LMFDB - Embedded newform 207.4.a.c.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	\(q-0.236068 q^{2} -7.94427 q^{4} +15.2361 q^{5} -20.6525 q^{7} +3.76393 q^{8} -3.59675 q^{10} +7.63932 q^{11} +55.0820 q^{13} +4.87539 q^{14} +62.6656 q^{16} +72.7639 q^{17} -96.7902 q^{19} -121.039 q^{20} -1.80340 q^{22} +23.0000 q^{23} +107.138 q^{25} -13.0031 q^{26} +164.069 q^{28} +228.748 q^{29} +336.413 q^{31} -44.9048 q^{32} -17.1772 q^{34} -314.663 q^{35} +213.108 q^{37} +22.8491 q^{38} +57.3475 q^{40} +325.108 q^{41} -297.787 q^{43} -60.6888 q^{44} -5.42956 q^{46} +494.545 q^{47} +83.5248 q^{49} -25.2918 q^{50} -437.587 q^{52} -220.403 q^{53} +116.393 q^{55} -77.7345 q^{56} -54.0000 q^{58} -502.768 q^{59} -69.3963 q^{61} -79.4164 q^{62} -490.724 q^{64} +839.234 q^{65} -425.820 q^{67} -578.056 q^{68} +74.2817 q^{70} +140.604 q^{71} -273.266 q^{73} -50.3081 q^{74} +768.928 q^{76} -157.771 q^{77} -234.521 q^{79} +954.778 q^{80} -76.7477 q^{82} +233.358 q^{83} +1108.64 q^{85} +70.2980 q^{86} +28.7539 q^{88} -880.581 q^{89} -1137.58 q^{91} -182.718 q^{92} -116.746 q^{94} -1474.70 q^{95} +20.4319 q^{97} -19.7175 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{4} + 26 q^{5} - 10 q^{7} + 12 q^{8} + 42 q^{10} + 60 q^{11} - 24 q^{13} + 50 q^{14} + 18 q^{16} + 150 q^{17} - 46 q^{19} - 14 q^{20} + 220 q^{22} + 46 q^{23} + 98 q^{25} - 348 q^{26}+ \cdots - 992 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^4 + 26 * q^5 - 10 * q^7 + 12 * q^8 + 42 * q^10 + 60 * q^11 - 24 * q^13 + 50 * q^14 + 18 * q^16 + 150 * q^17 - 46 * q^19 - 14 * q^20 + 220 * q^22 + 46 * q^23 + 98 * q^25 - 348 * q^26 + 270 * q^28 + 216 * q^29 + 324 * q^31 - 300 * q^32 + 310 * q^34 - 200 * q^35 + 140 * q^37 + 238 * q^38 + 146 * q^40 + 364 * q^41 - 126 * q^43 + 460 * q^44 + 92 * q^46 - 120 * q^47 - 146 * q^49 - 64 * q^50 - 1224 * q^52 - 490 * q^53 + 680 * q^55 + 10 * q^56 - 108 * q^58 + 32 * q^59 - 908 * q^61 - 132 * q^62 - 1214 * q^64 - 12 * q^65 - 874 * q^67 + 190 * q^68 + 560 * q^70 - 488 * q^71 + 652 * q^73 - 360 * q^74 + 1274 * q^76 + 400 * q^77 - 1198 * q^79 + 474 * q^80 + 88 * q^82 + 100 * q^83 + 1940 * q^85 + 798 * q^86 + 460 * q^88 - 102 * q^89 - 1980 * q^91 + 46 * q^92 - 2720 * q^94 - 928 * q^95 + 1624 * q^97 - 992 * 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 $

$2$	−0.236068	−0.0834626	−0.0417313	−	0.999129i	$-0.513287\pi$
$2$	−0.236068	−0.0834626	−0.0417313	+	0.999129i	$0.513287\pi$
$3$	0	0
$3$	0	0
$4$	−7.94427	−0.993034
$5$	15.2361	1.36276	0.681378	−	0.731932i	$-0.261381\pi$
$5$	15.2361	1.36276	0.681378	+	0.731932i	$0.261381\pi$
$6$	0	0
$7$	−20.6525	−1.11513	−0.557564	−	0.830134i	$-0.688264\pi$
