\pi$
$307$	1664.61	0.309461	0.154731	+	0.987957i	$0.450549\pi$
$308$	0	0
$309$	0	0
$310$	−831.177	−0.152283
$311$	−545.623	−0.0994838	−0.0497419	−	0.998762i	$-0.515840\pi$
$311$	−545.623	−0.0994838	−0.0497419	+	0.998762i	$0.515840\pi$
$312$	0	0
$313$	213.564	0.0385667	0.0192833	−	0.999814i	$-0.493862\pi$
$313$	213.564	0.0385667	0.0192833	+	0.999814i	$0.493862\pi$
$314$	−1552.58	−0.279035
$315$	0	0
$316$	5835.66	1.03887
$317$	2683.93	0.475535	0.237767	−	0.971322i	$-0.423584\pi$
$317$	2683.93	0.475535	0.237767	+	0.971322i	$0.423584\pi$
$318$	0	0
$319$	9333.37	1.63815
$320$	−1796.28	−0.313798
$321$	0	0
$322$	0	0
$323$	1137.37	0.195929
$324$	0	0
$325$	1430.64	0.244177
$326$	2456.77	0.417387
$327$	0	0
$328$	−2682.91	−0.451642
$329$	0	0
$330$	0	0
$331$	−5067.52	−0.841500	−0.420750	−	0.907177i	$-0.638233\pi$
$331$	−5067.52	−0.841500	−0.420750	+	0.907177i	$0.638233\pi$
$332$	4650.80	0.768812
$333$	0	0
$334$	778.187	0.127487
$335$	3273.34	0.533856
$336$	0	0
$337$	−9353.21	−1.51187	−0.755937	−	0.654644i	$-0.772818\pi$
$337$	−9353.21	−1.51187	−0.755937	+	0.654644i	$0.772818\pi$
$338$	−694.782	−0.111808
$339$	0	0
$340$	−1401.93	−0.223618
$341$	12300.7	1.95344
$342$	0	0
$343$	0	0
$344$	−1776.81	−0.278486
$345$	0	0
$346$	2090.77	0.324858
$347$	−2349.34	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$347$	−2349.34	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$348$	0	0
$349$	−10472.6	−1.60626	−0.803128	−	0.595806i	$-0.796833\pi$
$349$	−10472.6	−1.60626	−0.803128	+	0.595806i	$0.796833\pi$
$350$	0	0
$351$	0	0
$352$	−5500.49	−0.832889
$353$	7174.63	1.08178	0.540888	−	0.841095i	$-0.318088\pi$
$353$	7174.63	1.08178	0.540888	+	0.841095i	$0.318088\pi$
$354$	0	0
$355$	−3898.29	−0.582816
$356$	−3471.14	−0.516769
$357$	0	0
$358$	554.222	0.0818199
$359$	7534.13	1.10762	0.553811	−	0.832643i	$-0.313173\pi$
$359$	7534.13	1.10762	0.553811	+	0.832643i	$0.313173\pi$
$360$	0	0
$361$	−5912.46	−0.862000
$362$	187.273	0.0271902
$363$	0	0
$364$	0	0
$365$	−4148.36	−0.594891
$366$	0	0
$367$	5453.06	0.775606	0.387803	−	0.921742i	$-0.373234\pi$
$367$	5453.06	0.775606	0.387803	+	0.921742i	$0.373234\pi$
$368$	−2879.49	−0.407891
$369$	0	0
$370$	−1116.60	−0.156891
$371$	0	0
$372$	0	0
$373$	8231.29	1.14263	0.571314	−	0.820732i	$-0.306434\pi$
$373$	8231.29	1.14263	0.571314	+	0.820732i	$0.306434\pi$
$374$	−1136.80	−0.157173
$375$	0	0
$376$	3130.46	0.429365
$377$	11196.9	1.52963
$378$	0	0
$379$	1670.06	0.226346	0.113173	−	0.993575i	$-0.463898\pi$
$379$	1670.06	0.226346	0.113173	+	0.993575i	$0.463898\pi$
$380$	−1166.71	−0.157502
$381$	0	0
$382$	3155.95	0.422703
$383$	−3220.14	−0.429612	−0.214806	−	0.976657i	$-0.568912\pi$
$383$	−3220.14	−0.429612	−0.214806	+	0.976657i	$0.568912\pi$
$384$	0	0
$385$	0	0
$386$	2287.68	0.301658
$387$	0	0
$388$	10710.1	1.40134
$389$	−3522.23	−0.459085	−0.229543	−	0.973299i	$-0.573723\pi$
$389$	−3522.23	−0.459085	−0.229543	+	0.973299i	$0.573723\pi$
$390$	0	0
$391$	−1964.07	−0.254034
$392$	0	0
$393$	0	0
$394$	−419.090	−0.0535875
$395$	−3847.13	−0.490051
$396$	0	0
$397$	−5455.16	−0.689639	−0.344819	−	0.938669i	$-0.612060\pi$
