LMFDB - Embedded newform 207.4.a.h.1.1

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			1.1
Root			$4.57357$ 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)

$f(q)$	$=$	\(q-3.57357 q^{2} +4.77043 q^{4} +14.0958 q^{5} +0.770211 q^{7} +11.5411 q^{8} -50.3723 q^{10} +34.6566 q^{11} -15.6261 q^{13} -2.75241 q^{14} -79.4064 q^{16} -0.878967 q^{17} -23.7345 q^{19} +67.2429 q^{20} -123.848 q^{22} -23.0000 q^{23} +73.6905 q^{25} +55.8410 q^{26} +3.67424 q^{28} +208.088 q^{29} +28.7376 q^{31} +191.436 q^{32} +3.14106 q^{34} +10.8567 q^{35} +378.729 q^{37} +84.8171 q^{38} +162.680 q^{40} -85.2339 q^{41} +405.033 q^{43} +165.327 q^{44} +82.1922 q^{46} +302.256 q^{47} -342.407 q^{49} -263.339 q^{50} -74.5432 q^{52} +49.6962 q^{53} +488.512 q^{55} +8.88908 q^{56} -743.618 q^{58} +337.780 q^{59} +439.911 q^{61} -102.696 q^{62} -48.8596 q^{64} -220.261 q^{65} -324.861 q^{67} -4.19306 q^{68} -38.7973 q^{70} +88.8411 q^{71} +224.481 q^{73} -1353.42 q^{74} -113.224 q^{76} +26.6929 q^{77} -1085.07 q^{79} -1119.29 q^{80} +304.590 q^{82} -463.847 q^{83} -12.3897 q^{85} -1447.41 q^{86} +399.976 q^{88} +1296.24 q^{89} -12.0354 q^{91} -109.720 q^{92} -1080.14 q^{94} -334.556 q^{95} +195.994 q^{97} +1223.62 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.57357	−1.26345	−0.631725	−	0.775193i	$-0.717653\pi$
$2$	−3.57357	−1.26345	−0.631725	+	0.775193i	$0.717653\pi$
$3$	0	0
$3$	0	0
$4$	4.77043	0.596304
$5$	14.0958	1.26076	0.630382	−	0.776285i	$-0.282898\pi$
$5$	14.0958	1.26076	0.630382	+	0.776285i	$0.282898\pi$
$6$	0	0
$7$	0.770211	0.0415875	0.0207937	−	0.999784i	$-0.493381\pi$
$7$	0.770211	0.0415875	0.0207937	+	0.999784i	$0.493381\pi$
$8$	11.5411	0.510049
$9$	0	0
$10$	−50.3723	−1.59291
$11$	34.6566	0.949943	0.474971	−	0.880001i	$-0.342458\pi$
$11$	34.6566	0.949943	0.474971	+	0.880001i	$0.342458\pi$
$12$	0	0
$13$	−15.6261	−0.333377	−0.166688	−	0.986010i	$-0.553307\pi$
$13$	−15.6261	−0.333377	−0.166688	+	0.986010i	$0.553307\pi$
$14$	−2.75241	−0.0525437
$15$	0	0
$16$	−79.4064	−1.24073
$17$	−0.878967	−0.0125401	−0.00627003	−	0.999980i	$-0.501996\pi$
$17$	−0.878967	−0.0125401	−0.00627003	+	0.999980i	$0.501996\pi$
$18$	0	0
$19$	−23.7345	−0.286583	−0.143291	−	0.989681i	$-0.545769\pi$
$19$	−23.7345	−0.286583	−0.143291	+	0.989681i	$0.545769\pi$
$20$	67.2429	0.751799
$21$	0	0
$22$	−123.848	−1.20020
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	73.6905	0.589524
$26$	55.8410	0.421204
$27$	0	0
$28$	3.67424	0.0247988
$29$	208.088	1.33245	0.666224	−	0.745752i	$-0.267910\pi$
$29$	208.088	1.33245	0.666224	+	0.745752i	$0.267910\pi$
$30$	0	0
$31$	28.7376	0.166498	0.0832489	−	0.996529i	$-0.473470\pi$
$31$	28.7376	0.166498	0.0832489	+	0.996529i	$0.473470\pi$
$32$	191.436	1.05754
$33$	0	0
$34$	3.14106	0.0158437
$35$	10.8567	0.0524320
$36$	0	0
$37$	378.729	1.68278	0.841388	−	0.540431i	$-0.181739\pi$
$37$	378.729	1.68278	0.841388	+	0.540431i	$0.181739\pi$
$38$	84.8171	0.362083
$39$	0	0
$40$	162.680	0.643051
$41$	−85.2339	−0.324666	−0.162333	−	0.986736i	$-0.551902\pi$
$41$	−85.2339	−0.324666	−0.162333	+	0.986736i	$0.551902\pi$
$42$	0	0
$43$	405.033	1.43644	0.718220	−	0.695816i	$-0.244957\pi$
$43$	405.033	1.43644	0.718220	+	0.695816i	$0.244957\pi$
$44$	165.327	0.566455
$45$	0	0
$46$	82.1922	0.263447
$47$	302.256	0.938055	0.469028	−	0.883183i	$-0.344604\pi$
$47$	302.256	0.938055	0.469028	+	0.883183i	$0.344604\pi$
$48$	0	0
$49$	−342.407	−0.998270
$50$	−263.339	−0.744834
$51$	0	0
$52$	−74.5432	−0.198794
$53$	49.6962	0.128798	0.0643990	−	0.997924i	$-0.479487\pi$
$53$	49.6962	0.128798	0.0643990	+	0.997924i	$0.479487\pi$
$54$	0	0
$55$	488.512	1.19765
$56$	8.88908	0.0212117
$57$	0	0
$58$	−743.618	−1.68348
$59$	337.780	0.745343	0.372672	−	0.927963i	$-0.378442\pi$
$59$	337.780	0.745343	0.372672	+	0.927963i	$0.378442\pi$
$60$	0	0
$61$	439.911	0.923359	0.461679	−	0.887047i	$-0.347247\pi$
$61$	439.911	0.923359	0.461679	+	0.887047i	$0.347247\pi$
$62$	−102.696	−0.210361
$63$	0	0
$64$	−48.8596	−0.0954288
$65$	−220.261	−0.420309
$66$	0	0
$67$	−324.861	−0.592359	−0.296180	−	0.955132i	$-0.595713\pi$
$67$	−324.861	−0.592359	−0.296180	+	0.955132i	$0.595713\pi$
$68$	−4.19306	−0.00747769
$69$	0	0
$70$	−38.7973	−0.0662452
$71$	88.8411	0.148500	0.0742500	−	0.997240i	$-0.476344\pi$
$71$	88.8411	0.148500	0.0742500	+	0.997240i	$0.476344\pi$
$72$	0	0
$73$	224.481	0.359911	0.179955	−	0.983675i	$-0.442405\pi$
$73$	224.481	0.359911	0.179955	+	0.983675i	$0.442405\pi$
$74$	−1353.42	−2.12610
$75$	0	0
$76$	−113.224	−0.170891
$77$	26.6929	0.0395057
$78$	0	0
$79$	−1085.07	−1.54532	−0.772661	−	0.634819i	$-0.781075\pi$
$79$	−1085.07	−1.54532	−0.772661	+	0.634819i	$0.781075\pi$
$80$	−1119.29	−1.56426
$81$	0	0
$82$	304.590	0.410199
$83$	−463.847	−0.613420	−0.306710	−	0.951803i	$-0.599228\pi$
$83$	−463.847	−0.613420	−0.306710	+	0.951803i	$0.599228\pi$
