LMFDB - Embedded newform 207.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 26x^{3} + 10x^{2} + 144x + 56 $ x^5 - x^4 - 26x^3 + 10x^2 + 144*x + 56
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.18948$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.18948 q^{2} -3.20619 q^{4} -11.9773 q^{5} -36.1543 q^{7} +24.5357 q^{8} +26.2241 q^{10} -21.1816 q^{11} -42.7538 q^{13} +79.1589 q^{14} -28.0708 q^{16} +104.421 q^{17} +3.53221 q^{19} +38.4016 q^{20} +46.3766 q^{22} -23.0000 q^{23} +18.4565 q^{25} +93.6084 q^{26} +115.918 q^{28} -56.5533 q^{29} -36.8766 q^{31} -134.825 q^{32} -228.628 q^{34} +433.032 q^{35} +45.4764 q^{37} -7.73370 q^{38} -293.872 q^{40} +458.159 q^{41} -23.5875 q^{43} +67.9123 q^{44} +50.3580 q^{46} +191.004 q^{47} +964.131 q^{49} -40.4100 q^{50} +137.077 q^{52} +273.107 q^{53} +253.699 q^{55} -887.070 q^{56} +123.822 q^{58} -846.995 q^{59} -386.932 q^{61} +80.7404 q^{62} +519.763 q^{64} +512.076 q^{65} +489.918 q^{67} -334.794 q^{68} -948.113 q^{70} -767.497 q^{71} -1093.47 q^{73} -99.5696 q^{74} -11.3250 q^{76} +765.805 q^{77} +592.548 q^{79} +336.213 q^{80} -1003.13 q^{82} +124.425 q^{83} -1250.69 q^{85} +51.6444 q^{86} -519.705 q^{88} -345.013 q^{89} +1545.73 q^{91} +73.7424 q^{92} -418.198 q^{94} -42.3065 q^{95} +1447.26 q^{97} -2110.94 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 16 q^{4} + 20 q^{5} - 10 q^{7} + 48 q^{8} + 50 q^{10} + 46 q^{11} + 54 q^{13} + 164 q^{14} - 60 q^{16} + 250 q^{17} - 28 q^{19} + 242 q^{20} - 10 q^{22} - 115 q^{23} + 239 q^{25} + 368 q^{26}+ \cdots - 2400 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 16 * q^4 + 20 * q^5 - 10 * q^7 + 48 * q^8 + 50 * q^10 + 46 * q^11 + 54 * q^13 + 164 * q^14 - 60 * q^16 + 250 * q^17 - 28 * q^19 + 242 * q^20 - 10 * q^22 - 115 * q^23 + 239 * q^25 + 368 * q^26 + 224 * q^28 + 460 * q^29 - 360 * q^31 + 268 * q^32 - 20 * q^34 + 964 * q^35 - 92 * q^37 + 266 * q^38 - 278 * q^40 + 788 * q^41 + 108 * q^43 + 354 * q^44 - 92 * q^46 + 724 * q^47 + 149 * q^49 + 128 * q^50 - 288 * q^52 + 580 * q^53 - 672 * q^55 - 1172 * q^56 - 48 * q^58 - 408 * q^59 - 620 * q^61 - 1088 * q^62 + 696 * q^64 + 184 * q^65 + 100 * q^67 - 1584 * q^68 + 728 * q^70 - 124 * q^71 - 1946 * q^73 - 2866 * q^74 + 1362 * q^76 - 396 * q^77 - 198 * q^79 - 2398 * q^80 + 76 * q^82 - 578 * q^83 + 376 * q^85 - 3910 * q^86 + 1366 * q^88 + 930 * q^89 + 1972 * q^91 - 368 * q^92 + 976 * q^94 - 908 * q^95 + 2618 * q^97 - 2400 * q^98

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 $

$2$	−2.18948	−0.774097	−0.387048	−	0.922059i	$-0.626505\pi$
$2$	−2.18948	−0.774097	−0.387048	+	0.922059i	$0.626505\pi$
$3$	0	0
$3$	0	0
$4$	−3.20619	−0.400774
$5$	−11.9773	−1.07129	−0.535643	−	0.844445i	$-0.679931\pi$
$5$	−11.9773	−1.07129	−0.535643	+	0.844445i	$0.679931\pi$
$6$	0	0
$7$	−36.1543	−1.95215	−0.976074	−	0.217440i	$-0.930229\pi$
$7$	−36.1543	−1.95215	−0.976074	+	0.217440i	$0.930229\pi$
$8$	24.5357	1.08433
$9$	0	0
$10$	26.2241	0.829278
$11$	−21.1816	−0.580590	−0.290295	−	0.956937i	$-0.593753\pi$
$11$	−21.1816	−0.580590	−0.290295	+	0.956937i	$0.593753\pi$
$12$	0	0
$13$	−42.7538	−0.912136	−0.456068	−	0.889945i	$-0.650743\pi$
$13$	−42.7538	−0.912136	−0.456068	+	0.889945i	$0.650743\pi$
$14$	79.1589	1.51115
$15$	0	0
$16$	−28.0708	−0.438606
$17$	104.421	1.48976	0.744878	−	0.667201i	$-0.232508\pi$
$17$	104.421	1.48976	0.744878	+	0.667201i	$0.232508\pi$
$18$	0	0
$19$	3.53221	0.0426497	0.0213249	−	0.999773i	$-0.493212\pi$
$19$	3.53221	0.0426497	0.0213249	+	0.999773i	$0.493212\pi$
$20$	38.4016	0.429343
$21$	0	0
$22$	46.3766	0.449433
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	18.4565	0.147652
$26$	93.6084	0.706082
$27$	0	0
$28$	115.918	0.782370
$29$	−56.5533	−0.362127	−0.181063	−	0.983471i	$-0.557954\pi$
$29$	−56.5533	−0.362127	−0.181063	+	0.983471i	$0.557954\pi$
$30$	0	0
$31$	−36.8766	−0.213653	−0.106826	−	0.994278i	$-0.534069\pi$
$31$	−36.8766	−0.213653	−0.106826	+	0.994278i	$0.534069\pi$
$32$	−134.825	−0.744811
$33$	0	0
$34$	−228.628	−1.15322
$35$	433.032	2.09131
$36$	0	0
$37$	45.4764	0.202062	0.101031	−	0.994883i	$-0.467786\pi$
$37$	45.4764	0.202062	0.101031	+	0.994883i	$0.467786\pi$
$38$	−7.73370	−0.0330150
$39$	0	0
$40$	−293.872	−1.16163
$41$	458.159	1.74518	0.872590	−	0.488454i	$-0.162439\pi$
$41$	458.159	1.74518	0.872590	+	0.488454i	$0.162439\pi$
$42$	0	0
$43$	−23.5875	−0.0836527	−0.0418264	−	0.999125i	$-0.513318\pi$
$43$	−23.5875	−0.0836527	−0.0418264	+	0.999125i	$0.513318\pi$
$44$	67.9123	0.232686
$45$	0	0
$46$	50.3580	0.161410
$47$	191.004	0.592782	0.296391	−	0.955067i	$-0.404217\pi$
$47$	191.004	0.592782	0.296391	+	0.955067i	$0.404217\pi$
$48$	0	0
$49$	964.131	2.81088
$50$	−40.4100	−0.114297
$51$	0	0
$52$	137.077	0.365560
$53$	273.107	0.707813	0.353906	−	0.935281i	$-0.384853\pi$
$53$	273.107	0.707813	0.353906	+	0.935281i	$0.384853\pi$
$54$	0	0
$55$	253.699	0.621978
$56$	−887.070	−2.11678
$57$	0	0
$58$	123.822	0.280321
$59$	−846.995	−1.86897	−0.934486	−	0.355999i	$-0.884141\pi$
$59$	−846.995	−1.86897	−0.934486	+	0.355999i	$0.884141\pi$
$60$	0	0