$7$	−20.6525	−1.11513	−0.557564	+	0.830134i	$0.688264\pi$
$8$	3.76393	0.166344
$9$	0	0
$10$	−3.59675	−0.113739
$11$	7.63932	0.209395	0.104697	−	0.994504i	$-0.466613\pi$
$11$	7.63932	0.209395	0.104697	+	0.994504i	$0.466613\pi$
$12$	0	0
$13$	55.0820	1.17515	0.587577	−	0.809168i	$-0.300082\pi$
$13$	55.0820	1.17515	0.587577	+	0.809168i	$0.300082\pi$
$14$	4.87539	0.0930716
$15$	0	0
$16$	62.6656	0.979150
$17$	72.7639	1.03811	0.519054	−	0.854741i	$-0.326284\pi$
$17$	72.7639	1.03811	0.519054	+	0.854741i	$0.326284\pi$
$18$	0	0
$19$	−96.7902	−1.16869	−0.584347	−	0.811504i	$-0.698649\pi$
$19$	−96.7902	−1.16869	−0.584347	+	0.811504i	$0.698649\pi$
$20$	−121.039	−1.35326
$21$	0	0
$22$	−1.80340	−0.0174766
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	107.138	0.857102
$26$	−13.0031	−0.0980815
$27$	0	0
$28$	164.069	1.10736
$29$	228.748	1.46474	0.732369	−	0.680908i	$-0.238415\pi$
$29$	228.748	1.46474	0.732369	+	0.680908i	$0.238415\pi$
$30$	0	0
$31$	336.413	1.94908	0.974542	−	0.224204i	$-0.0719783\pi$
$31$	336.413	1.94908	0.974542	+	0.224204i	$0.0719783\pi$
$32$	−44.9048	−0.248066
$33$	0	0
$34$	−17.1772	−0.0866433
$35$	−314.663	−1.51965
$36$	0	0
$37$	213.108	0.946886	0.473443	−	0.880824i	$-0.343011\pi$
$37$	213.108	0.946886	0.473443	+	0.880824i	$0.343011\pi$
$38$	22.8491	0.0975424
$39$	0	0
$40$	57.3475	0.226686
$41$	325.108	1.23838	0.619188	−	0.785243i	$-0.287462\pi$
$41$	325.108	1.23838	0.619188	+	0.785243i	$0.287462\pi$
$42$	0	0
$43$	−297.787	−1.05610	−0.528048	−	0.849215i	$-0.677076\pi$
$43$	−297.787	−1.05610	−0.528048	+	0.849215i	$0.677076\pi$
$44$	−60.6888	−0.207936
$45$	0	0
$46$	−5.42956	−0.0174032
$47$	494.545	1.53483	0.767413	−	0.641154i	$-0.221544\pi$
$47$	494.545	1.53483	0.767413	+	0.641154i	$0.221544\pi$
$48$	0	0
$49$	83.5248	0.243512
$50$	−25.2918	−0.0715360
$51$	0	0
$52$	−437.587	−1.16697
$53$	−220.403	−0.571221	−0.285611	−	0.958346i	$-0.592196\pi$
$53$	−220.403	−0.571221	−0.285611	+	0.958346i	$0.592196\pi$
$54$	0	0
$55$	116.393	0.285354
$56$	−77.7345	−0.185495
$57$	0	0
$58$	−54.0000	−0.122251
$59$	−502.768	−1.10940	−0.554702	−	0.832049i	$-0.687168\pi$
$59$	−502.768	−1.10940	−0.554702	+	0.832049i	$0.687168\pi$
$60$	0	0
$61$	−69.3963	−0.145660	−0.0728302	−	0.997344i	$-0.523203\pi$
$61$	−69.3963	−0.145660	−0.0728302	+	0.997344i	$0.523203\pi$
$62$	−79.4164	−0.162676
$63$	0	0
$64$	−490.724	−0.958446
$65$	839.234	1.60145
$66$	0	0
$67$	−425.820	−0.776450	−0.388225	−	0.921565i	$-0.626912\pi$
$67$	−425.820	−0.776450	−0.388225	+	0.921565i	$0.626912\pi$
$68$	−578.056	−1.03088
$69$	0	0
$70$	74.2817	0.126834
$71$	140.604	0.235022	0.117511	−	0.993072i	$-0.462508\pi$
$71$	140.604	0.235022	0.117511	+	0.993072i	$0.462508\pi$
$72$	0	0