$84$	0	0
$85$	−12.3897	−0.0158100
$86$	−1447.41	−1.81487
$87$	0	0
$88$	399.976	0.484517
$89$	1296.24	1.54384	0.771918	−	0.635722i	$-0.219297\pi$
$89$	1296.24	1.54384	0.771918	+	0.635722i	$0.219297\pi$
$90$	0	0
$91$	−12.0354	−0.0138643
$92$	−109.720	−0.124338
$93$	0	0
$94$	−1080.14	−1.18519
$95$	−334.556	−0.361313
$96$	0	0
$97$	195.994	0.205156	0.102578	−	0.994725i	$-0.467291\pi$
$97$	195.994	0.205156	0.102578	+	0.994725i	$0.467291\pi$
$98$	1223.62	1.26126
$99$	0	0
$100$	351.536	0.351536
$101$	173.946	0.171369	0.0856847	−	0.996322i	$-0.472692\pi$
$101$	173.946	0.171369	0.0856847	+	0.996322i	$0.472692\pi$
$102$	0	0
$103$	−1750.76	−1.67483	−0.837413	−	0.546571i	$-0.815933\pi$
$103$	−1750.76	−1.67483	−0.837413	+	0.546571i	$0.815933\pi$
$104$	−180.342	−0.170038
$105$	0	0
$106$	−177.593	−0.162730
$107$	−427.158	−0.385934	−0.192967	−	0.981205i	$-0.561811\pi$
$107$	−427.158	−0.385934	−0.192967	+	0.981205i	$0.561811\pi$
$108$	0	0
$109$	1368.12	1.20222	0.601110	−	0.799166i	$-0.294725\pi$
$109$	1368.12	1.20222	0.601110	+	0.799166i	$0.294725\pi$
$110$	−1745.73	−1.51317
$111$	0	0
$112$	−61.1597	−0.0515987
$113$	339.060	0.282266	0.141133	−	0.989991i	$-0.454925\pi$
$113$	339.060	0.282266	0.141133	+	0.989991i	$0.454925\pi$
$114$	0	0
$115$	−324.203	−0.262887
$116$	992.671	0.794544
$117$	0	0
$118$	−1207.08	−0.941704
$119$	−0.676991	−0.000521509 0
$120$	0	0
$121$	−129.917	−0.0976085
$122$	−1572.06	−1.16662
$123$	0	0
$124$	137.091	0.0992833
$125$	−723.246	−0.517513
$126$	0	0
$127$	411.307	0.287382	0.143691	−	0.989623i	$-0.454103\pi$
$127$	411.307	0.287382	0.143691	+	0.989623i	$0.454103\pi$
$128$	−1356.89	−0.936975
$129$	0	0
$130$	787.121	0.531039
$131$	1230.44	0.820641	0.410320	−	0.911941i	$-0.365417\pi$
$131$	1230.44	0.820641	0.410320	+	0.911941i	$0.365417\pi$
$132$	0	0
$133$	−18.2806	−0.0119183
$134$	1160.91	0.748416
$135$	0	0
$136$	−10.1442	−0.00639604
$137$	2032.60	1.26757	0.633784	−	0.773510i	$-0.281501\pi$
$137$	2032.60	1.26757	0.633784	+	0.773510i	$0.281501\pi$
$138$	0	0
$139$	−292.753	−0.178640	−0.0893202	−	0.996003i	$-0.528469\pi$
$139$	−292.753	−0.178640	−0.0893202	+	0.996003i	$0.528469\pi$
$140$	51.7912	0.0312654
$141$	0	0
$142$	−317.480	−0.187622
$143$	−541.547	−0.316689
$144$	0	0
$145$	2933.16	1.67990
$146$	−802.199	−0.454729
$147$	0	0
$148$	1806.70	1.00345
$149$	−1089.45	−0.599002	−0.299501	−	0.954096i	$-0.596820\pi$
$149$	−1089.45	−0.599002	−0.299501	+	0.954096i	$0.596820\pi$
$150$	0	0
$151$	−292.288	−0.157523	−0.0787617	−	0.996893i	$-0.525097\pi$
$151$	−292.288	−0.157523	−0.0787617	+	0.996893i	$0.525097\pi$
$152$	−273.922	−0.146171
$153$	0	0
$154$	−95.3892	−0.0499135
$155$	405.079	0.209914
$156$	0	0
$157$	2191.65	1.11410	0.557048	−	0.830481i	$-0.311934\pi$
$157$	2191.65	1.11410	0.557048	+	0.830481i	$0.311934\pi$
$158$	3877.59	1.95244
$159$	0	0
$160$	2698.44	1.33331
$161$	−17.7149	−0.00867159
$162$	0	0
$163$	2654.50	1.27556	0.637781	−	0.770218i	$-0.279853\pi$
$163$	2654.50	1.27556	0.637781	+	0.770218i	$0.279853\pi$
$164$	−406.603	−0.193600
$165$	0	0
$166$	1657.59	0.775025
$167$	4092.12	1.89616	0.948078	−	0.318038i	$-0.103024\pi$
$167$	4092.12	1.89616	0.948078	+	0.318038i	$0.103024\pi$
$168$	0	0
$169$	−1952.83	−0.888860
$170$	44.2756	0.0199752
$171$	0	0
$172$	1932.18	0.856555
$173$	2402.61	1.05588	0.527940	−	0.849282i	$-0.322964\pi$
$173$	2402.61	1.05588	0.527940	+	0.849282i	$0.322964\pi$
$174$	0	0
$175$	56.7573	0.0245168
$176$	−2751.96	−1.17862
$177$	0	0
$178$	−4632.22	−1.95056
$179$	−2487.90	−1.03885	−0.519426	−	0.854515i	$-0.673854\pi$
$179$	−2487.90	−1.03885	−0.519426	+	0.854515i	$0.673854\pi$
$180$	0	0
$181$	−2223.03	−0.912907	−0.456454	−	0.889747i	$-0.650881\pi$
$181$	−2223.03	−0.912907	−0.456454	+	0.889747i	$0.650881\pi$
$182$	43.0093	0.0175168
$183$	0	0
$184$	−265.445	−0.106353
$185$	5338.48	2.12158
$186$	0	0
$187$	−30.4621	−0.0119123
$188$	1441.89	0.559366
$189$	0	0
$190$	1195.56	0.456501
$191$	−3732.12	−1.41386	−0.706928	−	0.707286i	$-0.749919\pi$
$191$	−3732.12	−1.41386	−0.706928	+	0.707286i	$0.749919\pi$
$192$	0	0
$193$	−4405.10	−1.64293	−0.821466	−	0.570258i	$-0.806843\pi$
$193$	−4405.10	−1.64293	−0.821466	+	0.570258i	$0.806843\pi$
$194$	−700.399	−0.259205
$195$	0	0
$196$	−1633.43	−0.595273
$197$	3105.42	1.12311	0.561554	−	0.827440i	$-0.310204\pi$
$197$	3105.42	1.12311	0.561554	+	0.827440i	$0.310204\pi$
$198$	0	0
$199$	−3960.09	−1.41067	−0.705334	−	0.708875i	$-0.749203\pi$
$199$	−3960.09	−1.41067	−0.705334	+	0.708875i	$0.749203\pi$
$200$	850.469	0.300686
$201$	0	0
$202$	−621.610	−0.216517
$203$	160.272	0.0554132
$204$	0	0
$205$	−1201.44	−0.409327
$206$	6256.45	2.11606
$207$	0	0
$208$	1240.81	0.413629
$209$	−822.559	−0.272237
$210$	0	0
$211$	−4579.76	−1.49423	−0.747117	−	0.664692i	$-0.768563\pi$
$211$	−4579.76	−1.49423	−0.747117	+	0.664692i	$0.768563\pi$