$61$	−386.932	−0.812156	−0.406078	−	0.913838i	$-0.633104\pi$
$61$	−386.932	−0.812156	−0.406078	+	0.913838i	$0.633104\pi$
$62$	80.7404	0.165388
$63$	0	0
$64$	519.763	1.01516
$65$	512.076	0.977158
$66$	0	0
$67$	489.918	0.893329	0.446665	−	0.894702i	$-0.352612\pi$
$67$	489.918	0.893329	0.446665	+	0.894702i	$0.352612\pi$
$68$	−334.794	−0.597055
$69$	0	0
$70$	−948.113	−1.61887
$71$	−767.497	−1.28289	−0.641445	−	0.767169i	$-0.721665\pi$
$71$	−767.497	−1.28289	−0.641445	+	0.767169i	$0.721665\pi$
$72$	0	0
$73$	−1093.47	−1.75317	−0.876586	−	0.481246i	$-0.840184\pi$
$73$	−1093.47	−1.75317	−0.876586	+	0.481246i	$0.840184\pi$
$74$	−99.5696	−0.156415
$75$	0	0
$76$	−11.3250	−0.0170929
$77$	765.805	1.13340
$78$	0	0
$79$	592.548	0.843884	0.421942	−	0.906623i	$-0.361349\pi$
$79$	592.548	0.843884	0.421942	+	0.906623i	$0.361349\pi$
$80$	336.213	0.469872
$81$	0	0
$82$	−1003.13	−1.35094
$83$	124.425	0.164547	0.0822735	−	0.996610i	$-0.473782\pi$
$83$	124.425	0.164547	0.0822735	+	0.996610i	$0.473782\pi$
$84$	0	0
$85$	−1250.69	−1.59595
$86$	51.6444	0.0647553
$87$	0	0
$88$	−519.705	−0.629554
$89$	−345.013	−0.410913	−0.205457	−	0.978666i	$-0.565868\pi$
$89$	−345.013	−0.410913	−0.205457	+	0.978666i	$0.565868\pi$
$90$	0	0
$91$	1545.73	1.78062
$92$	73.7424	0.0835672
$93$	0	0
$94$	−418.198	−0.458871
$95$	−42.3065	−0.0456900
$96$	0	0
$97$	1447.26	1.51492	0.757460	−	0.652882i	$-0.226440\pi$
$97$	1447.26	1.51492	0.757460	+	0.652882i	$0.226440\pi$
$98$	−2110.94	−2.17589
$99$	0	0
$100$	−59.1749	−0.0591749
$101$	−1146.59	−1.12960	−0.564800	−	0.825228i	$-0.691047\pi$
$101$	−1146.59	−1.12960	−0.564800	+	0.825228i	$0.691047\pi$
$102$	0	0
$103$	−1121.66	−1.07301	−0.536505	−	0.843897i	$-0.680256\pi$
$103$	−1121.66	−1.07301	−0.536505	+	0.843897i	$0.680256\pi$
$104$	−1048.99	−0.989061
$105$	0	0
$106$	−597.961	−0.547916
$107$	−224.533	−0.202864	−0.101432	−	0.994842i	$-0.532342\pi$
$107$	−224.533	−0.202864	−0.101432	+	0.994842i	$0.532342\pi$
$108$	0	0
$109$	−1213.28	−1.06615	−0.533077	−	0.846067i	$-0.678965\pi$
$109$	−1213.28	−1.06615	−0.533077	+	0.846067i	$0.678965\pi$
$110$	−555.468	−0.481471
$111$	0	0
$112$	1014.88	0.856224
$113$	1094.65	0.911288	0.455644	−	0.890162i	$-0.349409\pi$
$113$	1094.65	0.911288	0.455644	+	0.890162i	$0.349409\pi$
$114$	0	0
$115$	275.479	0.223378
$116$	181.321	0.145131
$117$	0	0
$118$	1854.48	1.44677
$119$	−3775.27	−2.90822
$120$	0	0
$121$	−882.340	−0.662915
$122$	847.178	0.628688
$123$	0	0
$124$	118.233	0.0856264
$125$	1276.11	0.913108
$126$	0	0
$127$	1241.01	0.867103	0.433551	−	0.901129i	$-0.357260\pi$
$127$	1241.01	0.867103	0.433551	+	0.901129i	$0.357260\pi$
$128$	−59.4073	−0.0410228
$129$	0	0
$130$	−1121.18	−0.756415
$131$	2689.20	1.79356	0.896781	−	0.442475i	$-0.145899\pi$
$131$	2689.20	1.79356	0.896781	+	0.442475i	$0.145899\pi$
$132$	0	0
$133$	−127.705	−0.0832586
$134$	−1072.66	−0.691523
$135$	0	0
$136$	2562.05	1.61539
$137$	1446.37	0.901986	0.450993	−	0.892527i	$-0.351070\pi$
$137$	1446.37	0.901986	0.450993	+	0.892527i	$0.351070\pi$
$138$	0	0
$139$	−1783.89	−1.08854	−0.544272	−	0.838909i	$-0.683194\pi$
$139$	−1783.89	−1.08854	−0.544272	+	0.838909i	$0.683194\pi$
$140$	−1388.38	−0.838141
$141$	0	0
$142$	1680.42	0.993081
$143$	905.594	0.529577
$144$	0	0
$145$	677.357	0.387941
$146$	2394.14	1.35712
$147$	0	0
$148$	−145.806	−0.0809810
$149$	2362.42	1.29891	0.649454	−	0.760401i	$-0.274998\pi$
$149$	2362.42	1.29891	0.649454	+	0.760401i	$0.274998\pi$
$150$	0	0
$151$	1821.89	0.981876	0.490938	−	0.871194i	$-0.336654\pi$
$151$	1821.89	0.981876	0.490938	+	0.871194i	$0.336654\pi$
$152$	86.6653	0.0462466
$153$	0	0
$154$	−1676.71	−0.877360
$155$	441.683	0.228883
$156$	0	0
$157$	−850.404	−0.432290	−0.216145	−	0.976361i	$-0.569348\pi$
$157$	−850.404	−0.432290	−0.216145	+	0.976361i	$0.569348\pi$
$158$	−1297.37	−0.653248
$159$	0	0
$160$	1614.85	0.797905
$161$	831.548	0.407051
$162$	0	0
$163$	−2633.80	−1.26561	−0.632807	−	0.774310i	$-0.718097\pi$
$163$	−2633.80	−1.26561	−0.632807	+	0.774310i	$0.718097\pi$
$164$	−1468.94	−0.699422
$165$	0	0
$166$	−272.425	−0.127375
$167$	2305.75	1.06841	0.534205	−	0.845355i	$-0.320611\pi$
$167$	2305.75	1.06841	0.534205	+	0.845355i	$0.320611\pi$
$168$	0	0
$169$	−369.113	−0.168008
$170$	2738.35	1.23542
$171$	0	0
$172$	75.6262	0.0335258
$173$	−1991.39	−0.875160	−0.437580	−	0.899179i	$-0.644164\pi$
$173$	−1991.39	−0.875160	−0.437580	+	0.899179i	$0.644164\pi$
$174$	0	0
$175$	−667.280	−0.288238
$176$	594.584	0.254651
$177$	0	0
$178$	755.397	0.318087
$179$	3526.49	1.47253	0.736264	−	0.676694i	$-0.236588\pi$
$179$	3526.49	1.47253	0.736264	+	0.676694i	$0.236588\pi$
$180$	0	0
$181$	4407.36	1.80993	0.904963	−	0.425491i	$-0.139898\pi$
$181$	4407.36	1.80993	0.904963	+	0.425491i	$0.139898\pi$
$182$	−3384.35	−1.37838
$183$	0	0
$184$	−564.321	−0.226099
$185$	−544.686	−0.216466
$186$	0	0
$187$	−2211.81	−0.864938
$188$	−612.394	−0.237572
$189$	0	0
$190$	92.6290	0.0353685
$191$	1355.35	0.513454	0.256727	−	0.966484i	$-0.417356\pi$
$191$	1355.35	0.513454	0.256727	+	0.966484i	$0.417356\pi$
$192$	0	0