$73$	−273.266	−0.438129	−0.219064	−	0.975710i	$-0.570300\pi$
$73$	−273.266	−0.438129	−0.219064	+	0.975710i	$0.570300\pi$
$74$	−50.3081	−0.0790296
$75$	0	0
$76$	768.928	1.16055
$77$	−157.771	−0.233502
$78$	0	0
$79$	−234.521	−0.333996	−0.166998	−	0.985957i	$-0.553407\pi$
$79$	−234.521	−0.333996	−0.166998	+	0.985957i	$0.553407\pi$
$80$	954.778	1.33434
$81$	0	0
$82$	−76.7477	−0.103358
$83$	233.358	0.308606	0.154303	−	0.988024i	$-0.450687\pi$
$83$	233.358	0.308606	0.154303	+	0.988024i	$0.450687\pi$
$84$	0	0
$85$	1108.64	1.41469
$86$	70.2980	0.0881445
$87$	0	0
$88$	28.7539	0.0348315
$89$	−880.581	−1.04878	−0.524390	−	0.851478i	$-0.675707\pi$
$89$	−880.581	−1.04878	−0.524390	+	0.851478i	$0.675707\pi$
$90$	0	0
$91$	−1137.58	−1.31045
$92$	−182.718	−0.207062
$93$	0	0
$94$	−116.746	−0.128101
$95$	−1474.70	−1.59265
$96$	0	0
$97$	20.4319	0.0213871	0.0106936	−	0.999943i	$-0.496596\pi$
$97$	20.4319	0.0213871	0.0106936	+	0.999943i	$0.496596\pi$
$98$	−19.7175	−0.0203242
$99$	0	0
$100$	−851.132	−0.851132
$101$	41.0433	0.0404353	0.0202176	−	0.999796i	$-0.493564\pi$
$101$	41.0433	0.0404353	0.0202176	+	0.999796i	$0.493564\pi$
$102$	0	0
$103$	−165.066	−0.157907	−0.0789535	−	0.996878i	$-0.525158\pi$
$103$	−165.066	−0.157907	−0.0789535	+	0.996878i	$0.525158\pi$
$104$	207.325	0.195480
$105$	0	0
$106$	52.0301	0.0476756
$107$	969.423	0.875866	0.437933	−	0.899008i	$-0.355711\pi$
$107$	969.423	0.875866	0.437933	+	0.899008i	$0.355711\pi$
$108$	0	0
$109$	1385.74	1.21771	0.608854	−	0.793282i	$-0.291629\pi$
$109$	1385.74	1.21771	0.608854	+	0.793282i	$0.291629\pi$
$110$	−27.4767	−0.0238164
$111$	0	0
$112$	−1294.20	−1.09188
$113$	−2090.30	−1.74017	−0.870083	−	0.492905i	$-0.835935\pi$
$113$	−2090.30	−1.74017	−0.870083	+	0.492905i	$0.835935\pi$
$114$	0	0
$115$	350.430	0.284154
$116$	−1817.23	−1.45453
$117$	0	0
$118$	118.687	0.0925937
$119$	−1502.76	−1.15762
$120$	0	0
$121$	−1272.64	−0.956154
$122$	16.3822	0.0121572
$123$	0	0
$124$	−2672.56	−1.93551
$125$	−272.150	−0.194735
$126$	0	0
$127$	−547.234	−0.382355	−0.191178	−	0.981555i	$-0.561231\pi$
$127$	−547.234	−0.382355	−0.191178	+	0.981555i	$0.561231\pi$
$128$	475.083	0.328061
$129$	0	0
$130$	−198.116	−0.133661
$131$	1850.64	1.23428	0.617141	−	0.786853i	$-0.288291\pi$
$131$	1850.64	1.23428	0.617141	+	0.786853i	$0.288291\pi$
$132$	0	0
$133$	1998.96	1.30325
$134$	100.522	0.0648046
$135$	0	0
$136$	273.878	0.172683
$137$	2264.90	1.41244	0.706218	−	0.707995i	$-0.250400\pi$
$137$	2264.90	1.41244	0.706218	+	0.707995i	$0.250400\pi$
$138$	0	0
$139$	−1479.84	−0.903011	−0.451506	−	0.892268i	$-0.649113\pi$
$139$	−1479.84	−0.903011	−0.451506	+	0.892268i	$0.649113\pi$
$140$	2499.76	1.50906
$141$	0	0
$142$	−33.1920	−0.0196156