$212$	237.072	0.0768028
$213$	0	0
$214$	1526.48	0.487608
$215$	5709.25	1.81101
$216$	0	0
$217$	22.1340	0.00692422
$218$	−4889.08	−1.51895
$219$	0	0
$220$	2330.41	0.714166
$221$	13.7348	0.00418056
$222$	0	0
$223$	−2471.89	−0.742286	−0.371143	−	0.928576i	$-0.621034\pi$
$223$	−2471.89	−0.742286	−0.371143	+	0.928576i	$0.621034\pi$
$224$	147.446	0.0439806
$225$	0	0
$226$	−1211.66	−0.356629
$227$	−512.013	−0.149707	−0.0748535	−	0.997195i	$-0.523849\pi$
$227$	−512.013	−0.149707	−0.0748535	+	0.997195i	$0.523849\pi$
$228$	0	0
$229$	−2588.38	−0.746921	−0.373460	−	0.927646i	$-0.621829\pi$
$229$	−2588.38	−0.746921	−0.373460	+	0.927646i	$0.621829\pi$
$230$	1158.56	0.332145
$231$	0	0
$232$	2401.56	0.679614
$233$	−1787.24	−0.502514	−0.251257	−	0.967920i	$-0.580844\pi$
$233$	−1787.24	−0.502514	−0.251257	+	0.967920i	$0.580844\pi$
$234$	0	0
$235$	4260.53	1.18267
$236$	1611.36	0.444451
$237$	0	0
$238$	2.41928	0.000658901 0
$239$	−3814.71	−1.03244	−0.516220	−	0.856456i	$-0.672661\pi$
$239$	−3814.71	−1.03244	−0.516220	+	0.856456i	$0.672661\pi$
$240$	0	0
$241$	−4567.63	−1.22086	−0.610430	−	0.792071i	$-0.709003\pi$
$241$	−4567.63	−1.22086	−0.610430	+	0.792071i	$0.709003\pi$
$242$	464.268	0.123323
$243$	0	0
$244$	2098.57	0.550603
$245$	−4826.48	−1.25858
$246$	0	0
$247$	370.878	0.0955400
$248$	331.664	0.0849220
$249$	0	0
$250$	2584.57	0.653851
$251$	−3741.36	−0.940847	−0.470424	−	0.882441i	$-0.655899\pi$
$251$	−3741.36	−0.940847	−0.470424	+	0.882441i	$0.655899\pi$
$252$	0	0
$253$	−797.103	−0.198077
$254$	−1469.84	−0.363093
$255$	0	0
$256$	5239.81	1.27925
$257$	−5140.06	−1.24758	−0.623790	−	0.781592i	$-0.714408\pi$
$257$	−5140.06	−1.24758	−0.623790	+	0.781592i	$0.714408\pi$
$258$	0	0
$259$	291.702	0.0699825
$260$	−1050.74	−0.250632
$261$	0	0
$262$	−4397.06	−1.03684
$263$	−7922.61	−1.85752	−0.928762	−	0.370675i	$-0.879126\pi$
$263$	−7922.61	−1.85752	−0.928762	+	0.370675i	$0.879126\pi$
$264$	0	0
$265$	700.505	0.162384
$266$	65.3271	0.0150581
$267$	0	0
$268$	−1549.73	−0.353226
$269$	6413.47	1.45367	0.726833	−	0.686814i	$-0.240991\pi$
$269$	6413.47	1.45367	0.726833	+	0.686814i	$0.240991\pi$
$270$	0	0
$271$	6613.47	1.48244	0.741218	−	0.671265i	$-0.234249\pi$
$271$	6613.47	1.48244	0.741218	+	0.671265i	$0.234249\pi$
$272$	69.7957	0.0155588
$273$	0	0
$274$	−7263.65	−1.60151
$275$	2553.87	0.560014
$276$	0	0
$277$	6732.51	1.46035	0.730176	−	0.683259i	$-0.239438\pi$
$277$	6732.51	1.46035	0.730176	+	0.683259i	$0.239438\pi$
$278$	1046.18	0.225703
$279$	0	0
$280$	125.298	0.0267429
$281$	−4032.08	−0.855991	−0.427996	−	0.903781i	$-0.640780\pi$
$281$	−4032.08	−0.855991	−0.427996	+	0.903781i	$0.640780\pi$
$282$	0	0
$283$	−136.001	−0.0285669	−0.0142834	−	0.999898i	$-0.504547\pi$
$283$	−136.001	−0.0285669	−0.0142834	+	0.999898i	$0.504547\pi$
$284$	423.811	0.0885512
$285$	0	0
$286$	1935.26	0.400120
$287$	−65.6481	−0.0135020
$288$	0	0
$289$	−4912.23	−0.999843
$290$	−10481.9	−2.12247
$291$	0	0
$292$	1070.87	0.214616
$293$	5675.38	1.13160	0.565801	−	0.824542i	$-0.308567\pi$
$293$	5675.38	1.13160	0.565801	+	0.824542i	$0.308567\pi$
$294$	0	0
$295$	4761.27	0.939702
$296$	4370.95	0.858299
$297$	0	0
$298$	3893.24	0.756809
$299$	359.400	0.0695138
$300$	0	0
$301$	311.961	0.0597379
$302$	1044.51	0.199023
$303$	0	0
$304$	1884.67	0.355571
$305$	6200.89	1.16414
$306$	0	0
$307$	7409.87	1.37754	0.688768	−	0.724982i	$-0.258152\pi$
$307$	7409.87	1.37754	0.688768	+	0.724982i	$0.258152\pi$
$308$	127.337	0.0235574
$309$	0	0
$310$	−1447.58	−0.265216
$311$	−7379.57	−1.34552	−0.672761	−	0.739860i	$-0.734892\pi$
$311$	−7379.57	−1.34552	−0.672761	+	0.739860i	$0.734892\pi$
$312$	0	0
$313$	−3768.81	−0.680594	−0.340297	−	0.940318i	$-0.610528\pi$
$313$	−3768.81	−0.680594	−0.340297	+	0.940318i	$0.610528\pi$
$314$	−7832.03	−1.40760
$315$	0	0
$316$	−5176.28	−0.921482
$317$	−5262.96	−0.932483	−0.466242	−	0.884657i	$-0.654392\pi$
$317$	−5262.96	−0.932483	−0.466242	+	0.884657i	$0.654392\pi$
$318$	0	0
$319$	7211.63	1.26575
$320$	−688.713	−0.120313
$321$	0	0
$322$	63.3054	0.0109561
$323$	20.8619	0.00359376
$324$	0	0
$325$	−1151.49	−0.196533
$326$	−9486.05	−1.61161
$327$	0	0
$328$	−983.692	−0.165595
$329$	232.801	0.0390114
$330$	0	0
$331$	2968.99	0.493022	0.246511	−	0.969140i	$-0.420716\pi$
$331$	2968.99	0.493022	0.246511	+	0.969140i	$0.420716\pi$
$332$	−2212.75	−0.365785
$333$	0	0
$334$	−14623.5	−2.39570
$335$	−4579.16	−0.746825
$336$	0	0
$337$	6534.46	1.05625	0.528123	−	0.849168i	$-0.322896\pi$
$337$	6534.46	1.05625	0.528123	+	0.849168i	$0.322896\pi$
$338$	6978.57	1.12303
$339$	0	0
$340$	−59.1043	−0.00942759
$341$	995.949	0.158163
$342$	0	0
$343$	−527.908	−0.0831031
$344$	4674.52	0.732655
$345$	0	0
$346$	−8585.92	−1.33405
$347$	−6855.73	−1.06062	−0.530310	−	0.847804i	$-0.677924\pi$
$347$	−6855.73	−1.06062	−0.530310	+	0.847804i	$0.677924\pi$
$348$	0	0