$193$	−177.379	−0.0661555	−0.0330778	−	0.999453i	$-0.510531\pi$
$193$	−177.379	−0.0661555	−0.0330778	+	0.999453i	$0.510531\pi$
$194$	−3168.75	−1.17269
$195$	0	0
$196$	−3091.19	−1.12653
$197$	3005.80	1.08708	0.543538	−	0.839384i	$-0.317084\pi$
$197$	3005.80	1.08708	0.543538	+	0.839384i	$0.317084\pi$
$198$	0	0
$199$	2384.12	0.849274	0.424637	−	0.905364i	$-0.360402\pi$
$199$	2384.12	0.849274	0.424637	+	0.905364i	$0.360402\pi$
$200$	452.842	0.160104
$201$	0	0
$202$	2510.43	0.874420
$203$	2044.64	0.706925
$204$	0	0
$205$	−5487.52	−1.86958
$206$	2455.84	0.830613
$207$	0	0
$208$	1200.13	0.400069
$209$	−74.8179	−0.0247620
$210$	0	0
$211$	−12.8956	−0.00420745	−0.00210372	−	0.999998i	$-0.500670\pi$
$211$	−12.8956	−0.00420745	−0.00210372	+	0.999998i	$0.500670\pi$
$212$	−875.632	−0.283673
$213$	0	0
$214$	491.609	0.157036
$215$	282.516	0.0896159
$216$	0	0
$217$	1333.25	0.417081
$218$	2656.44	0.825307
$219$	0	0
$220$	−813.408	−0.249273
$221$	−4464.40	−1.35886
$222$	0	0
$223$	−559.310	−0.167956	−0.0839780	−	0.996468i	$-0.526763\pi$
$223$	−559.310	−0.167956	−0.0839780	+	0.996468i	$0.526763\pi$
$224$	4874.51	1.45398
$225$	0	0
$226$	−2396.70	−0.705426
$227$	2210.81	0.646415	0.323208	−	0.946328i	$-0.395239\pi$
$227$	2210.81	0.646415	0.323208	+	0.946328i	$0.395239\pi$
$228$	0	0
$229$	−4550.81	−1.31322	−0.656608	−	0.754232i	$-0.728009\pi$
$229$	−4550.81	−1.31322	−0.656608	+	0.754232i	$0.728009\pi$
$230$	−603.154	−0.172917
$231$	0	0
$232$	−1387.57	−0.392667
$233$	5736.88	1.61303	0.806515	−	0.591214i	$-0.201351\pi$
$233$	5736.88	1.61303	0.806515	+	0.591214i	$0.201351\pi$
$234$	0	0
$235$	−2287.71	−0.635038
$236$	2715.63	0.749036
$237$	0	0
$238$	8265.87	2.25125
$239$	−4471.62	−1.21023	−0.605115	−	0.796138i	$-0.706873\pi$
$239$	−4471.62	−1.21023	−0.605115	+	0.796138i	$0.706873\pi$
$240$	0	0
$241$	4600.43	1.22963	0.614813	−	0.788673i	$-0.289232\pi$
$241$	4600.43	1.22963	0.614813	+	0.788673i	$0.289232\pi$
$242$	1931.86	0.513160
$243$	0	0
$244$	1240.58	0.325491
$245$	−11547.7	−3.01125
$246$	0	0
$247$	−151.016	−0.0389024
$248$	−904.792	−0.231671
$249$	0	0
$250$	−2794.01	−0.706834
$251$	−1378.02	−0.346533	−0.173267	−	0.984875i	$-0.555432\pi$
$251$	−1378.02	−0.346533	−0.173267	+	0.984875i	$0.555432\pi$
$252$	0	0
$253$	487.177	0.121061
$254$	−2717.17	−0.671222
$255$	0	0
$256$	−4028.03	−0.983407
$257$	−1097.23	−0.266317	−0.133158	−	0.991095i	$-0.542512\pi$
$257$	−1097.23	−0.266317	−0.133158	+	0.991095i	$0.542512\pi$
$258$	0	0
$259$	−1644.17	−0.394454
$260$	−1641.82	−0.391619
$261$	0	0
$262$	−5887.94	−1.38839
$263$	−4755.57	−1.11499	−0.557493	−	0.830182i	$-0.688237\pi$
$263$	−4755.57	−1.11499	−0.557493	+	0.830182i	$0.688237\pi$
$264$	0	0
$265$	−3271.09	−0.758269
$266$	279.606	0.0644502
$267$	0	0
$268$	−1570.77	−0.358023
$269$	2510.79	0.569092	0.284546	−	0.958662i	$-0.408157\pi$
$269$	2510.79	0.569092	0.284546	+	0.958662i	$0.408157\pi$
$270$	0	0
$271$	−6224.93	−1.39534	−0.697671	−	0.716418i	$-0.745780\pi$
$271$	−6224.93	−1.39534	−0.697671	+	0.716418i	$0.745780\pi$
$272$	−2931.19	−0.653416
$273$	0	0
$274$	−3166.80	−0.698225
$275$	−390.937	−0.0857251
$276$	0	0
$277$	6768.99	1.46826	0.734132	−	0.679007i	$-0.237589\pi$
$277$	6768.99	1.46826	0.734132	+	0.679007i	$0.237589\pi$
$278$	3905.79	0.842639
$279$	0	0
$280$	10624.7	2.26768
$281$	−3095.89	−0.657243	−0.328622	−	0.944462i	$-0.606584\pi$
$281$	−3095.89	−0.657243	−0.328622	+	0.944462i	$0.606584\pi$
$282$	0	0
$283$	3575.40	0.751008	0.375504	−	0.926821i	$-0.377470\pi$
$283$	3575.40	0.751008	0.375504	+	0.926821i	$0.377470\pi$
$284$	2460.74	0.514149
$285$	0	0
$286$	−1982.78	−0.409944
$287$	−16564.4	−3.40685
$288$	0	0
$289$	5990.78	1.21937
$290$	−1483.06	−0.300304
$291$	0	0
$292$	3505.89	0.702625
$293$	8075.58	1.61017	0.805086	−	0.593158i	$-0.202119\pi$
$293$	8075.58	1.61017	0.805086	+	0.593158i	$0.202119\pi$
$294$	0	0
$295$	10144.7	2.00220
$296$	1115.80	0.219102
$297$	0	0
$298$	−5172.47	−1.00548
$299$	983.337	0.190194
$300$	0	0
$301$	852.791	0.163302
$302$	−3988.99	−0.760067
$303$	0	0
$304$	−99.1520	−0.0187064
$305$	4634.41	0.870051
$306$	0	0
$307$	5012.21	0.931797	0.465899	−	0.884838i	$-0.345731\pi$
$307$	5012.21	0.931797	0.465899	+	0.884838i	$0.345731\pi$
$308$	−2455.32	−0.454236
$309$	0	0
$310$	−967.054	−0.177177
$311$	4019.10	0.732805	0.366403	−	0.930456i	$-0.380589\pi$
$311$	4019.10	0.732805	0.366403	+	0.930456i	$0.380589\pi$
$312$	0	0
$313$	8919.30	1.61070	0.805349	−	0.592801i	$-0.201978\pi$
$313$	8919.30	1.61070	0.805349	+	0.592801i	$0.201978\pi$
$314$	1861.94	0.334635
$315$	0	0
$316$	−1899.82	−0.338207
$317$	−6767.22	−1.19901	−0.599503	−	0.800372i	$-0.704635\pi$
$317$	−6767.22	−1.19901	−0.599503	+	0.800372i	$0.704635\pi$
$318$	0	0
$319$	1197.89	0.210247
$320$	−6225.37	−1.08753
$321$	0	0
$322$	−1820.66	−0.315097
$323$	368.838	0.0635377
$324$	0	0
$325$	−789.084	−0.134678
$326$	5766.64	0.979707
$327$	0	0
$328$	11241.2	1.89236
$329$	−6905.60	−1.15720
$330$	0	0
$331$	−11218.1	−1.86284	−0.931420	−	0.363946i	$-0.881429\pi$
$331$	−11218.1	−1.86284	−0.931420	+	0.363946i	$0.881429\pi$
$332$	−398.930	−0.0659462
$333$	0	0