$143$	420.789	0.246071
$144$	0	0
$145$	3485.22	1.99608
$146$	64.5094	0.0365674
$147$	0	0
$148$	−1692.99	−0.940290
$149$	350.199	0.192546	0.0962732	−	0.995355i	$-0.469308\pi$
$149$	350.199	0.192546	0.0962732	+	0.995355i	$0.469308\pi$
$150$	0	0
$151$	3098.84	1.67006	0.835032	−	0.550201i	$-0.185449\pi$
$151$	3098.84	1.67006	0.835032	+	0.550201i	$0.185449\pi$
$152$	−364.312	−0.194405
$153$	0	0
$154$	37.2447	0.0194887
$155$	5125.62	2.65613
$156$	0	0
$157$	3570.96	1.81525	0.907623	−	0.419786i	$-0.137895\pi$
$157$	3570.96	1.81525	0.907623	+	0.419786i	$0.137895\pi$
$158$	55.3629	0.0278762
$159$	0	0
$160$	−684.173	−0.338054
$161$	−475.007	−0.232520
$162$	0	0
$163$	−793.187	−0.381149	−0.190574	−	0.981673i	$-0.561035\pi$
$163$	−793.187	−0.381149	−0.190574	+	0.981673i	$0.561035\pi$
$164$	−2582.75	−1.22975
$165$	0	0
$166$	−55.0883	−0.0257571
$167$	−2864.84	−1.32747	−0.663737	−	0.747966i	$-0.731031\pi$
$167$	−2864.84	−1.32747	−0.663737	+	0.747966i	$0.731031\pi$
$168$	0	0
$169$	837.031	0.380988
$170$	−261.714	−0.118074
$171$	0	0
$172$	2365.70	1.04874
$173$	675.225	0.296742	0.148371	−	0.988932i	$-0.452597\pi$
$173$	675.225	0.296742	0.148371	+	0.988932i	$0.452597\pi$
$174$	0	0
$175$	−2212.66	−0.955779
$176$	478.723	0.205029
$177$	0	0
$178$	207.877	0.0875339
$179$	2171.84	0.906878	0.453439	−	0.891287i	$-0.350197\pi$
$179$	2171.84	0.906878	0.453439	+	0.891287i	$0.350197\pi$
$180$	0	0
$181$	−3093.12	−1.27022	−0.635110	−	0.772422i	$-0.719045\pi$
$181$	−3093.12	−1.27022	−0.635110	+	0.772422i	$0.719045\pi$
$182$	268.546	0.109374
$183$	0	0
$184$	86.5704	0.0346851
$185$	3246.93	1.29037
$186$	0	0
$187$	555.867	0.217374
$188$	−3928.80	−1.52413
$189$	0	0
$190$	348.130	0.132926
$191$	854.865	0.323853	0.161926	−	0.986803i	$-0.448229\pi$
$191$	854.865	0.323853	0.161926	+	0.986803i	$0.448229\pi$
$192$	0	0
$193$	4203.75	1.56784	0.783919	−	0.620863i	$-0.213218\pi$
$193$	4203.75	1.56784	0.783919	+	0.620863i	$0.213218\pi$
$194$	−4.82333	−0.00178502
$195$	0	0
$196$	−663.543	−0.241816
$197$	−3816.81	−1.38039	−0.690195	−	0.723624i	$-0.742475\pi$
$197$	−3816.81	−1.38039	−0.690195	+	0.723624i	$0.742475\pi$
$198$	0	0
$199$	−1748.28	−0.622775	−0.311388	−	0.950283i	$-0.600794\pi$
$199$	−1748.28	−0.622775	−0.311388	+	0.950283i	$0.600794\pi$
$200$	403.259	0.142574
$201$	0	0
$202$	−9.68901	−0.00337483
$203$	−4724.21	−1.63337
$204$	0	0
$205$	4953.37	1.68760
$206$	38.9667	0.0131793
$207$	0	0
$208$	3451.75	1.15065
$209$	−739.412	−0.244719
$210$	0	0
$211$	803.562	0.262178	0.131089	−	0.991371i	$-0.458153\pi$
$211$	803.562	0.262178	0.131089	+	0.991371i	$0.458153\pi$
$212$	1750.94	0.567242
$213$	0	0
$214$	−228.850	−0.0731021
$215$	−4537.11	−1.43920