$349$	9490.70	1.45566	0.727830	−	0.685757i	$-0.240529\pi$
$349$	9490.70	1.45566	0.727830	+	0.685757i	$0.240529\pi$
$350$	−202.826	−0.0309758
$351$	0	0
$352$	6634.53	1.00461
$353$	12731.2	1.91958	0.959789	−	0.280721i	$-0.0905735\pi$
$353$	12731.2	1.91958	0.959789	+	0.280721i	$0.0905735\pi$
$354$	0	0
$355$	1252.28	0.187223
$356$	6183.64	0.920596
$357$	0	0
$358$	8890.71	1.31254
$359$	−5770.57	−0.848353	−0.424177	−	0.905579i	$-0.639436\pi$
$359$	−5770.57	−0.848353	−0.424177	+	0.905579i	$0.639436\pi$
$360$	0	0
$361$	−6295.67	−0.917870
$362$	7944.15	1.15341
$363$	0	0
$364$	−57.4140	−0.00826734
$365$	3164.23	0.453762
$366$	0	0
$367$	−600.196	−0.0853678	−0.0426839	−	0.999089i	$-0.513591\pi$
$367$	−600.196	−0.0853678	−0.0426839	+	0.999089i	$0.513591\pi$
$368$	1826.35	0.258709
$369$	0	0
$370$	−19077.5	−2.68051
$371$	38.2765	0.00535639
$372$	0	0
$373$	650.318	0.0902740	0.0451370	−	0.998981i	$-0.485628\pi$
$373$	650.318	0.0902740	0.0451370	+	0.998981i	$0.485628\pi$
$374$	108.858	0.0150506
$375$	0	0
$376$	3488.37	0.478454
$377$	−3251.60	−0.444207
$378$	0	0
$379$	−3681.71	−0.498989	−0.249494	−	0.968376i	$-0.580264\pi$
$379$	−3681.71	−0.498989	−0.249494	+	0.968376i	$0.580264\pi$
$380$	−1595.98	−0.215453
$381$	0	0
$382$	13337.0	1.78634
$383$	−9834.02	−1.31200	−0.655998	−	0.754762i	$-0.727752\pi$
$383$	−9834.02	−1.31200	−0.655998	+	0.754762i	$0.727752\pi$
$384$	0	0
$385$	376.257	0.0498074
$386$	15741.9	2.07576
$387$	0	0
$388$	934.976	0.122336
$389$	−9694.04	−1.26352	−0.631758	−	0.775166i	$-0.717666\pi$
$389$	−9694.04	−1.26352	−0.631758	+	0.775166i	$0.717666\pi$
$390$	0	0
$391$	20.2163	0.00261478
$392$	−3951.75	−0.509167
$393$	0	0
$394$	−11097.5	−1.41899
$395$	−15295.0	−1.94829
$396$	0	0
$397$	−388.579	−0.0491240	−0.0245620	−	0.999698i	$-0.507819\pi$
$397$	−388.579	−0.0491240	−0.0245620	+	0.999698i	$0.507819\pi$
$398$	14151.7	1.78231
$399$	0	0
$400$	−5851.50	−0.731438
$401$	−1900.42	−0.236664	−0.118332	−	0.992974i	$-0.537755\pi$
$401$	−1900.42	−0.236664	−0.118332	+	0.992974i	$0.537755\pi$
$402$	0	0
$403$	−449.056	−0.0555064
$404$	829.800	0.102188
$405$	0	0
$406$	−572.743	−0.0700117
$407$	13125.5	1.59854
$408$	0	0
$409$	−12654.5	−1.52989	−0.764945	−	0.644096i	$-0.777234\pi$
$409$	−12654.5	−1.52989	−0.764945	+	0.644096i	$0.777234\pi$
$410$	4293.42	0.517163
$411$	0	0
$412$	−8351.86	−0.998706
$413$	260.162	0.0309970
$414$	0	0
$415$	−6538.28	−0.773377
$416$	−2991.39	−0.352561
$417$	0	0
$418$	2939.48	0.343958
$419$	−5617.52	−0.654973	−0.327486	−	0.944856i	$-0.606202\pi$
$419$	−5617.52	−0.654973	−0.327486	+	0.944856i	$0.606202\pi$
$420$	0	0
$421$	9653.27	1.11751	0.558755	−	0.829333i	$-0.311279\pi$
$421$	9653.27	1.11751	0.558755	+	0.829333i	$0.311279\pi$
$422$	16366.1	1.88789
$423$	0	0
$424$	573.548	0.0656933
$425$	−64.7716	−0.00739266
$426$	0	0
$427$	338.825	0.0384002
$428$	−2037.73	−0.230134
$429$	0	0
$430$	−20402.4	−2.28812
$431$	10873.1	1.21518	0.607588	−	0.794253i	$-0.292137\pi$
$431$	10873.1	1.21518	0.607588	+	0.794253i	$0.292137\pi$
$432$	0	0
$433$	−9450.87	−1.04891	−0.524457	−	0.851437i	$-0.675732\pi$
$433$	−9450.87	−1.04891	−0.524457	+	0.851437i	$0.675732\pi$
$434$	−79.0976	−0.00874840
$435$	0	0
$436$	6526.52	0.716890
$437$	545.894	0.0597566
$438$	0	0
$439$	−9684.70	−1.05291	−0.526453	−	0.850204i	$-0.676478\pi$
$439$	−9684.70	−1.05291	−0.526453	+	0.850204i	$0.676478\pi$
$440$	5637.96	0.610862
$441$	0	0
$442$	−49.0824	−0.00528192
$443$	3263.76	0.350036	0.175018	−	0.984565i	$-0.444002\pi$
$443$	3263.76	0.350036	0.175018	+	0.984565i	$0.444002\pi$
$444$	0	0
$445$	18271.5	1.94641
$446$	8833.47	0.937840
$447$	0	0
$448$	−37.6322	−0.00396865
$449$	5206.00	0.547186	0.273593	−	0.961846i	$-0.411788\pi$
$449$	5206.00	0.547186	0.273593	+	0.961846i	$0.411788\pi$
$450$	0	0
$451$	−2953.92	−0.308414
$452$	1617.46	0.168317
$453$	0	0
$454$	1829.72	0.189147
$455$	−169.648	−0.0174796
$456$	0	0
$457$	12120.6	1.24065	0.620324	−	0.784346i	$-0.287001\pi$
$457$	12120.6	1.24065	0.620324	+	0.784346i	$0.287001\pi$
$458$	9249.76	0.943696
$459$	0	0
$460$	−1546.59	−0.156761
$461$	15435.0	1.55939	0.779696	−	0.626158i	$-0.215374\pi$
$461$	15435.0	1.55939	0.779696	+	0.626158i	$0.215374\pi$
$462$	0	0
$463$	−1631.70	−0.163783	−0.0818913	−	0.996641i	$-0.526096\pi$
$463$	−1631.70	−0.163783	−0.0818913	+	0.996641i	$0.526096\pi$
$464$	−16523.5	−1.65320
$465$	0	0
$466$	6386.83	0.634902
$467$	−1772.49	−0.175634	−0.0878172	−	0.996137i	$-0.527989\pi$
$467$	−1772.49	−0.175634	−0.0878172	+	0.996137i	$0.527989\pi$
$468$	0	0
$469$	−250.211	−0.0246347
$470$	−15225.3	−1.49424
$471$	0	0
$472$	3898.35	0.380162
$473$	14037.1	1.36454
$474$	0	0
$475$	−1749.01	−0.168947
$476$	−3.22954	−0.000310978 0
$477$	0	0
$478$	13632.2	1.30444
$479$	−8581.09	−0.818539	−0.409270	−	0.912414i	$-0.634216\pi$
$479$	−8581.09	−0.818539	−0.409270	+	0.912414i	$0.634216\pi$