$397$	−5455.16	−0.689639	−0.344819	+	0.938669i	$0.612060\pi$
$398$	−2614.11	−0.329229
$399$	0	0
$400$	1354.97	0.169372
$401$	1161.80	0.144682	0.0723409	−	0.997380i	$-0.476953\pi$
$401$	1161.80	0.144682	0.0723409	+	0.997380i	$0.476953\pi$
$402$	0	0
$403$	14756.8	1.82404
$404$	13832.4	1.70344
$405$	0	0
$406$	0	0
$407$	16524.8	2.01255
$408$	0	0
$409$	−7398.65	−0.894473	−0.447237	−	0.894416i	$-0.647592\pi$
$409$	−7398.65	−0.894473	−0.447237	+	0.894416i	$0.647592\pi$
$410$	860.765	0.103683
$411$	0	0
$412$	−3092.74	−0.369826
$413$	0	0
$414$	0	0
$415$	−3066.02	−0.362662
$416$	−6598.74	−0.777716
$417$	0	0
$418$	−946.069	−0.110703
$419$	−2134.46	−0.248867	−0.124433	−	0.992228i	$-0.539711\pi$
$419$	−2134.46	−0.248867	−0.124433	+	0.992228i	$0.539711\pi$
$420$	0	0
$421$	−3902.36	−0.451756	−0.225878	−	0.974156i	$-0.572525\pi$
$421$	−3902.36	−0.451756	−0.225878	+	0.974156i	$0.572525\pi$
$422$	1011.88	0.116724
$423$	0	0
$424$	4945.58	0.566459
$425$	924.214	0.105485
$426$	0	0
$427$	0	0
$428$	−2807.06	−0.317020
$429$	0	0
$430$	570.059	0.0639319
$431$	3618.13	0.404360	0.202180	−	0.979348i	$-0.435197\pi$
$431$	3618.13	0.404360	0.202180	+	0.979348i	$0.435197\pi$
$432$	0	0
$433$	4222.37	0.468624	0.234312	−	0.972161i	$-0.424716\pi$
$433$	4222.37	0.468624	0.234312	+	0.972161i	$0.424716\pi$
$434$	0	0
$435$	0	0
$436$	−7399.14	−0.812740
$437$	−1634.54	−0.178926
$438$	0	0
$439$	13518.0	1.46965	0.734826	−	0.678256i	$-0.237264\pi$
$439$	13518.0	1.46965	0.734826	+	0.678256i	$0.237264\pi$
$440$	2396.15	0.259618
$441$	0	0
$442$	−1363.78	−0.146762
$443$	−16591.2	−1.77939	−0.889695	−	0.456555i	$-0.849083\pi$
$443$	−16591.2	−1.77939	−0.889695	+	0.456555i	$0.849083\pi$
$444$	0	0
$445$	2288.33	0.243769
$446$	−3045.23	−0.323309
$447$	0	0
$448$	0	0
$449$	−8354.32	−0.878095	−0.439048	−	0.898464i	$-0.644684\pi$
$449$	−8354.32	−0.878095	−0.439048	+	0.898464i	$0.644684\pi$
$450$	0	0
$451$	−12738.6	−1.33002
$452$	15008.5	1.56181
$453$	0	0
$454$	2375.54	0.245572
$455$	0	0
$456$	0	0
$457$	−1280.34	−0.131054	−0.0655269	−	0.997851i	$-0.520873\pi$
$457$	−1280.34	−0.131054	−0.0655269	+	0.997851i	$0.520873\pi$
$458$	−2163.94	−0.220774
$459$	0	0
$460$	2014.73	0.204212
$461$	−6986.72	−0.705865	−0.352932	−	0.935649i	$-0.614815\pi$
$461$	−6986.72	−0.705865	−0.352932	+	0.935649i	$0.614815\pi$
$462$	0	0
$463$	−5587.32	−0.560831	−0.280416	−	0.959879i	$-0.590472\pi$
$463$	−5587.32	−0.560831	−0.280416	+	0.959879i	$0.590472\pi$
$464$	10604.7	1.06102
$465$	0	0
$466$	1491.73	0.148289
$467$	−9933.83	−0.984332	−0.492166	−	0.870501i	$-0.663795\pi$
$467$	−9933.83	−0.984332	−0.492166	+	0.870501i	$0.663795\pi$
$468$	0	0
$469$	0	0
$470$	−1004.36	−0.0985691
$471$	0	0
$472$	−992.243	−0.0967620
$473$	−8436.41	−0.820099
$474$	0	0
$475$	769.148	0.0742967
$476$	0	0
$477$	0	0
$478$	606.729	0.0580568
$479$	5068.62	0.483489	0.241744	−	0.970340i	$-0.422280\pi$
$479$	5068.62	0.483489	0.241744	+	0.970340i	$0.422280\pi$
$480$	0	0
$481$	19824.3	1.87923
$482$	−3634.68	−0.343476
$483$	0	0
$484$	−7162.86	−0.672695
$485$	−7060.56	−0.661038
$486$	0	0