$216$	0	0
$217$	−6947.77	−2.17348
$218$	−327.130	−0.101633
$219$	0	0
$220$	−924.659	−0.283366
$221$	4007.99	1.21994
$222$	0	0
$223$	2439.37	0.732521	0.366261	−	0.930512i	$-0.380638\pi$
$223$	2439.37	0.732521	0.366261	+	0.930512i	$0.380638\pi$
$224$	927.395	0.276626
$225$	0	0
$226$	493.453	0.145239
$227$	−1743.66	−0.509828	−0.254914	−	0.966964i	$-0.582047\pi$
$227$	−1743.66	−0.509828	−0.254914	+	0.966964i	$0.582047\pi$
$228$	0	0
$229$	−3074.87	−0.887308	−0.443654	−	0.896198i	$-0.646318\pi$
$229$	−3074.87	−0.887308	−0.443654	+	0.896198i	$0.646318\pi$
$230$	−82.7252	−0.0237163
$231$	0	0
$232$	860.991	0.243650
$233$	−2141.41	−0.602097	−0.301048	−	0.953609i	$-0.597337\pi$
$233$	−2141.41	−0.602097	−0.301048	+	0.953609i	$0.597337\pi$
$234$	0	0
$235$	7534.92	2.09159
$236$	3994.12	1.10168
$237$	0	0
$238$	354.752	0.0966184
$239$	6489.96	1.75649	0.878244	−	0.478213i	$-0.158715\pi$
$239$	6489.96	1.75649	0.878244	+	0.478213i	$0.158715\pi$
$240$	0	0
$241$	−2275.08	−0.608095	−0.304048	−	0.952657i	$-0.598338\pi$
$241$	−2275.08	−0.608095	−0.304048	+	0.952657i	$0.598338\pi$
$242$	300.430	0.0798031
$243$	0	0
$244$	551.303	0.144646
$245$	1272.59	0.331848
$246$	0	0
$247$	−5331.40	−1.37340
$248$	1266.24	0.324218
$249$	0	0
$250$	64.2459	0.0162531
$251$	5666.83	1.42505	0.712525	−	0.701647i	$-0.247552\pi$
$251$	5666.83	1.42505	0.712525	+	0.701647i	$0.247552\pi$
$252$	0	0
$253$	175.704	0.0436618
$254$	129.184	0.0319124
$255$	0	0
$256$	3813.64	0.931065
$257$	−5186.02	−1.25873	−0.629367	−	0.777108i	$-0.716686\pi$
$257$	−5186.02	−1.25873	−0.629367	+	0.777108i	$0.716686\pi$
$258$	0	0
$259$	−4401.22	−1.05590
$260$	−6667.10	−1.59029
$261$	0	0
$262$	−436.876	−0.103016
$263$	7104.47	1.66570	0.832852	−	0.553495i	$-0.186706\pi$
$263$	7104.47	1.66570	0.832852	+	0.553495i	$0.186706\pi$
$264$	0	0
$265$	−3358.08	−0.778435
$266$	−471.890	−0.108772
$267$	0	0
$268$	3382.83	0.771041
$269$	−4632.95	−1.05010	−0.525048	−	0.851073i	$-0.675953\pi$
$269$	−4632.95	−1.05010	−0.525048	+	0.851073i	$0.675953\pi$
$270$	0	0
$271$	−7501.74	−1.68154	−0.840771	−	0.541390i	$-0.817898\pi$
$271$	−7501.74	−1.68154	−0.840771	+	0.541390i	$0.817898\pi$
$272$	4559.80	1.01646
$273$	0	0
$274$	−534.671	−0.117886
$275$	818.460	0.179473
$276$	0	0
$277$	−6324.35	−1.37182	−0.685909	−	0.727688i	$-0.740595\pi$
$277$	−6324.35	−1.37182	−0.685909	+	0.727688i	$0.740595\pi$
$278$	349.343	0.0753677
$279$	0	0
$280$	−1184.37	−0.252784
$281$	−2655.90	−0.563835	−0.281918	−	0.959439i	$-0.590970\pi$
$281$	−2655.90	−0.563835	−0.281918	+	0.959439i	$0.590970\pi$
$282$	0	0
$283$	1984.20	0.416780	0.208390	−	0.978046i	$-0.433178\pi$
$283$	1984.20	0.416780	0.208390	+	0.978046i	$0.433178\pi$