$334$	−5048.39	−0.827052
$335$	−5867.91	−0.957010
$336$	0	0
$337$	2513.85	0.406344	0.203172	−	0.979143i	$-0.434875\pi$
$337$	2513.85	0.406344	0.203172	+	0.979143i	$0.434875\pi$
$338$	808.164	0.130054
$339$	0	0
$340$	4009.94	0.639617
$341$	781.105	0.124045
$342$	0	0
$343$	−22456.6	−3.53510
$344$	−578.737	−0.0907076
$345$	0	0
$346$	4360.11	0.677459
$347$	−384.562	−0.0594938	−0.0297469	−	0.999557i	$-0.509470\pi$
$347$	−384.562	−0.0594938	−0.0297469	+	0.999557i	$0.509470\pi$
$348$	0	0
$349$	−2600.52	−0.398861	−0.199431	−	0.979912i	$-0.563909\pi$
$349$	−2600.52	−0.398861	−0.199431	+	0.979912i	$0.563909\pi$
$350$	1460.99	0.223124
$351$	0	0
$352$	2855.81	0.432430
$353$	5746.10	0.866385	0.433193	−	0.901301i	$-0.357387\pi$
$353$	5746.10	0.866385	0.433193	+	0.901301i	$0.357387\pi$
$354$	0	0
$355$	9192.57	1.37434
$356$	1106.18	0.164683
$357$	0	0
$358$	−7721.18	−1.13988
$359$	−10987.6	−1.61533	−0.807664	−	0.589643i	$-0.799268\pi$
$359$	−10987.6	−1.61533	−0.807664	+	0.589643i	$0.799268\pi$
$360$	0	0
$361$	−6846.52	−0.998181
$362$	−9649.81	−1.40106
$363$	0	0
$364$	−4955.91	−0.713628
$365$	13096.9	1.87815
$366$	0	0
$367$	2627.60	0.373732	0.186866	−	0.982385i	$-0.440167\pi$
$367$	2627.60	0.373732	0.186866	+	0.982385i	$0.440167\pi$
$368$	645.628	0.0914557
$369$	0	0
$370$	1192.58	0.167565
$371$	−9873.97	−1.38175
$372$	0	0
$373$	−3613.59	−0.501620	−0.250810	−	0.968036i	$-0.580697\pi$
$373$	−3613.59	−0.501620	−0.250810	+	0.968036i	$0.580697\pi$
$374$	4842.70	0.669546
$375$	0	0
$376$	4686.41	0.642774
$377$	2417.87	0.330309
$378$	0	0
$379$	6016.03	0.815364	0.407682	−	0.913124i	$-0.366337\pi$
$379$	6016.03	0.815364	0.407682	+	0.913124i	$0.366337\pi$
$380$	135.643	0.0183114
$381$	0	0
$382$	−2967.51	−0.397463
$383$	−4403.22	−0.587451	−0.293726	−	0.955890i	$-0.594895\pi$
$383$	−4403.22	−0.587451	−0.293726	+	0.955890i	$0.594895\pi$
$384$	0	0
$385$	−9172.30	−1.21419
$386$	388.367	0.0512108
$387$	0	0
$388$	−4640.20	−0.607140
$389$	8511.27	1.10935	0.554677	−	0.832066i	$-0.312842\pi$
$389$	8511.27	1.10935	0.554677	+	0.832066i	$0.312842\pi$
$390$	0	0
$391$	−2401.69	−0.310636
$392$	23655.6	3.04793
$393$	0	0
$394$	−6581.12	−0.841503
$395$	−7097.14	−0.904040
$396$	0	0
$397$	927.638	0.117272	0.0586358	−	0.998279i	$-0.481325\pi$
$397$	927.638	0.117272	0.0586358	+	0.998279i	$0.481325\pi$
$398$	−5219.97	−0.657420
$399$	0	0
$400$	−518.087	−0.0647609
$401$	1954.86	0.243444	0.121722	−	0.992564i	$-0.461158\pi$
$401$	1954.86	0.243444	0.121722	+	0.992564i	$0.461158\pi$
$402$	0	0
$403$	1576.61	0.194880
$404$	3676.18	0.452715
$405$	0	0
$406$	−4476.70	−0.547228
$407$	−963.264	−0.117315
$408$	0	0
$409$	−3081.52	−0.372546	−0.186273	−	0.982498i	$-0.559641\pi$
$409$	−3081.52	−0.372546	−0.186273	+	0.982498i	$0.559641\pi$
$410$	12014.8	1.44724
$411$	0	0
$412$	3596.24	0.430034
$413$	30622.5	3.64851
$414$	0	0
$415$	−1490.28	−0.176277
$416$	5764.29	0.679369
$417$	0	0
$418$	163.812	0.0191682
$419$	−1349.08	−0.157296	−0.0786481	−	0.996902i	$-0.525060\pi$
$419$	−1349.08	−0.157296	−0.0786481	+	0.996902i	$0.525060\pi$
$420$	0	0
$421$	9632.71	1.11513	0.557565	−	0.830134i	$-0.311736\pi$
$421$	9632.71	1.11513	0.557565	+	0.830134i	$0.311736\pi$
$422$	28.2347	0.00325697
$423$	0	0
$424$	6700.86	0.767506
$425$	1927.24	0.219965
$426$	0	0
$427$	13989.2	1.58545
$428$	719.895	0.0813024
$429$	0	0
$430$	−618.562	−0.0693714
$431$	−8791.24	−0.982504	−0.491252	−	0.871018i	$-0.663461\pi$
$431$	−8791.24	−0.982504	−0.491252	+	0.871018i	$0.663461\pi$
$432$	0	0
$433$	10918.2	1.21177	0.605885	−	0.795552i	$-0.292819\pi$
$433$	10918.2	1.21177	0.605885	+	0.795552i	$0.292819\pi$
$434$	−2919.11	−0.322861
$435$	0	0
$436$	3890.00	0.427287
$437$	−81.2409	−0.00889309
$438$	0	0
$439$	−5800.72	−0.630645	−0.315323	−	0.948985i	$-0.602113\pi$
$439$	−5800.72	−0.630645	−0.315323	+	0.948985i	$0.602113\pi$
$440$	6224.68	0.674432
$441$	0	0
$442$	9774.70	1.05189
$443$	8203.93	0.879866	0.439933	−	0.898031i	$-0.355002\pi$
$443$	8203.93	0.879866	0.439933	+	0.898031i	$0.355002\pi$
$444$	0	0
$445$	4132.33	0.440205
$446$	1224.60	0.130014
$447$	0	0
$448$	−18791.7	−1.98175
$449$	−10030.6	−1.05429	−0.527144	−	0.849776i	$-0.676737\pi$
$449$	−10030.6	−1.05429	−0.527144	+	0.849776i	$0.676737\pi$
$450$	0	0
$451$	−9704.53	−1.01323
$452$	−3509.64	−0.365221
$453$	0	0
$454$	−4840.51	−0.500388
$455$	−18513.7	−1.90756
$456$	0	0
$457$	11691.9	1.19677	0.598385	−	0.801208i	$-0.295809\pi$
$457$	11691.9	1.19677	0.598385	+	0.801208i	$0.295809\pi$
$458$	9963.90	1.01656
$459$	0	0
$460$	−883.237	−0.0895242
$461$	−2132.60	−0.215456	−0.107728	−	0.994180i	$-0.534358\pi$
$461$	−2132.60	−0.215456	−0.107728	+	0.994180i	$0.534358\pi$
$462$	0	0
$463$	13503.1	1.35538	0.677690	−	0.735348i	$-0.262981\pi$
$463$	13503.1	1.35538	0.677690	+	0.735348i	$0.262981\pi$
$464$	1587.50	0.158831
$465$	0	0
$466$	−12560.8	−1.24864
$467$	−10319.2	−1.02252	−0.511261	−	0.859425i	$-0.670821\pi$
$467$	−10319.2	−1.02252	−0.511261	+	0.859425i	$0.670821\pi$
$468$	0	0
$469$	−17712.6	−1.74391
$470$	5008.90	0.491581
$471$	0	0
$472$	−20781.6	−2.02659
$473$	499.622	0.0485680