$487$	264.353	0.0245975	0.0122988	−	0.999924i	$-0.496085\pi$
$487$	264.353	0.0245975	0.0122988	+	0.999924i	$0.496085\pi$
$488$	825.500	0.0765751
$489$	0	0
$490$	0	0
$491$	7459.47	0.685623	0.342812	−	0.939404i	$-0.388621\pi$
$491$	7459.47	0.685623	0.342812	+	0.939404i	$0.388621\pi$
$492$	0	0
$493$	7233.37	0.660800
$494$	−1134.97	−0.103370
$495$	0	0
$496$	13976.3	1.26523
$497$	0	0
$498$	0	0
$499$	7206.67	0.646523	0.323261	−	0.946310i	$-0.395221\pi$
$499$	7206.67	0.646523	0.323261	+	0.946310i	$0.395221\pi$
$500$	−948.054	−0.0847965
$501$	0	0
$502$	235.827	0.0209670
$503$	10886.7	0.965037	0.482519	−	0.875886i	$-0.339722\pi$
$503$	10886.7	0.965037	0.482519	+	0.875886i	$0.339722\pi$
$504$	0	0
$505$	−9118.95	−0.803541
$506$	1633.72	0.143533
$507$	0	0
$508$	−10560.4	−0.922330
$509$	7504.97	0.653540	0.326770	−	0.945104i	$-0.394040\pi$
$509$	7504.97	0.653540	0.326770	+	0.945104i	$0.394040\pi$
$510$	0	0
$511$	0	0
$512$	−10605.8	−0.915459
$513$	0	0
$514$	−4046.11	−0.347210
$515$	2038.87	0.174453
$516$	0	0
$517$	14863.7	1.26442
$518$	0	0
$519$	0	0
$520$	2874.58	0.242420
$521$	22264.8	1.87224	0.936120	−	0.351681i	$-0.114390\pi$
$521$	22264.8	1.87224	0.936120	+	0.351681i	$0.114390\pi$
$522$	0	0
$523$	11286.7	0.943655	0.471828	−	0.881691i	$-0.343595\pi$
$523$	11286.7	0.943655	0.471828	+	0.881691i	$0.343595\pi$
$524$	−13477.7	−1.12362
$525$	0	0
$526$	−2724.27	−0.225824
$527$	9533.09	0.787985
$528$	0	0
$529$	−9344.40	−0.768012
$530$	−1586.70	−0.130042
$531$	0	0
$532$	0	0
$533$	−15282.1	−1.24191
$534$	0	0
$535$	1850.54	0.149544
$536$	6577.10	0.530014
$537$	0	0
$538$	632.781	0.0507084
$539$	0	0
$540$	0	0
$541$	16406.8	1.30385	0.651924	−	0.758284i	$-0.273962\pi$
$541$	16406.8	1.30385	0.651924	+	0.758284i	$0.273962\pi$
$542$	−2504.08	−0.198450
$543$	0	0
$544$	−4262.88	−0.335974
$545$	4877.85	0.383384
$546$	0	0
$547$	−8692.48	−0.679458	−0.339729	−	0.940523i	$-0.610335\pi$
$547$	−8692.48	−0.679458	−0.339729	+	0.940523i	$0.610335\pi$
$548$	−15020.8	−1.17091
$549$	0	0
$550$	−768.764	−0.0596004
$551$	6019.74	0.465426
$552$	0	0
$553$	0	0
$554$	−2330.36	−0.178714
$555$	0	0
$556$	−16549.9	−1.26236
$557$	13935.7	1.06010	0.530049	−	0.847967i	$-0.322174\pi$
$557$	13935.7	1.06010	0.530049	+	0.847967i	$0.322174\pi$
$558$	0	0
$559$	−10120.9	−0.765773
$560$	0	0
$561$	0	0
$562$	46.8553	0.00351685
$563$	−22263.6	−1.66661	−0.833304	−	0.552815i	$-0.813553\pi$
$563$	−22263.6	−1.66661	−0.833304	+	0.552815i	$0.813553\pi$
$564$	0	0
$565$	−9894.26	−0.736734
$566$	−1123.65	−0.0834462
$567$	0	0
$568$	−7832.81	−0.578622
$569$	−9525.66	−0.701822	−0.350911	−	0.936409i	$-0.614128\pi$
$569$	−9525.66	−0.701822	−0.350911	+	0.936409i	$0.614128\pi$
$570$	0	0
$571$	7669.99	0.562135	0.281068	−	0.959688i	$-0.409311\pi$
$571$	7669.99	0.562135	0.281068	+	0.959688i	$0.409311\pi$
$572$	−20703.5	−1.51339
$573$	0	0
$574$	0	0
$575$	−1328.20	−0.0963304
$576$	0	0
$577$	−3907.15	−0.281901	−0.140950	−	0.990017i	$-0.545016\pi$
$577$	−3907.15	−0.281901	−0.140950	+	0.990017i	$0.545016\pi$
$578$	2286.13	0.164516
$579$	0	0
$580$	−7419.95	−0.531201
$581$	0	0
$582$	0	0
$583$	23481.9	1.66813