$480$	0	0
$481$	−5918.06	−0.560998
$482$	16322.8	1.54249
$483$	0	0
$484$	−619.760	−0.0582044
$485$	2762.68	0.258654
$486$	0	0
$487$	18759.7	1.74555	0.872775	−	0.488122i	$-0.162318\pi$
$487$	18759.7	1.74555	0.872775	+	0.488122i	$0.162318\pi$
$488$	5077.06	0.470958
$489$	0	0
$490$	17247.8	1.59016
$491$	9373.98	0.861592	0.430796	−	0.902449i	$-0.358233\pi$
$491$	9373.98	0.861592	0.430796	+	0.902449i	$0.358233\pi$
$492$	0	0
$493$	−182.903	−0.0167090
$494$	−1325.36	−0.120710
$495$	0	0
$496$	−2281.95	−0.206578
$497$	68.4264	0.00617574
$498$	0	0
$499$	−12711.3	−1.14035	−0.570175	−	0.821523i	$-0.693125\pi$
$499$	−12711.3	−1.14035	−0.570175	+	0.821523i	$0.693125\pi$
$500$	−3450.20	−0.308595
$501$	0	0
$502$	13370.0	1.18871
$503$	−7048.98	−0.624848	−0.312424	−	0.949943i	$-0.601141\pi$
$503$	−7048.98	−0.624848	−0.312424	+	0.949943i	$0.601141\pi$
$504$	0	0
$505$	2451.91	0.216056
$506$	2848.51	0.250260
$507$	0	0
$508$	1962.11	0.171367
$509$	−10031.3	−0.873535	−0.436767	−	0.899574i	$-0.643877\pi$
$509$	−10031.3	−0.873535	−0.436767	+	0.899574i	$0.643877\pi$
$510$	0	0
$511$	172.898	0.0149678
$512$	−7869.76	−0.679292
$513$	0	0
$514$	18368.4	1.57625
$515$	−24678.2	−2.11156
$516$	0	0
$517$	10475.2	0.891099
$518$	−1042.42	−0.0884193
$519$	0	0
$520$	−2542.06	−0.214378
$521$	−1292.93	−0.108722	−0.0543610	−	0.998521i	$-0.517312\pi$
$521$	−1292.93	−0.108722	−0.0543610	+	0.998521i	$0.517312\pi$
$522$	0	0
$523$	−6448.73	−0.539165	−0.269582	−	0.962977i	$-0.586886\pi$
$523$	−6448.73	−0.539165	−0.269582	+	0.962977i	$0.586886\pi$
$524$	5869.73	0.489352
$525$	0	0
$526$	28312.0	2.34689
$527$	−25.2594	−0.00208789
$528$	0	0
$529$	529.000	0.0434783
$530$	−2503.31	−0.205164
$531$	0	0
$532$	−87.2064	−0.00710691
$533$	1331.87	0.108236
$534$	0	0
$535$	−6021.12	−0.486572
$536$	−3749.25	−0.302132
$537$	0	0
$538$	−22919.0	−1.83663
$539$	−11866.7	−0.948300
$540$	0	0
$541$	−14040.6	−1.11581	−0.557903	−	0.829906i	$-0.688394\pi$
$541$	−14040.6	−1.11581	−0.557903	+	0.829906i	$0.688394\pi$
$542$	−23633.7	−1.87298
$543$	0	0
$544$	−168.266	−0.0132617
$545$	19284.7	1.51572
$546$	0	0
$547$	2398.58	0.187488	0.0937440	−	0.995596i	$-0.470116\pi$
$547$	2398.58	0.187488	0.0937440	+	0.995596i	$0.470116\pi$
$548$	9696.39	0.755856
$549$	0	0
$550$	−9126.43	−0.707550
$551$	−4938.87	−0.381857
$552$	0	0
$553$	−835.737	−0.0642661
$554$	−24059.1	−1.84508
$555$	0	0
$556$	−1396.56	−0.106524
$557$	−5086.11	−0.386904	−0.193452	−	0.981110i	$-0.561968\pi$
$557$	−5086.11	−0.386904	−0.193452	+	0.981110i	$0.561968\pi$
$558$	0	0
$559$	−6329.07	−0.478875
$560$	−862.093	−0.0650537
$561$	0	0
$562$	14408.9	1.08150
$563$	12808.3	0.958806	0.479403	−	0.877595i	$-0.340853\pi$
$563$	12808.3	0.958806	0.479403	+	0.877595i	$0.340853\pi$
$564$	0	0
$565$	4779.31	0.355871
$566$	486.010	0.0360928
$567$	0	0
$568$	1025.32	0.0757423
$569$	11625.9	0.856560	0.428280	−	0.903646i	$-0.359120\pi$
$569$	11625.9	0.856560	0.428280	+	0.903646i	$0.359120\pi$
$570$	0	0
$571$	−5508.34	−0.403707	−0.201854	−	0.979416i	$-0.564697\pi$
$571$	−5508.34	−0.403707	−0.201854	+	0.979416i	$0.564697\pi$
$572$	−2583.42	−0.188843
$573$	0	0
$574$	234.598	0.0170591
$575$	−1694.88	−0.122924
$576$	0	0
$577$	−16634.3	−1.20016	−0.600082	−	0.799939i	$-0.704865\pi$
$577$	−16634.3	−1.20016	−0.600082	+	0.799939i	$0.704865\pi$
$578$	17554.2	1.26325
$579$	0	0
$580$	13992.4	1.00173
$581$	−357.260	−0.0255106
$582$	0	0
$583$	1722.30	0.122351
$584$	2590.75	0.183572
$585$	0	0
$586$	−20281.4	−1.42972
$587$	−6224.23	−0.437652	−0.218826	−	0.975764i	$-0.570223\pi$
$587$	−6224.23	−0.437652	−0.218826	+	0.975764i	$0.570223\pi$
$588$	0	0
$589$	−682.074	−0.0477154
$590$	−17014.8	−1.18727
$591$	0	0
$592$	−30073.6	−2.08786
$593$	−13717.8	−0.949954	−0.474977	−	0.879998i	$-0.657544\pi$
$593$	−13717.8	−0.949954	−0.474977	+	0.879998i	$0.657544\pi$
$594$	0	0
$595$	−9.54270	−0.000657500 0
$596$	−5197.16	−0.357188
$597$	0	0
$598$	−1284.34	−0.0878272
$599$	−7465.64	−0.509245	−0.254623	−	0.967040i	$-0.581951\pi$
$599$	−7465.64	−0.509245	−0.254623	+	0.967040i	$0.581951\pi$
$600$	0	0
$601$	27528.4	1.86840	0.934199	−	0.356752i	$-0.116116\pi$
$601$	27528.4	1.86840	0.934199	+	0.356752i	$0.116116\pi$
$602$	−1114.81	−0.0754759
$603$	0	0
$604$	−1394.34	−0.0939319
$605$	−1831.28	−0.123061
$606$	0	0
$607$	−29065.0	−1.94351	−0.971757	−	0.235985i	$-0.924168\pi$
$607$	−29065.0	−1.94351	−0.971757	+	0.235985i	$0.924168\pi$
$608$	−4543.64	−0.303074
$609$	0	0
$610$	−22159.3	−1.47083
$611$	−4723.08	−0.312726
$612$	0	0
$613$	−22786.9	−1.50139	−0.750696	−	0.660648i	$-0.770282\pi$
$613$	−22786.9	−1.50139	−0.750696	+	0.660648i	$0.770282\pi$
$614$	−26479.7	−1.74045
$615$	0	0
$616$	308.066	0.0201499
$617$	5693.08	0.371467	0.185733	−	0.982600i	$-0.440534\pi$
$617$	5693.08	0.371467	0.185733	+	0.982600i	$0.440534\pi$
$618$	0	0
$619$	−20940.6	−1.35973	−0.679866	−	0.733336i	$-0.737962\pi$