$474$	0	0
$475$	65.1921	0.00629730
$476$	12104.2	1.16554
$477$	0	0
$478$	9790.51	0.936836
$479$	−6969.19	−0.664782	−0.332391	−	0.943142i	$-0.607855\pi$
$479$	−6969.19	−0.664782	−0.332391	+	0.943142i	$0.607855\pi$
$480$	0	0
$481$	−1944.29	−0.184308
$482$	−10072.5	−0.951849
$483$	0	0
$484$	2828.95	0.265679
$485$	−17334.3	−1.62291
$486$	0	0
$487$	−8355.82	−0.777491	−0.388746	−	0.921345i	$-0.627092\pi$
$487$	−8355.82	−0.777491	−0.388746	+	0.921345i	$0.627092\pi$
$488$	−9493.64	−0.880649
$489$	0	0
$490$	25283.5	2.33100
$491$	454.131	0.0417407	0.0208703	−	0.999782i	$-0.493356\pi$
$491$	454.131	0.0417407	0.0208703	+	0.999782i	$0.493356\pi$
$492$	0	0
$493$	−5905.36	−0.539481
$494$	330.645	0.0301142
$495$	0	0
$496$	1035.15	0.0937093
$497$	27748.3	2.50439
$498$	0	0
$499$	−17517.2	−1.57150	−0.785748	−	0.618547i	$-0.787721\pi$
$499$	−17517.2	−1.57150	−0.785748	+	0.618547i	$0.787721\pi$
$500$	−4091.44	−0.365950
$501$	0	0
$502$	3017.14	0.268250
$503$	19111.9	1.69415	0.847076	−	0.531471i	$-0.178361\pi$
$503$	19111.9	1.69415	0.847076	+	0.531471i	$0.178361\pi$
$504$	0	0
$505$	13733.1	1.21012
$506$	−1066.66	−0.0937133
$507$	0	0
$508$	−3978.92	−0.347512
$509$	3042.66	0.264958	0.132479	−	0.991186i	$-0.457706\pi$
$509$	3042.66	0.264958	0.132479	+	0.991186i	$0.457706\pi$
$510$	0	0
$511$	39533.8	3.42245
$512$	9294.54	0.802275
$513$	0	0
$514$	2402.36	0.206155
$515$	13434.4	1.14950
$516$	0	0
$517$	−4045.76	−0.344163
$518$	3599.87	0.305346
$519$	0	0
$520$	12564.2	1.05957
$521$	15630.7	1.31438	0.657192	−	0.753723i	$-0.271744\pi$
$521$	15630.7	1.31438	0.657192	+	0.753723i	$0.271744\pi$
$522$	0	0
$523$	−10090.2	−0.843619	−0.421810	−	0.906684i	$-0.638605\pi$
$523$	−10090.2	−0.843619	−0.421810	+	0.906684i	$0.638605\pi$
$524$	−8622.10	−0.718813
$525$	0	0
$526$	10412.2	0.863107
$527$	−3850.69	−0.318290
$528$	0	0
$529$	529.000	0.0434783
$530$	7161.97	0.586974
$531$	0	0
$532$	409.445	0.0333679
$533$	−19588.0	−1.59184
$534$	0	0
$535$	2689.30	0.217325
$536$	12020.5	0.968668
$537$	0	0
$538$	−5497.32	−0.440532
$539$	−20421.8	−1.63197
$540$	0	0
$541$	8104.99	0.644105	0.322052	−	0.946722i	$-0.395627\pi$
$541$	8104.99	0.644105	0.322052	+	0.946722i	$0.395627\pi$
$542$	13629.3	1.08013
$543$	0	0
$544$	−14078.6	−1.10959
$545$	14531.8	1.14216
$546$	0	0
$547$	9989.69	0.780856	0.390428	−	0.920634i	$-0.372327\pi$
$547$	9989.69	0.780856	0.390428	+	0.920634i	$0.372327\pi$
$548$	−4637.36	−0.361493
$549$	0	0
$550$	855.948	0.0663595
$551$	−199.758	−0.0154446
$552$	0	0
$553$	−21423.1	−1.64739
$554$	−14820.5	−1.13658
$555$	0	0
$556$	5719.50	0.436260
$557$	−11784.8	−0.896481	−0.448241	−	0.893913i	$-0.647949\pi$
$557$	−11784.8	−0.896481	−0.448241	+	0.893913i	$0.647949\pi$
$558$	0	0
$559$	1008.46	0.0763027
$560$	−12155.5	−0.917260
$561$	0	0
$562$	6778.38	0.508770
$563$	−15284.1	−1.14413	−0.572067	−	0.820207i	$-0.693858\pi$
$563$	−15284.1	−1.14413	−0.572067	+	0.820207i	$0.693858\pi$
$564$	0	0
$565$	−13110.9	−0.976250
$566$	−7828.25	−0.581353
$567$	0	0
$568$	−18831.1	−1.39108
$569$	−14150.0	−1.04253	−0.521263	−	0.853396i	$-0.674539\pi$
$569$	−14150.0	−1.04253	−0.521263	+	0.853396i	$0.674539\pi$
$570$	0	0
$571$	−11692.1	−0.856914	−0.428457	−	0.903562i	$-0.640943\pi$
$571$	−11692.1	−0.856914	−0.428457	+	0.903562i	$0.640943\pi$
$572$	−2903.51	−0.212241
$573$	0	0
$574$	36267.3	2.63723
$575$	−424.498	−0.0307875
$576$	0	0
$577$	−15258.6	−1.10091	−0.550455	−	0.834865i	$-0.685546\pi$
$577$	−15258.6	−1.10091	−0.550455	+	0.834865i	$0.685546\pi$
$578$	−13116.7	−0.943913
$579$	0	0
$580$	−2171.74	−0.155477
$581$	−4498.49	−0.321220
$582$	0	0
$583$	−5784.84	−0.410949
$584$	−26829.2	−1.90102
$585$	0	0
$586$	−17681.3	−1.24643
$587$	18011.8	1.26648	0.633242	−	0.773954i	$-0.281724\pi$
$587$	18011.8	1.26648	0.633242	+	0.773954i	$0.281724\pi$
$588$	0	0
$589$	−130.256	−0.00911223
$590$	−22211.7	−1.54990
$591$	0	0
$592$	−1276.56	−0.0886255
$593$	20155.3	1.39575	0.697874	−	0.716221i	$-0.254130\pi$
$593$	20155.3	1.39575	0.697874	+	0.716221i	$0.254130\pi$
$594$	0	0
$595$	45217.7	3.11554
$596$	−7574.38	−0.520568
$597$	0	0
$598$	−2152.99	−0.147228
$599$	−4201.56	−0.286596	−0.143298	−	0.989680i	$-0.545771\pi$
$599$	−4201.56	−0.286596	−0.143298	+	0.989680i	$0.545771\pi$
$600$	0	0
$601$	−4514.35	−0.306396	−0.153198	−	0.988195i	$-0.548957\pi$
$601$	−4514.35	−0.306396	−0.153198	+	0.988195i	$0.548957\pi$
$602$	−1867.17	−0.126412
$603$	0	0
$604$	−5841.33	−0.393511
$605$	10568.1	0.710171
$606$	0	0
$607$	12457.9	0.833032	0.416516	−	0.909128i	$-0.363251\pi$
$607$	12457.9	0.833032	0.416516	+	0.909128i	$0.363251\pi$
$608$	−476.231	−0.0317660
$609$	0	0
$610$	−10146.9	−0.673504
$611$	−8166.13	−0.540698
$612$	0	0
$613$	633.774	0.0417584	0.0208792	−	0.999782i	$-0.493353\pi$
$613$	633.774	0.0417584	0.0208792	+	0.999782i	$0.493353\pi$
$614$	−10974.1	−0.721301
$615$	0	0
$616$	18789.6	1.22898
$617$	−15940.6	−1.04010	−0.520051	−	0.854135i	$-0.674087\pi$
$617$	−15940.6	−1.04010	−0.520051	+	0.854135i	$0.674087\pi$
$618$	0	0
$619$	−13100.8	−0.850675	−0.425337	−	0.905035i	$-0.639845\pi$
$619$	−13100.8	−0.850675	−0.425337	+	0.905035i	$0.639845\pi$