$584$	−8335.28	−0.590610
$585$	0	0
$586$	−3216.34	−0.226733
$587$	−19891.6	−1.39866	−0.699331	−	0.714798i	$-0.746519\pi$
$587$	−19891.6	−1.39866	−0.699331	+	0.714798i	$0.746519\pi$
$588$	0	0
$589$	7933.61	0.555007
$590$	318.344	0.0222136
$591$	0	0
$592$	18775.8	1.30351
$593$	4780.36	0.331038	0.165519	−	0.986207i	$-0.447070\pi$
$593$	4780.36	0.331038	0.165519	+	0.986207i	$0.447070\pi$
$594$	0	0
$595$	0	0
$596$	5082.74	0.349324
$597$	0	0
$598$	1959.92	0.134025
$599$	−11085.6	−0.756170	−0.378085	−	0.925771i	$-0.623417\pi$
$599$	−11085.6	−0.756170	−0.378085	+	0.925771i	$0.623417\pi$
$600$	0	0
$601$	1573.44	0.106792	0.0533958	−	0.998573i	$-0.482995\pi$
$601$	1573.44	0.106792	0.0533958	+	0.998573i	$0.482995\pi$
$602$	0	0
$603$	0	0
$604$	−25399.1	−1.71105
$605$	4722.08	0.317322
$606$	0	0
$607$	−8542.73	−0.571233	−0.285617	−	0.958344i	$-0.592198\pi$
$607$	−8542.73	−0.571233	−0.285617	+	0.958344i	$0.592198\pi$
$608$	−3547.65	−0.236639
$609$	0	0
$610$	−264.848	−0.0175793
$611$	17831.4	1.18066
$612$	0	0
$613$	15068.2	0.992817	0.496409	−	0.868089i	$-0.334652\pi$
$613$	15068.2	0.992817	0.496409	+	0.868089i	$0.334652\pi$
$614$	−1073.09	−0.0705316
$615$	0	0
$616$	0	0
$617$	−2524.58	−0.164725	−0.0823627	−	0.996602i	$-0.526247\pi$
$617$	−2524.58	−0.164725	−0.0823627	+	0.996602i	$0.526247\pi$
$618$	0	0
$619$	21238.9	1.37910	0.689551	−	0.724237i	$-0.257808\pi$
$619$	21238.9	1.37910	0.689551	+	0.724237i	$0.257808\pi$
$620$	−9778.99	−0.633441
$621$	0	0
$622$	351.735	0.0226741
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−137.674	−0.00879002
$627$	0	0
$628$	−18266.4	−1.16068
$629$	12806.8	0.811827
$630$	0	0
$631$	−8885.83	−0.560601	−0.280300	−	0.959912i	$-0.590434\pi$
$631$	−8885.83	−0.560601	−0.280300	+	0.959912i	$0.590434\pi$
$632$	−7730.02	−0.486525
$633$	0	0
$634$	−1730.19	−0.108383
$635$	6961.92	0.435079
$636$	0	0
$637$	0	0
$638$	−6016.74	−0.373362
$639$	0	0
$640$	5770.41	0.356399
$641$	−3655.63	−0.225255	−0.112628	−	0.993637i	$-0.535927\pi$
$641$	−3655.63	−0.225255	−0.112628	+	0.993637i	$0.535927\pi$
$642$	0	0
$643$	−4221.22	−0.258894	−0.129447	−	0.991586i	$-0.541320\pi$
$643$	−4221.22	−0.258894	−0.129447	+	0.991586i	$0.541320\pi$
$644$	0	0
$645$	0	0
$646$	−733.205	−0.0446557
$647$	103.679	0.00629989	0.00314995	−	0.999995i	$-0.498997\pi$
$647$	103.679	0.00629989	0.00314995	+	0.999995i	$0.498997\pi$
$648$	0	0
$649$	−4711.24	−0.284949
$650$	−922.259	−0.0556523
$651$	0	0
$652$	28904.5	1.73618
$653$	−4476.11	−0.268245	−0.134122	−	0.990965i	$-0.542821\pi$
$653$	−4476.11	−0.268245	−0.134122	+	0.990965i	$0.542821\pi$
$654$	0	0
$655$	8885.09	0.530029
$656$	−14473.8	−0.861445
$657$	0	0
$658$	0	0
$659$	−12022.0	−0.710641	−0.355321	−	0.934745i	$-0.615628\pi$
$659$	−12022.0	−0.710641	−0.355321	+	0.934745i	$0.615628\pi$
$660$	0	0
$661$	−13451.0	−0.791504	−0.395752	−	0.918357i	$-0.629516\pi$
$661$	−13451.0	−0.791504	−0.395752	+	0.918357i	$0.629516\pi$
$662$	3266.77	0.191792
$663$	0	0
$664$	−6160.53	−0.360053
$665$	0	0
$666$	0	0
$667$	−10395.2	−0.603454
$668$	9155.55	0.530297
$669$	0	0
$670$	−2110.15	−0.121675
$671$	3919.53	0.225502