$284$	−1116.99	−0.233385
$285$	0	0
$286$	−99.3349	−0.0205377
$287$	−6714.29	−1.38095
$288$	0	0
$289$	381.590	0.0776694
$290$	−822.748	−0.166598
$291$	0	0
$292$	2170.90	0.435077
$293$	1005.18	0.200420	0.100210	−	0.994966i	$-0.468048\pi$
$293$	1005.18	0.200420	0.100210	+	0.994966i	$0.468048\pi$
$294$	0	0
$295$	−7660.20	−1.51185
$296$	802.125	0.157509
$297$	0	0
$298$	−82.6708	−0.0160704
$299$	1266.89	0.245037
$300$	0	0
$301$	6150.04	1.17768
$302$	−731.536	−0.139388
$303$	0	0
$304$	−6065.42	−1.14433
$305$	−1057.33	−0.198500
$306$	0	0
$307$	−7512.58	−1.39663	−0.698315	−	0.715791i	$-0.746067\pi$
$307$	−7512.58	−1.39663	−0.698315	+	0.715791i	$0.746067\pi$
$308$	1253.37	0.231876
$309$	0	0
$310$	−1209.99	−0.221687
$311$	594.508	0.108397	0.0541984	−	0.998530i	$-0.482740\pi$
$311$	594.508	0.108397	0.0541984	+	0.998530i	$0.482740\pi$
$312$	0	0
$313$	1280.17	0.231180	0.115590	−	0.993297i	$-0.463124\pi$
$313$	1280.17	0.231180	0.115590	+	0.993297i	$0.463124\pi$
$314$	−842.989	−0.151505
$315$	0	0
$316$	1863.10	0.331669
$317$	6849.61	1.21360	0.606802	−	0.794853i	$-0.292452\pi$
$317$	6849.61	1.21360	0.606802	+	0.794853i	$0.292452\pi$
$318$	0	0
$319$	1747.48	0.306708
$320$	−7476.71	−1.30613
$321$	0	0
$322$	112.134	0.0194068
$323$	−7042.84	−1.21323
$324$	0	0
$325$	5901.37	1.00723
$326$	187.246	0.0318117
$327$	0	0
$328$	1223.69	0.205996
$329$	−10213.6	−1.71153
$330$	0	0
$331$	−3516.94	−0.584014	−0.292007	−	0.956416i	$-0.594323\pi$
$331$	−3516.94	−0.584014	−0.292007	+	0.956416i	$0.594323\pi$
$332$	−1853.86	−0.306457
$333$	0	0
$334$	676.297	0.110794
$335$	−6487.82	−1.05811
$336$	0	0
$337$	−316.930	−0.0512294	−0.0256147	−	0.999672i	$-0.508154\pi$
$337$	−316.930	−0.0512294	−0.0256147	+	0.999672i	$0.508154\pi$
$338$	−197.596	−0.0317983
$339$	0	0
$340$	−8807.31	−1.40483
$341$	2569.97	0.408128
$342$	0	0
$343$	5358.81	0.843581
$344$	−1120.85	−0.175675
$345$	0	0
$346$	−159.399	−0.0247669
$347$	7439.30	1.15090	0.575450	−	0.817837i	$-0.304827\pi$
$347$	7439.30	1.15090	0.575450	+	0.817837i	$0.304827\pi$
$348$	0	0
$349$	7957.28	1.22047	0.610234	−	0.792221i	$-0.291075\pi$
$349$	7957.28	1.22047	0.610234	+	0.792221i	$0.291075\pi$
$350$	522.338	0.0797719
$351$	0	0
$352$	−343.042	−0.0519438
$353$	−6627.43	−0.999270	−0.499635	−	0.866236i	$-0.666533\pi$
$353$	−6627.43	−0.999270	−0.499635	+	0.866236i	$0.666533\pi$
$354$	0	0
$355$	2142.25	0.320278
$356$	6995.58	1.04147
$357$	0	0
$358$	−512.702	−0.0756904
$359$	−2297.30	−0.337736	−0.168868	−	0.985639i	$-0.554011\pi$
$359$	−2297.30	−0.337736	−0.168868	+	0.985639i	$0.554011\pi$
$360$	0	0
$361$	2509.35	0.365848
$362$	730.186	0.106016
$363$	0	0
$364$	9037.25	1.30132
$365$	−4163.50	−0.597062
$366$	0	0