$620$	−1416.12	−0.0917303
$621$	0	0
$622$	−8799.73	−0.567262
$623$	12473.7	0.802163
$624$	0	0
$625$	−17591.4	−1.12585
$626$	−19528.6	−1.24684
$627$	0	0
$628$	2726.56	0.173251
$629$	4748.70	0.301022
$630$	0	0
$631$	−4711.66	−0.297255	−0.148628	−	0.988893i	$-0.547486\pi$
$631$	−4711.66	−0.297255	−0.148628	+	0.988893i	$0.547486\pi$
$632$	14538.6	0.915053
$633$	0	0
$634$	14816.7	0.928147
$635$	−14864.0	−0.928914
$636$	0	0
$637$	−41220.3	−2.56390
$638$	−2622.75	−0.162752
$639$	0	0
$640$	711.541	0.0439471
$641$	−629.519	−0.0387902	−0.0193951	−	0.999812i	$-0.506174\pi$
$641$	−629.519	−0.0387902	−0.0193951	+	0.999812i	$0.506174\pi$
$642$	0	0
$643$	−5126.76	−0.314432	−0.157216	−	0.987564i	$-0.550252\pi$
$643$	−5126.76	−0.314432	−0.157216	+	0.987564i	$0.550252\pi$
$644$	−2666.10	−0.163135
$645$	0	0
$646$	−807.562	−0.0491843
$647$	25194.9	1.53093	0.765466	−	0.643476i	$-0.222508\pi$
$647$	25194.9	1.53093	0.765466	+	0.643476i	$0.222508\pi$
$648$	0	0
$649$	17940.7	1.08511
$650$	1727.68	0.104254
$651$	0	0
$652$	8444.46	0.507225
$653$	17041.5	1.02126	0.510631	−	0.859800i	$-0.329412\pi$
$653$	17041.5	1.02126	0.510631	+	0.859800i	$0.329412\pi$
$654$	0	0
$655$	−32209.5	−1.92142
$656$	−12860.9	−0.765446
$657$	0	0
$658$	15119.6	0.895783
$659$	−8605.48	−0.508683	−0.254341	−	0.967115i	$-0.581859\pi$
$659$	−8605.48	−0.508683	−0.254341	+	0.967115i	$0.581859\pi$
$660$	0	0
$661$	−15024.1	−0.884066	−0.442033	−	0.896999i	$-0.645743\pi$
$661$	−15024.1	−0.884066	−0.442033	+	0.896999i	$0.645743\pi$
$662$	24561.7	1.44202
$663$	0	0
$664$	3052.85	0.178424
$665$	1529.56	0.0891937
$666$	0	0
$667$	1300.73	0.0755087
$668$	−7392.68	−0.428191
$669$	0	0
$670$	12847.7	0.740819
$671$	8195.84	0.471530
$672$	0	0
$673$	2865.62	0.164133	0.0820664	−	0.996627i	$-0.473848\pi$
$673$	2865.62	0.164133	0.0820664	+	0.996627i	$0.473848\pi$
$674$	−5504.01	−0.314550
$675$	0	0
$676$	1183.45	0.0673331
$677$	−6533.41	−0.370900	−0.185450	−	0.982654i	$-0.559374\pi$
$677$	−6533.41	−0.370900	−0.185450	+	0.982654i	$0.559374\pi$
$678$	0	0
$679$	−52324.7	−2.95735
$680$	−30686.5	−1.73055
$681$	0	0
$682$	−1710.21	−0.0960225
$683$	−24152.8	−1.35312	−0.676559	−	0.736388i	$-0.736530\pi$
$683$	−24152.8	−1.35312	−0.676559	+	0.736388i	$0.736530\pi$
$684$	0	0
$685$	−17323.7	−0.966285
$686$	49168.1	2.73651
$687$	0	0
$688$	662.121	0.0366906
$689$	−11676.3	−0.645622
$690$	0	0
$691$	−3290.80	−0.181169	−0.0905846	−	0.995889i	$-0.528874\pi$
$691$	−3290.80	−0.181169	−0.0905846	+	0.995889i	$0.528874\pi$
$692$	6384.79	0.350742
$693$	0	0
$694$	841.989	0.0460540
$695$	21366.2	1.16614
$696$	0	0
$697$	47841.5	2.59989
$698$	5693.78	0.308757
$699$	0	0
$700$	2139.43	0.115518
$701$	−8816.55	−0.475031	−0.237515	−	0.971384i	$-0.576333\pi$
$701$	−8816.55	−0.475031	−0.237515	+	0.971384i	$0.576333\pi$
$702$	0	0
$703$	160.632	0.00861788
$704$	−11009.4	−0.589393
$705$	0	0
$706$	−12580.9	−0.670666
$707$	41454.0	2.20515
$708$	0	0
$709$	26955.8	1.42785	0.713926	−	0.700221i	$-0.246915\pi$
$709$	26955.8	1.42785	0.713926	+	0.700221i	$0.246915\pi$
$710$	−20126.9	−1.06387
$711$	0	0
$712$	−8465.13	−0.445567
$713$	848.161	0.0445496
$714$	0	0
$715$	−10846.6	−0.567328
$716$	−11306.6	−0.590151
$717$	0	0
$718$	24057.1	1.25042
$719$	−24126.2	−1.25140	−0.625698	−	0.780065i	$-0.715186\pi$
$719$	−24126.2	−1.25140	−0.625698	+	0.780065i	$0.715186\pi$
$720$	0	0
$721$	40552.6	2.09467
$722$	14990.3	0.772689
$723$	0	0
$724$	−14130.8	−0.725371
$725$	−1043.77	−0.0534686
$726$	0	0
$727$	10884.1	0.555252	0.277626	−	0.960689i	$-0.410452\pi$
$727$	10884.1	0.555252	0.277626	+	0.960689i	$0.410452\pi$
$728$	37925.6	1.93079
$729$	0	0
$730$	−28675.4	−1.45387
$731$	−2463.04	−0.124622
$732$	0	0
$733$	18865.2	0.950619	0.475309	−	0.879819i	$-0.342336\pi$
$733$	18865.2	0.950619	0.475309	+	0.879819i	$0.342336\pi$
$734$	−5753.08	−0.289305
$735$	0	0
$736$	3100.98	0.155304
$737$	−10377.3	−0.518658
$738$	0	0
$739$	19750.5	0.983133	0.491567	−	0.870840i	$-0.336424\pi$
$739$	19750.5	0.983133	0.491567	+	0.870840i	$0.336424\pi$
$740$	1746.37	0.0867538
$741$	0	0
$742$	21618.8	1.06961
$743$	10392.5	0.513141	0.256570	−	0.966526i	$-0.417407\pi$
$743$	10392.5	0.513141	0.256570	+	0.966526i	$0.417407\pi$
$744$	0	0
$745$	−28295.5	−1.39150
$746$	7911.86	0.388303
$747$	0	0
$748$	7091.48	0.346645
$749$	8117.82	0.396019
$750$	0	0
$751$	12887.2	0.626177	0.313089	−	0.949724i	$-0.398636\pi$
$751$	12887.2	0.626177	0.313089	+	0.949724i	$0.398636\pi$
$752$	−5361.63	−0.259998
$753$	0	0
$754$	−5293.86	−0.255691
$755$	−21821.4	−1.05187
$756$	0	0
$757$	−10543.7	−0.506231	−0.253116	−	0.967436i	$-0.581455\pi$
$757$	−10543.7	−0.506231	−0.253116	+	0.967436i	$0.581455\pi$
$758$	−13172.0	−0.631170
$759$	0	0
$760$	−1038.02	−0.0495433
$761$	21165.3	1.00820	0.504101	−	0.863645i	$-0.331824\pi$
$761$	21165.3	1.00820	0.504101	+	0.863645i	$0.331824\pi$
$762$	0	0
$763$	43865.1	2.08129
$764$	−4345.52	−0.205779
$765$	0	0
$766$	9640.74	0.454744
$767$	36212.3	1.70476
$768$	0	0
$769$	14341.8	0.672535	0.336267	−	0.941767i	$-0.390835\pi$
$769$	14341.8	0.672535	0.336267	+	0.941767i	$0.390835\pi$