$672$	0	0
$673$	9774.83	0.559869	0.279935	−	0.960019i	$-0.409687\pi$
$673$	9774.83	0.559869	0.279935	+	0.960019i	$0.409687\pi$
$674$	6029.53	0.344583
$675$	0	0
$676$	−8174.27	−0.465081
$677$	23857.4	1.35438	0.677190	−	0.735808i	$-0.263198\pi$
$677$	23857.4	1.35438	0.677190	+	0.735808i	$0.263198\pi$
$678$	0	0
$679$	0	0
$680$	1857.02	0.104726
$681$	0	0
$682$	−7929.65	−0.445223
$683$	−11919.4	−0.667763	−0.333881	−	0.942615i	$-0.608358\pi$
$683$	−11919.4	−0.667763	−0.333881	+	0.942615i	$0.608358\pi$
$684$	0	0
$685$	9902.40	0.552338
$686$	0	0
$687$	0	0
$688$	−9585.59	−0.531173
$689$	28170.5	1.55763
$690$	0	0
$691$	8203.58	0.451634	0.225817	−	0.974170i	$-0.427495\pi$
$691$	8203.58	0.451634	0.225817	+	0.974170i	$0.427495\pi$
$692$	24598.4	1.35129
$693$	0	0
$694$	1514.50	0.0828380
$695$	10910.4	0.595477
$696$	0	0
$697$	−9872.44	−0.536507
$698$	6751.12	0.366094
$699$	0	0
$700$	0	0
$701$	−449.084	−0.0241964	−0.0120982	−	0.999927i	$-0.503851\pi$
$701$	−449.084	−0.0241964	−0.0120982	+	0.999927i	$0.503851\pi$
$702$	0	0
$703$	10658.0	0.571800
$704$	−17137.0	−0.917438
$705$	0	0
$706$	−4625.11	−0.246556
$707$	0	0
$708$	0	0
$709$	1897.64	0.100518	0.0502590	−	0.998736i	$-0.483995\pi$
$709$	1897.64	0.100518	0.0502590	+	0.998736i	$0.483995\pi$
$710$	2513.02	0.132834
$711$	0	0
$712$	4597.93	0.242015
$713$	−13700.2	−0.719601
$714$	0	0
$715$	13648.7	0.713891
$716$	6520.54	0.340341
$717$	0	0
$718$	−4856.86	−0.252446
$719$	6494.70	0.336873	0.168436	−	0.985713i	$-0.446128\pi$
$719$	6494.70	0.336873	0.168436	+	0.985713i	$0.446128\pi$
$720$	0	0
$721$	0	0
$722$	3811.45	0.196465
$723$	0	0
$724$	2203.31	0.113101
$725$	4891.57	0.250577
$726$	0	0
$727$	18311.2	0.934148	0.467074	−	0.884218i	$-0.345308\pi$
$727$	18311.2	0.934148	0.467074	+	0.884218i	$0.345308\pi$
$728$	0	0
$729$	0	0
$730$	2674.23	0.135586
$731$	−6538.23	−0.330814
$732$	0	0
$733$	−21233.3	−1.06994	−0.534972	−	0.844870i	$-0.679678\pi$
$733$	−21233.3	−1.06994	−0.534972	+	0.844870i	$0.679678\pi$
$734$	−3515.30	−0.176774
$735$	0	0
$736$	6126.27	0.306817
$737$	31228.5	1.56081
$738$	0	0
$739$	−22023.5	−1.09628	−0.548138	−	0.836388i	$-0.684663\pi$
$739$	−22023.5	−1.09628	−0.548138	+	0.836388i	$0.684663\pi$
$740$	−13137.1	−0.652608
$741$	0	0
$742$	0	0
$743$	−9436.77	−0.465951	−0.232975	−	0.972483i	$-0.574846\pi$
$743$	−9436.77	−0.465951	−0.232975	+	0.972483i	$0.574846\pi$
$744$	0	0
$745$	−3350.77	−0.164782
$746$	−5306.29	−0.260425
$747$	0	0
$748$	−13374.8	−0.653783
$749$	0	0
$750$	0	0
$751$	−39161.8	−1.90284	−0.951421	−	0.307893i	$-0.900376\pi$
$751$	−39161.8	−1.90284	−0.951421	+	0.307893i	$0.900376\pi$
$752$	16888.3	0.818954
$753$	0	0
$754$	−7218.07	−0.348629
$755$	16744.2	0.807132
$756$	0	0
$757$	20340.6	0.976607	0.488303	−	0.872674i	$-0.337616\pi$
$757$	20340.6	0.976607	0.488303	+	0.872674i	$0.337616\pi$
$758$	−1076.60	−0.0515883
$759$	0	0
$760$	1545.44	0.0737621
$761$	3307.90	0.157570	0.0787852	−	0.996892i	$-0.474896\pi$
$761$	3307.90	0.157570	0.0787852	+	0.996892i	$0.474896\pi$
$762$	0	0
$763$	0	0
$764$	37130.4	1.75829
$765$	0	0
$766$	2075.85	0.0979160