$619$	−20940.6	−1.35973	−0.679866	+	0.733336i	$0.737962\pi$
$620$	1932.40	0.125173
$621$	0	0
$622$	26371.5	1.70000
$623$	998.380	0.0642043
$624$	0	0
$625$	−19406.0	−1.24199
$626$	13468.1	0.859896
$627$	0	0
$628$	10455.1	0.664340
$629$	−332.891	−0.0211021
$630$	0	0
$631$	27598.8	1.74119	0.870595	−	0.491999i	$-0.163734\pi$
$631$	27598.8	1.74119	0.870595	+	0.491999i	$0.163734\pi$
$632$	−12522.9	−0.788190
$633$	0	0
$634$	18807.6	1.17815
$635$	5797.68	0.362321
$636$	0	0
$637$	5350.48	0.332800
$638$	−25771.3	−1.59921
$639$	0	0
$640$	−19126.3	−1.18130
$641$	21618.2	1.33209	0.666043	−	0.745914i	$-0.267987\pi$
$641$	21618.2	1.33209	0.666043	+	0.745914i	$0.267987\pi$
$642$	0	0
$643$	312.377	0.0191586	0.00957928	−	0.999954i	$-0.496951\pi$
$643$	312.377	0.0191586	0.00957928	+	0.999954i	$0.496951\pi$
$644$	−84.5076	−0.00517091
$645$	0	0
$646$	−74.5515	−0.00454054
$647$	−24658.0	−1.49831	−0.749156	−	0.662394i	$-0.769541\pi$
$647$	−24658.0	−1.49831	−0.749156	+	0.662394i	$0.769541\pi$
$648$	0	0
$649$	11706.3	0.708034
$650$	4114.95	0.248310
$651$	0	0
$652$	12663.1	0.760623
$653$	−5605.33	−0.335917	−0.167958	−	0.985794i	$-0.553717\pi$
$653$	−5605.33	−0.335917	−0.167958	+	0.985794i	$0.553717\pi$
$654$	0	0
$655$	17344.0	1.03463
$656$	6768.12	0.402821
$657$	0	0
$658$	−831.932	−0.0492889
$659$	7019.45	0.414930	0.207465	−	0.978242i	$-0.433479\pi$
$659$	7019.45	0.414930	0.207465	+	0.978242i	$0.433479\pi$
$660$	0	0
$661$	−25530.3	−1.50229	−0.751144	−	0.660138i	$-0.770498\pi$
$661$	−25530.3	−1.50229	−0.751144	+	0.660138i	$0.770498\pi$
$662$	−10609.9	−0.622908
$663$	0	0
$664$	−5353.30	−0.312874
$665$	−257.679	−0.0150261
$666$	0	0
$667$	−4786.03	−0.277835
$668$	19521.2	1.13069
$669$	0	0
$670$	16364.0	0.943575
$671$	15245.9	0.877138
$672$	0	0
$673$	5813.10	0.332955	0.166477	−	0.986045i	$-0.446761\pi$
$673$	5813.10	0.332955	0.166477	+	0.986045i	$0.446761\pi$
$674$	−23351.4	−1.33451
$675$	0	0
$676$	−9315.83	−0.530031
$677$	30771.4	1.74689	0.873443	−	0.486926i	$-0.161882\pi$
$677$	30771.4	1.74689	0.873443	+	0.486926i	$0.161882\pi$
$678$	0	0
$679$	150.957	0.00853194
$680$	−142.991	−0.00806390
$681$	0	0
$682$	−3559.10	−0.199831
$683$	23739.4	1.32996	0.664980	−	0.746861i	$-0.268440\pi$
$683$	23739.4	1.32996	0.664980	+	0.746861i	$0.268440\pi$
$684$	0	0
$685$	28651.1	1.59810
$686$	1886.52	0.104997
$687$	0	0
$688$	−32162.2	−1.78223
$689$	−776.556	−0.0429382
$690$	0	0
$691$	−1721.20	−0.0947575	−0.0473787	−	0.998877i	$-0.515087\pi$
$691$	−1721.20	−0.0947575	−0.0473787	+	0.998877i	$0.515087\pi$
$692$	11461.5	0.629626
$693$	0	0
$694$	24499.5	1.34004
$695$	−4126.58	−0.225223
$696$	0	0
$697$	74.9178	0.00407132
$698$	−33915.7	−1.83915
$699$	0	0
$700$	270.757	0.0146195
$701$	−14258.5	−0.768239	−0.384120	−	0.923283i	$-0.625495\pi$
$701$	−14258.5	−0.768239	−0.384120	+	0.923283i	$0.625495\pi$
$702$	0	0
$703$	−8988.96	−0.482255
$704$	−1693.31	−0.0906519
$705$	0	0
$706$	−45495.8	−2.42529
$707$	133.975	0.00712683
$708$	0	0
$709$	11544.3	0.611500	0.305750	−	0.952112i	$-0.401093\pi$
$709$	11544.3	0.611500	0.305750	+	0.952112i	$0.401093\pi$
$710$	−4475.13	−0.236547
$711$	0	0
$712$	14960.1	0.787432
$713$	−660.965	−0.0347172
$714$	0	0
$715$	−7633.52	−0.399269
$716$	−11868.4	−0.619472
$717$	0	0
$718$	20621.6	1.07185
$719$	−11736.1	−0.608737	−0.304368	−	0.952554i	$-0.598445\pi$
$719$	−11736.1	−0.608737	−0.304368	+	0.952554i	$0.598445\pi$
$720$	0	0
$721$	−1348.45	−0.0696518
$722$	22498.1	1.15968
$723$	0	0
$724$	−10604.8	−0.544371
$725$	15334.1	0.785510
$726$	0	0
$727$	15532.4	0.792387	0.396193	−	0.918167i	$-0.370331\pi$
$727$	15532.4	0.792387	0.396193	+	0.918167i	$0.370331\pi$
$728$	−138.901	−0.00707147
$729$	0	0
$730$	−11307.6	−0.573306
$731$	−356.011	−0.0180130
$732$	0	0
$733$	10102.8	0.509080	0.254540	−	0.967062i	$-0.418076\pi$
$733$	10102.8	0.509080	0.254540	+	0.967062i	$0.418076\pi$
$734$	2144.85	0.107858
$735$	0	0
$736$	−4403.03	−0.220513
$737$	−11258.6	−0.562707
$738$	0	0
$739$	−3063.89	−0.152513	−0.0762565	−	0.997088i	$-0.524297\pi$
$739$	−3063.89	−0.152513	−0.0762565	+	0.997088i	$0.524297\pi$
$740$	25466.9	1.26511
$741$	0	0
$742$	−136.784	−0.00676752
$743$	13681.0	0.675517	0.337758	−	0.941233i	$-0.390331\pi$
$743$	13681.0	0.675517	0.337758	+	0.941233i	$0.390331\pi$
$744$	0	0
$745$	−15356.6	−0.755200
$746$	−2323.96	−0.114057
$747$	0	0
$748$	−145.317	−0.00710338
$749$	−329.002	−0.0160500
$750$	0	0
$751$	−3652.90	−0.177492	−0.0887458	−	0.996054i	$-0.528286\pi$
$751$	−3652.90	−0.177492	−0.0887458	+	0.996054i	$0.528286\pi$
$752$	−24001.1	−1.16387
$753$	0	0
$754$	11619.8	0.561233
$755$	−4120.02	−0.198600
$756$	0	0
$757$	5446.96	0.261523	0.130762	−	0.991414i	$-0.458258\pi$
$757$	5446.96	0.261523	0.130762	+	0.991414i	$0.458258\pi$
$758$	13156.9	0.630447
$759$	0	0
$760$	−3861.14	−0.184287