$770$	20082.5	0.939902
$771$	0	0
$772$	568.711	0.0265134
$773$	5489.88	0.255443	0.127721	−	0.991810i	$-0.459234\pi$
$773$	5489.88	0.255443	0.127721	+	0.991810i	$0.459234\pi$
$774$	0	0
$775$	−680.611	−0.0315461
$776$	35509.6	1.64268
$777$	0	0
$778$	−18635.2	−0.858747
$779$	1618.31	0.0744315
$780$	0	0
$781$	16256.8	0.744833
$782$	5258.44	0.240462
$783$	0	0
$784$	−27063.9	−1.23287
$785$	10185.6	0.463106
$786$	0	0
$787$	−9357.97	−0.423857	−0.211929	−	0.977285i	$-0.567974\pi$
$787$	−9357.97	−0.423857	−0.211929	+	0.977285i	$0.567974\pi$
$788$	−9637.16	−0.435672
$789$	0	0
$790$	15539.0	0.699815
$791$	−39576.1	−1.77897
$792$	0	0
$793$	16542.8	0.740797
$794$	−2031.04	−0.0907795
$795$	0	0
$796$	−7643.94	−0.340367
$797$	−76.5677	−0.00340297	−0.00170148	−	0.999999i	$-0.500542\pi$
$797$	−76.5677	−0.00340297	−0.00170148	+	0.999999i	$0.500542\pi$
$798$	0	0
$799$	19944.8	0.883100
$800$	−2488.40	−0.109973
$801$	0	0
$802$	−4280.13	−0.188449
$803$	23161.5	1.01787
$804$	0	0
$805$	−9959.73	−0.436067
$806$	−3451.96	−0.150856
$807$	0	0
$808$	−28132.3	−1.22487
$809$	11216.3	0.487449	0.243724	−	0.969845i	$-0.421631\pi$
$809$	11216.3	0.487449	0.243724	+	0.969845i	$0.421631\pi$
$810$	0	0
$811$	−13688.9	−0.592701	−0.296351	−	0.955079i	$-0.595770\pi$
$811$	−13688.9	−0.592701	−0.296351	+	0.955079i	$0.595770\pi$
$812$	−6555.52	−0.283317
$813$	0	0
$814$	2109.04	0.0908132
$815$	31545.9	1.35583
$816$	0	0
$817$	−83.3162	−0.00356777
$818$	6746.91	0.288386
$819$	0	0
$820$	17594.0	0.749281
$821$	34053.7	1.44760	0.723802	−	0.690008i	$-0.242393\pi$
$821$	34053.7	1.44760	0.723802	+	0.690008i	$0.242393\pi$
$822$	0	0
$823$	27930.4	1.18298	0.591490	−	0.806313i	$-0.298540\pi$
$823$	27930.4	1.18298	0.591490	+	0.806313i	$0.298540\pi$
$824$	−27520.6	−1.16350
$825$	0	0
$826$	−67047.2	−2.82430
$827$	−17136.6	−0.720554	−0.360277	−	0.932845i	$-0.617318\pi$
$827$	−17136.6	−0.720554	−0.360277	+	0.932845i	$0.617318\pi$
$828$	0	0
$829$	8180.70	0.342735	0.171368	−	0.985207i	$-0.445181\pi$
$829$	8180.70	0.342735	0.171368	+	0.985207i	$0.445181\pi$
$830$	3262.93	0.136455
$831$	0	0
$832$	−22221.8	−0.925966
$833$	100676.	4.18752
$834$	0	0
$835$	−27616.7	−1.14457
$836$	239.881	0.00992398
$837$	0	0
$838$	2953.79	0.121762
$839$	12825.5	0.527753	0.263877	−	0.964556i	$-0.414999\pi$
$839$	12825.5	0.527753	0.263877	+	0.964556i	$0.414999\pi$
$840$	0	0
$841$	−21190.7	−0.868864
$842$	−21090.6	−0.863218
$843$	0	0
$844$	41.3458	0.00168623
$845$	4420.99	0.179984
$846$	0	0
$847$	31900.3	1.29411
$848$	−7666.32	−0.310451
$849$	0	0
$850$	−4219.66	−0.170274
$851$	−1045.96	−0.0421328
$852$	0	0
$853$	−10101.9	−0.405490	−0.202745	−	0.979232i	$-0.564986\pi$
$853$	−10101.9	−0.405490	−0.202745	+	0.979232i	$0.564986\pi$
$854$	−30629.1	−1.22729
$855$	0	0
$856$	−5509.07	−0.219972
$857$	−8138.65	−0.324400	−0.162200	−	0.986758i	$-0.551859\pi$
$857$	−8138.65	−0.324400	−0.162200	+	0.986758i	$0.551859\pi$
$858$	0	0
$859$	32834.0	1.30417	0.652084	−	0.758147i	$-0.273895\pi$
$859$	32834.0	1.30417	0.652084	+	0.758147i	$0.273895\pi$
$860$	−905.800	−0.0359157
$861$	0	0
$862$	19248.2	0.760553
$863$	6055.28	0.238846	0.119423	−	0.992843i	$-0.461896\pi$
$863$	6055.28	0.238846	0.119423	+	0.992843i	$0.461896\pi$
$864$	0	0
$865$	23851.6	0.937546
$866$	−23905.2	−0.938028
$867$	0	0
$868$	−4274.64	−0.167155
$869$	−12551.1	−0.489951
$870$	0	0
$871$	−20945.9	−0.814838
$872$	−29768.6	−1.15607
$873$	0	0
$874$	177.875	0.00688411
$875$	−46136.7	−1.78252
$876$	0	0
$877$	−17147.2	−0.660227	−0.330113	−	0.943941i	$-0.607087\pi$
$877$	−17147.2	−0.660227	−0.330113	+	0.943941i	$0.607087\pi$
$878$	12700.5	0.488180
$879$	0	0
$880$	−7121.53	−0.272803
$881$	33590.6	1.28456	0.642280	−	0.766470i	$-0.277989\pi$
$881$	33590.6	1.28456	0.642280	+	0.766470i	$0.277989\pi$
$882$	0	0
$883$	−46447.7	−1.77020	−0.885102	−	0.465398i	$-0.845911\pi$
$883$	−46447.7	−1.77020	−0.885102	+	0.465398i	$0.845911\pi$
$884$	14313.7	0.544596
$885$	0	0
$886$	−17962.3	−0.681101
$887$	19697.6	0.745638	0.372819	−	0.927904i	$-0.378391\pi$
$887$	19697.6	0.745638	0.372819	+	0.927904i	$0.378391\pi$
$888$	0	0
$889$	−44867.9	−1.69271
$890$	−9047.64	−0.340761
$891$	0	0
$892$	1793.26	0.0673124
$893$	674.666	0.0252820
$894$	0	0
$895$	−42238.0	−1.57750
$896$	2147.83	0.0800825
$897$	0	0
$898$	21961.9	0.816121
$899$	2085.49	0.0773693
$900$	0	0
$901$	28518.1	1.05447
$902$	21247.8	0.784341
$903$	0	0
$904$	26857.9	0.988142
$905$	−52788.4	−1.93895
$906$	0	0
$907$	33566.6	1.22884	0.614422	−	0.788978i	$-0.289389\pi$
$907$	33566.6	1.22884	0.614422	+	0.788978i	$0.289389\pi$
$908$	−7088.27	−0.259066
$909$	0	0
$910$	40535.4	1.47663
$911$	27310.8	0.993244	0.496622	−	0.867967i	$-0.334573\pi$
$911$	27310.8	0.993244	0.496622	+	0.867967i	$0.334573\pi$
$912$	0	0
$913$	−2635.52	−0.0955344
$914$	−25599.2	−0.926417
$915$	0	0
$916$	14590.8	0.526303
$917$	−97226.1	−3.50130
$918$	0	0
$919$	−18988.9	−0.681593	−0.340797	−	0.940137i	$-0.610697\pi$
$919$	−18988.9	−0.681593	−0.340797	+	0.940137i	$0.610697\pi$
$920$	6759.06	0.242217
$921$	0	0
$922$	4669.28	0.166784