$767$	−5651.91	−0.266074
$768$	0	0
$769$	−17234.0	−0.808159	−0.404079	−	0.914724i	$-0.632408\pi$
$769$	−17234.0	−0.808159	−0.404079	+	0.914724i	$0.632408\pi$
$770$	0	0
$771$	0	0
$772$	26915.1	1.25479
$773$	−8726.33	−0.406034	−0.203017	−	0.979175i	$-0.565075\pi$
$773$	−8726.33	−0.406034	−0.203017	+	0.979175i	$0.565075\pi$
$774$	0	0
$775$	6446.75	0.298806
$776$	−14186.7	−0.656281
$777$	0	0
$778$	2270.60	0.104634
$779$	−8216.03	−0.377882
$780$	0	0
$781$	−37190.7	−1.70395
$782$	1266.14	0.0578989
$783$	0	0
$784$	0	0
$785$	12042.0	0.547515
$786$	0	0
$787$	10642.1	0.482018	0.241009	−	0.970523i	$-0.422522\pi$
$787$	10642.1	0.482018	0.241009	+	0.970523i	$0.422522\pi$
$788$	−4930.69	−0.222904
$789$	0	0
$790$	2480.05	0.111691
$791$	0	0
$792$	0	0
$793$	4702.12	0.210564
$794$	3516.66	0.157181
$795$	0	0
$796$	−30755.5	−1.36947
$797$	29234.1	1.29928	0.649640	−	0.760242i	$-0.274920\pi$
$797$	29234.1	1.29928	0.649640	+	0.760242i	$0.274920\pi$
$798$	0	0
$799$	11519.3	0.510044
$800$	−2882.78	−0.127402
$801$	0	0
$802$	−748.950	−0.0329755
$803$	−39576.5	−1.73926
$804$	0	0
$805$	0	0
$806$	−9512.93	−0.415730
$807$	0	0
$808$	−18322.7	−0.797758
$809$	−36211.2	−1.57369	−0.786846	−	0.617150i	$-0.788287\pi$
$809$	−36211.2	−1.57369	−0.786846	+	0.617150i	$0.788287\pi$
$810$	0	0
$811$	27995.2	1.21214	0.606069	−	0.795412i	$-0.292746\pi$
$811$	27995.2	1.21214	0.606069	+	0.795412i	$0.292746\pi$
$812$	0	0
$813$	0	0
$814$	−10652.7	−0.458694
$815$	−19055.2	−0.818985
$816$	0	0
$817$	−5441.23	−0.233005
$818$	4769.52	0.203866
$819$	0	0
$820$	10127.1	0.431285
$821$	−44805.7	−1.90467	−0.952333	−	0.305060i	$-0.901323\pi$
$821$	−44805.7	−1.90467	−0.952333	+	0.305060i	$0.901323\pi$
$822$	0	0
$823$	−15850.9	−0.671360	−0.335680	−	0.941976i	$-0.608966\pi$
$823$	−15850.9	−0.671360	−0.335680	+	0.941976i	$0.608966\pi$
$824$	4096.70	0.173198
$825$	0	0
$826$	0	0
$827$	45013.9	1.89273	0.946363	−	0.323104i	$-0.104726\pi$
$827$	45013.9	1.89273	0.946363	+	0.323104i	$0.104726\pi$
$828$	0	0
$829$	−5655.83	−0.236954	−0.118477	−	0.992957i	$-0.537801\pi$
$829$	−5655.83	−0.236954	−0.118477	+	0.992957i	$0.537801\pi$
$830$	1976.50	0.0826571
$831$	0	0
$832$	−20558.7	−0.856664
$833$	0	0
$834$	0	0
$835$	−6035.75	−0.250151
$836$	−11130.7	−0.460483
$837$	0	0
$838$	1375.97	0.0567211
$839$	−22037.0	−0.906797	−0.453398	−	0.891308i	$-0.649789\pi$
$839$	−22037.0	−0.906797	−0.453398	+	0.891308i	$0.649789\pi$
$840$	0	0
$841$	13894.9	0.569719
$842$	2515.65	0.102963
$843$	0	0
$844$	11905.0	0.485530
$845$	5388.85	0.219387
$846$	0	0
$847$	0	0
$848$	26680.5	1.08044
$849$	0	0
$850$	−595.793	−0.0240418
$851$	−18404.8	−0.741374
$852$	0	0
$853$	44486.7	1.78569	0.892846	−	0.450362i	$-0.148705\pi$
$853$	44486.7	1.78569	0.892846	+	0.450362i	$0.148705\pi$
$854$	0	0
$855$	0	0
$856$	3718.29	0.148468
$857$	−38559.6	−1.53696	−0.768478	−	0.639877i	$-0.778985\pi$
$857$	−38559.6	−1.53696	−0.768478	+	0.639877i	$0.778985\pi$
$858$	0	0
$859$	−8426.01	−0.334682	−0.167341	−	0.985899i	$-0.553518\pi$
$859$	−8426.01	−0.334682	−0.167341	+	0.985899i	$0.553518\pi$
$860$	6706.88	0.265933
$861$	0	0
$862$	−2332.42	−0.0921608