$761$	26195.8	1.24783	0.623915	−	0.781492i	$-0.285541\pi$
$761$	26195.8	1.24783	0.623915	+	0.781492i	$0.285541\pi$
$762$	0	0
$763$	1053.74	0.0499974
$764$	−17803.8	−0.843088
$765$	0	0
$766$	35142.6	1.65764
$767$	−5278.18	−0.248480
$768$	0	0
$769$	15201.2	0.712834	0.356417	−	0.934327i	$-0.383998\pi$
$769$	15201.2	0.712834	0.356417	+	0.934327i	$0.383998\pi$
$770$	−1344.58	−0.0629291
$771$	0	0
$772$	−21014.2	−0.979687
$773$	28555.0	1.32866	0.664328	−	0.747441i	$-0.268718\pi$
$773$	28555.0	1.32866	0.664328	+	0.747441i	$0.268718\pi$
$774$	0	0
$775$	2117.69	0.0981544
$776$	2261.98	0.104640
$777$	0	0
$778$	34642.4	1.59639
$779$	2022.99	0.0930436
$780$	0	0
$781$	3078.93	0.141067
$782$	−72.2443	−0.00330364
$783$	0	0
$784$	27189.3	1.23858
$785$	30893.0	1.40461
$786$	0	0
$787$	2461.65	0.111497	0.0557486	−	0.998445i	$-0.482245\pi$
$787$	2461.65	0.111497	0.0557486	+	0.998445i	$0.482245\pi$
$788$	14814.2	0.669714
$789$	0	0
$790$	54657.7	2.46156
$791$	261.148	0.0117387
$792$	0	0
$793$	−6874.09	−0.307826
$794$	1388.62	0.0620656
$795$	0	0
$796$	−18891.3	−0.841188
$797$	35779.6	1.59019	0.795094	−	0.606487i	$-0.207422\pi$
$797$	35779.6	1.59019	0.795094	+	0.606487i	$0.207422\pi$
$798$	0	0
$799$	−265.673	−0.0117633
$800$	14107.0	0.623448
$801$	0	0
$802$	6791.29	0.299014
$803$	7779.75	0.341895
$804$	0	0
$805$	−249.704	−0.0109328
$806$	1604.74	0.0701296
$807$	0	0
$808$	2007.53	0.0874068
$809$	21167.0	0.919890	0.459945	−	0.887947i	$-0.347869\pi$
$809$	21167.0	0.919890	0.459945	+	0.887947i	$0.347869\pi$
$810$	0	0
$811$	42245.1	1.82913	0.914565	−	0.404438i	$-0.132533\pi$
$811$	42245.1	1.82913	0.914565	+	0.404438i	$0.132533\pi$
$812$	764.566	0.0330431
$813$	0	0
$814$	−46904.9	−2.01968
$815$	37417.2	1.60818
$816$	0	0
$817$	−9613.26	−0.411659
$818$	45221.8	1.93294
$819$	0	0
$820$	−5731.37	−0.244083
$821$	−16085.0	−0.683763	−0.341881	−	0.939743i	$-0.611064\pi$
$821$	−16085.0	−0.683763	−0.341881	+	0.939743i	$0.611064\pi$
$822$	0	0
$823$	−30884.6	−1.30810	−0.654052	−	0.756450i	$-0.726932\pi$
$823$	−30884.6	−1.30810	−0.654052	+	0.756450i	$0.726932\pi$
$824$	−20205.6	−0.854243
$825$	0	0
$826$	−929.709	−0.0391631
$827$	−34941.5	−1.46921	−0.734605	−	0.678496i	$-0.762632\pi$
$827$	−34941.5	−1.46921	−0.734605	+	0.678496i	$0.762632\pi$
$828$	0	0
$829$	−17009.2	−0.712609	−0.356305	−	0.934370i	$-0.615963\pi$
$829$	−17009.2	−0.712609	−0.356305	+	0.934370i	$0.615963\pi$
$830$	23365.0	0.977123
$831$	0	0
$832$	763.483	0.0318137
$833$	300.964	0.0125184
$834$	0	0
$835$	57681.6	2.39060
$836$	−3923.96	−0.162336
$837$	0	0
$838$	20074.6	0.827525
$839$	−10126.7	−0.416700	−0.208350	−	0.978054i	$-0.566809\pi$
$839$	−10126.7	−0.416700	−0.208350	+	0.978054i	$0.566809\pi$
$840$	0	0
$841$	18911.6	0.775417
$842$	−34496.7	−1.41192
$843$	0	0
$844$	−21847.4	−0.891018
$845$	−27526.6	−1.12064
$846$	0	0
$847$	−100.063	−0.00405929
$848$	−3946.19	−0.159803
$849$	0	0
$850$	231.466	0.00934026
$851$	−8710.78	−0.350883
$852$	0	0
$853$	−24818.1	−0.996198	−0.498099	−	0.867120i	$-0.665968\pi$
$853$	−24818.1	−0.996198	−0.498099	+	0.867120i	$0.665968\pi$
$854$	−1210.82	−0.0485167
$855$	0	0
$856$	−4929.87	−0.196845
$857$	36314.0	1.44745	0.723724	−	0.690090i	$-0.242429\pi$
$857$	36314.0	1.44745	0.723724	+	0.690090i	$0.242429\pi$
$858$	0	0
$859$	−22525.8	−0.894728	−0.447364	−	0.894352i	$-0.647637\pi$
$859$	−22525.8	−0.894728	−0.447364	+	0.894352i	$0.647637\pi$
$860$	27235.6	1.07991
$861$	0	0
$862$	−38856.0	−1.53531
$863$	−15662.3	−0.617789	−0.308894	−	0.951096i	$-0.599959\pi$
$863$	−15662.3	−0.617789	−0.308894	+	0.951096i	$0.599959\pi$
$864$	0	0
$865$	33866.7	1.33122
$866$	33773.4	1.32525
$867$	0	0
$868$	105.589	0.00412894
$869$	−37605.0	−1.46797
$870$	0	0
$871$	5076.30	0.197479
$872$	15789.6	0.613192
$873$	0	0
$874$	−1950.79	−0.0754995
$875$	−557.052	−0.0215221
$876$	0	0
$877$	−35339.3	−1.36069	−0.680343	−	0.732893i	$-0.738169\pi$
$877$	−35339.3	−1.36069	−0.680343	+	0.732893i	$0.738169\pi$
$878$	34609.0	1.33029
$879$	0	0
$880$	−38791.0	−1.48596
$881$	−24946.1	−0.953978	−0.476989	−	0.878909i	$-0.658272\pi$
$881$	−24946.1	−0.953978	−0.476989	+	0.878909i	$0.658272\pi$
$882$	0	0
$883$	25605.1	0.975853	0.487927	−	0.872885i	$-0.337753\pi$
$883$	25605.1	0.975853	0.487927	+	0.872885i	$0.337753\pi$
$884$	65.5210	0.00249289
$885$	0	0
$886$	−11663.3	−0.442253
$887$	15401.5	0.583011	0.291505	−	0.956569i	$-0.405844\pi$
$887$	15401.5	0.583011	0.291505	+	0.956569i	$0.405844\pi$
$888$	0	0
$889$	316.793	0.0119515
$890$	−65294.7	−2.45919
$891$	0	0
$892$	−11792.0	−0.442628
$893$	−7173.91	−0.268831
$894$	0	0
$895$	−35068.9	−1.30975
$896$	−1045.09	−0.0389665
$897$	0	0
$898$	−18604.0	−0.691341
$899$	5979.96	0.221850
$900$	0	0
$901$	−43.6813	−0.00161513
$902$	10556.1	0.389665
$903$	0	0
$904$	3913.12	0.143970
$905$	−31335.3	−1.15096
$906$	0	0
$907$	−28802.1	−1.05442	−0.527209	−	0.849736i	$-0.676762\pi$