$923$	32813.4	1.17017
$924$	0	0
$925$	839.334	0.0298347
$926$	−29564.7	−1.04920
$927$	0	0
$928$	7624.81	0.269716
$929$	8168.78	0.288492	0.144246	−	0.989542i	$-0.453924\pi$
$929$	8168.78	0.288492	0.144246	+	0.989542i	$0.453924\pi$
$930$	0	0
$931$	3405.52	0.119883
$932$	−18393.6	−0.646460
$933$	0	0
$934$	22593.8	0.791531
$935$	26491.6	0.926595
$936$	0	0
$937$	−21901.4	−0.763594	−0.381797	−	0.924246i	$-0.624695\pi$
$937$	−21901.4	−0.763594	−0.381797	+	0.924246i	$0.624695\pi$
$938$	38781.4	1.34996
$939$	0	0
$940$	7334.85	0.254507
$941$	−3043.08	−0.105421	−0.0527107	−	0.998610i	$-0.516786\pi$
$941$	−3043.08	−0.105421	−0.0527107	+	0.998610i	$0.516786\pi$
$942$	0	0
$943$	−10537.6	−0.363895
$944$	23775.8	0.819743
$945$	0	0
$946$	−1093.91	−0.0375963
$947$	3281.61	0.112606	0.0563030	−	0.998414i	$-0.482069\pi$
$947$	3281.61	0.112606	0.0563030	+	0.998414i	$0.482069\pi$
$948$	0	0
$949$	46750.2	1.59913
$950$	−142.737	−0.00487472
$951$	0	0
$952$	−92628.9	−3.15349
$953$	25885.0	0.879850	0.439925	−	0.898034i	$-0.355005\pi$
$953$	25885.0	0.879850	0.439925	+	0.898034i	$0.355005\pi$
$954$	0	0
$955$	−16233.5	−0.550056
$956$	14336.9	0.485029
$957$	0	0
$958$	15258.9	0.514606
$959$	−52292.6	−1.76081
$960$	0	0
$961$	−28431.1	−0.954353
$962$	4256.98	0.142672
$963$	0	0
$964$	−14749.9	−0.492802
$965$	2124.53	0.0708714
$966$	0	0
$967$	41321.7	1.37416	0.687082	−	0.726580i	$-0.258891\pi$
$967$	41321.7	1.37416	0.687082	+	0.726580i	$0.258891\pi$
$968$	−21648.8	−0.718822
$969$	0	0
$970$	37953.1	1.25629
$971$	−56834.5	−1.87838	−0.939189	−	0.343401i	$-0.888421\pi$
$971$	−56834.5	−1.87838	−0.939189	+	0.343401i	$0.888421\pi$
$972$	0	0
$973$	64495.3	2.12500
$974$	18294.9	0.601854
$975$	0	0
$976$	10861.5	0.356217
$977$	44716.5	1.46429	0.732144	−	0.681150i	$-0.238520\pi$
$977$	44716.5	1.46429	0.732144	+	0.681150i	$0.238520\pi$
$978$	0	0
$979$	7307.92	0.238572
$980$	37024.2	1.20683
$981$	0	0
$982$	−994.310	−0.0323113
$983$	−24317.3	−0.789014	−0.394507	−	0.918893i	$-0.629085\pi$
$983$	−24317.3	−0.789014	−0.394507	+	0.918893i	$0.629085\pi$
$984$	0	0
$985$	−36001.4	−1.16457
$986$	12929.6	0.417610
$987$	0	0
$988$	484.185	0.0155911
$989$	542.514	0.0174428
$990$	0	0
$991$	1887.26	0.0604954	0.0302477	−	0.999542i	$-0.490370\pi$
$991$	1887.26	0.0604954	0.0302477	+	0.999542i	$0.490370\pi$
$992$	4971.89	0.159131
$993$	0	0
$994$	−60754.3	−1.93864
$995$	−28555.4	−0.909815
$996$	0	0
$997$	17485.0	0.555423	0.277712	−	0.960664i	$-0.410424\pi$
$997$	17485.0	0.555423	0.277712	+	0.960664i	$0.410424\pi$
$998$	38353.4	1.21649
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.h.1.2	yes	5
3.2	odd	2		207.4.a.g.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.a.g.1.4	✓	5	3.2	odd	2
207.4.a.h.1.2	yes	5	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18948 q^{2} -3.20619 q^{4} -11.9773 q^{5} -36.1543 q^{7} +24.5357 q^{8} +26.2241 q^{10} -21.1816 q^{11} -42.7538 q^{13} +79.1589 q^{14} -28.0708 q^{16} +104.421 q^{17} +3.53221 q^{19} +38.4016 q^{20} +46.3766 q^{22} -23.0000 q^{23} +18.4565 q^{25} +93.6084 q^{26} +115.918 q^{28} -56.5533 q^{29} -36.8766 q^{31} -134.825 q^{32} -228.628 q^{34} +433.032 q^{35} +45.4764 q^{37} -7.73370 q^{38} -293.872 q^{40} +458.159 q^{41} -23.5875 q^{43} +67.9123 q^{44} +50.3580 q^{46} +191.004 q^{47} +964.131 q^{49} -40.4100 q^{50} +137.077 q^{52} +273.107 q^{53} +253.699 q^{55} -887.070 q^{56} +123.822 q^{58} -846.995 q^{59} -386.932 q^{61} +80.7404 q^{62} +519.763 q^{64} +512.076 q^{65} +489.918 q^{67} -334.794 q^{68} -948.113 q^{70} -767.497 q^{71} -1093.47 q^{73} -99.5696 q^{74} -11.3250 q^{76} +765.805 q^{77} +592.548 q^{79} +336.213 q^{80} -1003.13 q^{82} +124.425 q^{83} -1250.69 q^{85} +51.6444 q^{86} -519.705 q^{88} -345.013 q^{89} +1545.73 q^{91} +73.7424 q^{92} -418.198 q^{94} -42.3065 q^{95} +1447.26 q^{97} -2110.94 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 16 q^{4} + 20 q^{5} - 10 q^{7} + 48 q^{8} + 50 q^{10} + 46 q^{11} + 54 q^{13} + 164 q^{14} - 60 q^{16} + 250 q^{17} - 28 q^{19} + 242 q^{20} - 10 q^{22} - 115 q^{23} + 239 q^{25} + 368 q^{26}+ \cdots - 2400 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 16 * q^4 + 20 * q^5 - 10 * q^7 + 48 * q^8 + 50 * q^10 + 46 * q^11 + 54 * q^13 + 164 * q^14 - 60 * q^16 + 250 * q^17 - 28 * q^19 + 242 * q^20 - 10 * q^22 - 115 * q^23 + 239 * q^25 + 368 * q^26 + 224 * q^28 + 460 * q^29 - 360 * q^31 + 268 * q^32 - 20 * q^34 + 964 * q^35 - 92 * q^37 + 266 * q^38 - 278 * q^40 + 788 * q^41 + 108 * q^43 + 354 * q^44 - 92 * q^46 + 724 * q^47 + 149 * q^49 + 128 * q^50 - 288 * q^52 + 580 * q^53 - 672 * q^55 - 1172 * q^56 - 48 * q^58 - 408 * q^59 - 620 * q^61 - 1088 * q^62 + 696 * q^64 + 184 * q^65 + 100 * q^67 - 1584 * q^68 + 728 * q^70 - 124 * q^71 - 1946 * q^73 - 2866 * q^74 + 1362 * q^76 - 396 * q^77 - 198 * q^79 - 2398 * q^80 + 76 * q^82 - 578 * q^83 + 376 * q^85 - 3910 * q^86 + 1366 * q^88 + 930 * q^89 + 1972 * q^91 - 368 * q^92 + 976 * q^94 - 908 * q^95 + 2618 * q^97 - 2400 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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