$863$	20054.3	0.791029	0.395514	−	0.918460i	$-0.370566\pi$
$863$	20054.3	0.791029	0.395514	+	0.918460i	$0.370566\pi$
$864$	0	0
$865$	−16216.4	−0.637427
$866$	−2721.94	−0.106808
$867$	0	0
$868$	0	0
$869$	−36702.6	−1.43274
$870$	0	0
$871$	37463.8	1.45742
$872$	9801.04	0.380625
$873$	0	0
$874$	1053.70	0.0407803
$875$	0	0
$876$	0	0
$877$	47082.1	1.81283	0.906414	−	0.422391i	$-0.138809\pi$
$877$	47082.1	1.81283	0.906414	+	0.422391i	$0.138809\pi$
$878$	−8714.32	−0.334959
$879$	0	0
$880$	12926.8	0.495185
$881$	−9467.24	−0.362042	−0.181021	−	0.983479i	$-0.557940\pi$
$881$	−9467.24	−0.362042	−0.181021	+	0.983479i	$0.557940\pi$
$882$	0	0
$883$	−3049.49	−0.116221	−0.0581106	−	0.998310i	$-0.518508\pi$
$883$	−3049.49	−0.116221	−0.0581106	+	0.998310i	$0.518508\pi$
$884$	−16045.2	−0.610475
$885$	0	0
$886$	10695.5	0.405554
$887$	−19468.6	−0.736968	−0.368484	−	0.929634i	$-0.620123\pi$
$887$	−19468.6	−0.736968	−0.368484	+	0.929634i	$0.620123\pi$
$888$	0	0
$889$	0	0
$890$	−1475.17	−0.0555592
$891$	0	0
$892$	−35827.8	−1.34485
$893$	9586.61	0.359243
$894$	0	0
$895$	−4298.64	−0.160545
$896$	0	0
$897$	0	0
$898$	5385.60	0.200133
$899$	50455.6	1.87184
$900$	0	0
$901$	18198.5	0.672898
$902$	8211.92	0.303134
$903$	0	0
$904$	−19880.5	−0.731432
$905$	−1452.52	−0.0533518
$906$	0	0
$907$	−32014.2	−1.17201	−0.586006	−	0.810307i	$-0.699301\pi$
$907$	−32014.2	−1.17201	−0.586006	+	0.810307i	$0.699301\pi$
$908$	27948.8	1.02149
$909$	0	0
$910$	0	0
$911$	−20921.1	−0.760866	−0.380433	−	0.924809i	$-0.624225\pi$
$911$	−20921.1	−0.760866	−0.380433	+	0.924809i	$0.624225\pi$
$912$	0	0
$913$	−29250.6	−1.06030
$914$	825.366	0.0298694
$915$	0	0
$916$	−25459.2	−0.918337
$917$	0	0
$918$	0	0
$919$	43467.4	1.56024	0.780118	−	0.625632i	$-0.215159\pi$
$919$	43467.4	1.56024	0.780118	+	0.625632i	$0.215159\pi$
$920$	−2668.75	−0.0956372
$921$	0	0
$922$	4503.97	0.160879
$923$	−44616.4	−1.59108
$924$	0	0
$925$	8660.58	0.307847
$926$	3601.85	0.127823
$927$	0	0
$928$	−22562.1	−0.798099
$929$	−18542.1	−0.654840	−0.327420	−	0.944879i	$-0.606179\pi$
$929$	−18542.1	−0.654840	−0.327420	+	0.944879i	$0.606179\pi$
$930$	0	0
$931$	0	0
$932$	17550.5	0.616830
$933$	0	0
$934$	6403.82	0.224346
$935$	8817.25	0.308401
$936$	0	0
$937$	−13842.5	−0.482619	−0.241309	−	0.970448i	$-0.577577\pi$
$937$	−13842.5	−0.482619	−0.241309	+	0.970448i	$0.577577\pi$
$938$	0	0
$939$	0	0
$940$	−11816.5	−0.410012
$941$	−53393.2	−1.84970	−0.924850	−	0.380333i	$-0.875809\pi$
$941$	−53393.2	−1.84970	−0.924850	+	0.380333i	$0.875809\pi$
$942$	0	0
$943$	14187.9	0.489947
$944$	−5352.98	−0.184560
$945$	0	0
$946$	5438.52	0.186915
$947$	−22344.2	−0.766727	−0.383364	−	0.923598i	$-0.625234\pi$
$947$	−22344.2	−0.766727	−0.383364	+	0.923598i	$0.625234\pi$
$948$	0	0
$949$	−47478.5	−1.62404
$950$	−495.830	−0.0169335
$951$	0	0
$952$	0	0
$953$	8902.63	0.302607	0.151304	−	0.988487i	$-0.451653\pi$
$953$	8902.63	0.302607	0.151304	+	0.988487i	$0.451653\pi$
$954$	0	0
$955$	−24478.1	−0.829416
$956$	7138.31	0.241495
$957$	0	0
$958$	−3267.48	−0.110196
$959$	0	0
$960$	0	0
$961$	36706.0	1.23212
$962$	−12779.7	−0.428309