$907$	−28802.1	−1.05442	−0.527209	+	0.849736i	$0.676762\pi$
$908$	−2442.52	−0.0892709
$909$	0	0
$910$	606.249	0.0220846
$911$	35607.8	1.29499	0.647496	−	0.762069i	$-0.275816\pi$
$911$	35607.8	1.29499	0.647496	+	0.762069i	$0.275816\pi$
$912$	0	0
$913$	−16075.4	−0.582714
$914$	−43313.7	−1.56750
$915$	0	0
$916$	−12347.7	−0.445392
$917$	947.698	0.0341284
$918$	0	0
$919$	43275.2	1.55334	0.776670	−	0.629908i	$-0.216908\pi$
$919$	43275.2	1.55334	0.776670	+	0.629908i	$0.216908\pi$
$920$	−3741.65	−0.134085
$921$	0	0
$922$	−55158.1	−1.97021
$923$	−1388.24	−0.0495064
$924$	0	0
$925$	27908.8	0.992037
$926$	5830.99	0.206931
$927$	0	0
$928$	39835.6	1.40912
$929$	−9027.96	−0.318835	−0.159418	−	0.987211i	$-0.550962\pi$
$929$	−9027.96	−0.318835	−0.159418	+	0.987211i	$0.550962\pi$
$930$	0	0
$931$	8126.86	0.286087
$932$	−8525.90	−0.299652
$933$	0	0
$934$	6334.14	0.221905
$935$	−429.386	−0.0150186
$936$	0	0
$937$	−14374.0	−0.501152	−0.250576	−	0.968097i	$-0.580620\pi$
$937$	−14374.0	−0.501152	−0.250576	+	0.968097i	$0.580620\pi$
$938$	894.149	0.0311247
$939$	0	0
$940$	20324.6	0.705229
$941$	−535.779	−0.0185610	−0.00928050	−	0.999957i	$-0.502954\pi$
$941$	−535.779	−0.0185610	−0.00928050	+	0.999957i	$0.502954\pi$
$942$	0	0
$943$	1960.38	0.0676975
$944$	−26821.9	−0.924766
$945$	0	0
$946$	−50162.5	−1.72402
$947$	14640.6	0.502381	0.251191	−	0.967938i	$-0.419178\pi$
$947$	14640.6	0.502381	0.251191	+	0.967938i	$0.419178\pi$
$948$	0	0
$949$	−3507.76	−0.119986
$950$	6250.21	0.213457
$951$	0	0
$952$	−7.81321	−0.000265995 0
$953$	19789.0	0.672644	0.336322	−	0.941747i	$-0.390817\pi$
$953$	19789.0	0.672644	0.336322	+	0.941747i	$0.390817\pi$
$954$	0	0
$955$	−52607.0	−1.78254
$956$	−18197.8	−0.615648
$957$	0	0
$958$	30665.2	1.03418
$959$	1565.53	0.0527150
$960$	0	0
$961$	−28965.1	−0.972279
$962$	21148.6	0.708793
$963$	0	0
$964$	−21789.6	−0.728004
$965$	−62093.2	−2.07135
$966$	0	0
$967$	22988.6	0.764493	0.382246	−	0.924060i	$-0.375151\pi$
$967$	22988.6	0.764493	0.382246	+	0.924060i	$0.375151\pi$
$968$	−1499.38	−0.0497851
$969$	0	0
$970$	−9872.66	−0.326796
$971$	12281.6	0.405907	0.202954	−	0.979188i	$-0.434946\pi$
$971$	12281.6	0.405907	0.202954	+	0.979188i	$0.434946\pi$
$972$	0	0
$973$	−225.482	−0.00742921
$974$	−67039.2	−2.20542
$975$	0	0
$976$	−34931.8	−1.14563
$977$	13716.2	0.449152	0.224576	−	0.974457i	$-0.427900\pi$
$977$	13716.2	0.449152	0.224576	+	0.974457i	$0.427900\pi$
$978$	0	0
$979$	44923.4	1.46656
$980$	−23024.4	−0.750498
$981$	0	0
$982$	−33498.6	−1.08858
$983$	38091.7	1.23595	0.617975	−	0.786198i	$-0.287953\pi$
$983$	38091.7	1.23595	0.617975	+	0.786198i	$0.287953\pi$
$984$	0	0
$985$	43773.3	1.41597
$986$	653.616	0.0211109
$987$	0	0
$988$	1769.25	0.0569709
$989$	−9315.75	−0.299518
$990$	0	0
$991$	−9639.10	−0.308977	−0.154488	−	0.987995i	$-0.549373\pi$
$991$	−9639.10	−0.308977	−0.154488	+	0.987995i	$0.549373\pi$
$992$	5501.42	0.176079
$993$	0	0
$994$	−244.527	−0.00780274
$995$	−55820.4	−1.77852
$996$	0	0
$997$	−10135.6	−0.321962	−0.160981	−	0.986957i	$-0.551466\pi$
$997$	−10135.6	−0.321962	−0.160981	+	0.986957i	$0.551466\pi$
$998$	45424.7	1.44078
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.h.1.1	yes	5
3.2	odd	2		207.4.a.g.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.a.g.1.5	✓	5	3.2	odd	2
207.4.a.h.1.1	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)

\(f(q)\)	\(=\)	\(q-3.57357 q^{2} +4.77043 q^{4} +14.0958 q^{5} +0.770211 q^{7} +11.5411 q^{8} -50.3723 q^{10} +34.6566 q^{11} -15.6261 q^{13} -2.75241 q^{14} -79.4064 q^{16} -0.878967 q^{17} -23.7345 q^{19} +67.2429 q^{20} -123.848 q^{22} -23.0000 q^{23} +73.6905 q^{25} +55.8410 q^{26} +3.67424 q^{28} +208.088 q^{29} +28.7376 q^{31} +191.436 q^{32} +3.14106 q^{34} +10.8567 q^{35} +378.729 q^{37} +84.8171 q^{38} +162.680 q^{40} -85.2339 q^{41} +405.033 q^{43} +165.327 q^{44} +82.1922 q^{46} +302.256 q^{47} -342.407 q^{49} -263.339 q^{50} -74.5432 q^{52} +49.6962 q^{53} +488.512 q^{55} +8.88908 q^{56} -743.618 q^{58} +337.780 q^{59} +439.911 q^{61} -102.696 q^{62} -48.8596 q^{64} -220.261 q^{65} -324.861 q^{67} -4.19306 q^{68} -38.7973 q^{70} +88.8411 q^{71} +224.481 q^{73} -1353.42 q^{74} -113.224 q^{76} +26.6929 q^{77} -1085.07 q^{79} -1119.29 q^{80} +304.590 q^{82} -463.847 q^{83} -12.3897 q^{85} -1447.41 q^{86} +399.976 q^{88} +1296.24 q^{89} -12.0354 q^{91} -109.720 q^{92} -1080.14 q^{94} -334.556 q^{95} +195.994 q^{97} +1223.62 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 16 q^{4} + 20 q^{5} - 10 q^{7} + 48 q^{8} + 50 q^{10} + 46 q^{11} + 54 q^{13} + 164 q^{14} - 60 q^{16} + 250 q^{17} - 28 q^{19} + 242 q^{20} - 10 q^{22} - 115 q^{23} + 239 q^{25} + 368 q^{26}+ \cdots - 2400 q^{98}+O(q^{100}) \) 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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