$963$	0	0
$964$	−42762.8	−1.42873
$965$	−17743.6	−0.591905
$966$	0	0
$967$	−2225.57	−0.0740119	−0.0370059	−	0.999315i	$-0.511782\pi$
$967$	−2225.57	−0.0740119	−0.0370059	+	0.999315i	$0.511782\pi$
$968$	9488.05	0.315039
$969$	0	0
$970$	4551.58	0.150662
$971$	−11324.8	−0.374285	−0.187143	−	0.982333i	$-0.559923\pi$
$971$	−11324.8	−0.374285	−0.187143	+	0.982333i	$0.559923\pi$
$972$	0	0
$973$	0	0
$974$	−170.415	−0.00560621
$975$	0	0
$976$	4453.43	0.146056
$977$	57695.9	1.88931	0.944654	−	0.328067i	$-0.106397\pi$
$977$	57695.9	1.88931	0.944654	+	0.328067i	$0.106397\pi$
$978$	0	0
$979$	21831.3	0.712697
$980$	0	0
$981$	0	0
$982$	−4808.73	−0.156266
$983$	59657.3	1.93568	0.967839	−	0.251570i	$-0.0809469\pi$
$983$	59657.3	1.93568	0.967839	+	0.251570i	$0.0809469\pi$
$984$	0	0
$985$	3250.54	0.105148
$986$	−4662.98	−0.150608
$987$	0	0
$988$	−13353.1	−0.429980
$989$	9396.20	0.302105
$990$	0	0
$991$	2890.60	0.0926568	0.0463284	−	0.998926i	$-0.485248\pi$
$991$	2890.60	0.0926568	0.0463284	+	0.998926i	$0.485248\pi$
$992$	−29735.3	−0.951710
$993$	0	0
$994$	0	0
$995$	20275.5	0.646005
$996$	0	0
$997$	10619.6	0.337337	0.168669	−	0.985673i	$-0.446053\pi$
$997$	10619.6	0.337337	0.168669	+	0.985673i	$0.446053\pi$
$998$	−4645.77	−0.147354
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ca.1.3		6
3.2	odd	2		245.4.a.p.1.4	yes	6
7.6	odd	2		2205.4.a.bz.1.3		6
15.14	odd	2		1225.4.a.bi.1.3		6
21.2	odd	6		245.4.e.p.116.3		12
21.5	even	6		245.4.e.q.116.3		12
21.11	odd	6		245.4.e.p.226.3		12
21.17	even	6		245.4.e.q.226.3		12
21.20	even	2		245.4.a.o.1.4	✓	6
105.104	even	2		1225.4.a.bj.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.4	✓	6	21.20	even	2
245.4.a.p.1.4	yes	6	3.2	odd	2
245.4.e.p.116.3		12	21.2	odd	6
245.4.e.p.226.3		12	21.11	odd	6
245.4.e.q.116.3		12	21.5	even	6
245.4.e.q.226.3		12	21.17	even	6
1225.4.a.bi.1.3		6	15.14	odd	2
1225.4.a.bj.1.3		6	105.104	even	2
2205.4.a.bz.1.3		6	7.6	odd	2
2205.4.a.ca.1.3		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-0.644648 q^{2} -7.58443 q^{4} +5.00000 q^{5} +10.0465 q^{8} -3.22324 q^{10} +47.7013 q^{11} +57.2256 q^{13} +54.1990 q^{16} +36.9686 q^{17} +30.7659 q^{19} -37.9221 q^{20} -30.7506 q^{22} -53.1282 q^{23} +25.0000 q^{25} -36.8904 q^{26} +195.663 q^{29} +257.870 q^{31} -115.311 q^{32} -23.8317 q^{34} +346.423 q^{37} -19.8332 q^{38} +50.2324 q^{40} -267.050 q^{41} -176.859 q^{43} -361.787 q^{44} +34.2490 q^{46} +311.598 q^{47} -16.1162 q^{50} -434.024 q^{52} +492.270 q^{53} +238.507 q^{55} -126.134 q^{58} -98.7653 q^{59} +82.1682 q^{61} -166.235 q^{62} -359.257 q^{64} +286.128 q^{65} +654.668 q^{67} -280.386 q^{68} -779.658 q^{71} -829.673 q^{73} -223.321 q^{74} -233.342 q^{76} -769.426 q^{79} +270.995 q^{80} +172.153 q^{82} -613.203 q^{83} +184.843 q^{85} +114.012 q^{86} +479.230 q^{88} +457.666 q^{89} +402.947 q^{92} -200.871 q^{94} +153.830 q^{95} -1412.11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31}+ \cdots + 516 q^{97}+O(q^{100}) \) 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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