$367$	3946.33	0.561299	0.280649	−	0.959810i	$-0.409450\pi$
$367$	3946.33	0.561299	0.280649	+	0.959810i	$0.409450\pi$
$368$	1441.31	0.204167
$369$	0	0
$370$	−766.497	−0.107698
$371$	4551.87	0.636985
$372$	0	0
$373$	−8079.93	−1.12162	−0.560808	−	0.827946i	$-0.689510\pi$
$373$	−8079.93	−1.12162	−0.560808	+	0.827946i	$0.689510\pi$
$374$	−131.222	−0.0181426
$375$	0	0
$376$	1861.43	0.255309
$377$	12599.9	1.72129
$378$	0	0
$379$	−14073.0	−1.90734	−0.953672	−	0.300848i	$-0.902730\pi$
$379$	−14073.0	−1.90734	−0.953672	+	0.300848i	$0.902730\pi$
$380$	11715.4	1.58155
$381$	0	0
$382$	−201.806	−0.0270296
$383$	−368.909	−0.0492177	−0.0246088	−	0.999697i	$-0.507834\pi$
$383$	−368.909	−0.0492177	−0.0246088	+	0.999697i	$0.507834\pi$
$384$	0	0
$385$	−2403.81	−0.318206
$386$	−992.372	−0.130856
$387$	0	0
$388$	−162.317	−0.0212381
$389$	13191.9	1.71943	0.859715	−	0.510775i	$-0.170641\pi$
$389$	13191.9	1.71943	0.859715	+	0.510775i	$0.170641\pi$
$390$	0	0
$391$	1673.57	0.216461
$392$	314.382	0.0405068
$393$	0	0
$394$	901.027	0.115211
$395$	−3573.18	−0.455155
$396$	0	0
$397$	9006.02	1.13854	0.569269	−	0.822151i	$-0.307226\pi$
$397$	9006.02	1.13854	0.569269	+	0.822151i	$0.307226\pi$
$398$	412.713	0.0519784
$399$	0	0
$400$	6713.86	0.839232
$401$	−7510.15	−0.935259	−0.467630	−	0.883925i	$-0.654892\pi$
$401$	−7510.15	−0.935259	−0.467630	+	0.883925i	$0.654892\pi$
$402$	0	0
$403$	18530.3	2.29048
$404$	−326.059	−0.0401536
$405$	0	0
$406$	1115.23	0.136325
$407$	1628.00	0.198273
$408$	0	0
$409$	4657.49	0.563076	0.281538	−	0.959550i	$-0.409155\pi$
$409$	4657.49	0.563076	0.281538	+	0.959550i	$0.409155\pi$
$410$	−1169.33	−0.140852
$411$	0	0
$412$	1311.33	0.156807
$413$	10383.4	1.23713
$414$	0	0
$415$	3555.45	0.420555
$416$	−2473.45	−0.291516
$417$	0	0
$418$	174.551	0.0204249
$419$	−16461.2	−1.91929	−0.959647	−	0.281208i	$-0.909265\pi$
$419$	−16461.2	−1.91929	−0.959647	+	0.281208i	$0.909265\pi$
$420$	0	0
$421$	−14629.5	−1.69358	−0.846792	−	0.531923i	$-0.821469\pi$
$421$	−14629.5	−1.69358	−0.846792	+	0.531923i	$0.821469\pi$
$422$	−189.695	−0.0218820
$423$	0	0
$424$	−829.583	−0.0950191
$425$	7795.77	0.889765
$426$	0	0
$427$	1433.21	0.162430
$428$	−7701.36	−0.869764
$429$	0	0
$430$	1071.07	0.120119
$431$	−5198.55	−0.580987	−0.290493	−	0.956877i	$-0.593819\pi$
$431$	−5198.55	−0.580987	−0.290493	+	0.956877i	$0.593819\pi$
$432$	0	0
$433$	−3483.21	−0.386588	−0.193294	−	0.981141i	$-0.561917\pi$
$433$	−3483.21	−0.386588	−0.193294	+	0.981141i	$0.561917\pi$
$434$	1640.15	0.181404
$435$	0	0
$436$	−11008.7	−1.20923
$437$	−2226.18	−0.243690
$438$	0	0
$439$	−10248.2	−1.11416	−0.557082	−	0.830457i	$-0.688079\pi$
$439$	−10248.2	−1.11416	−0.557082	+	0.830457i	$0.688079\pi$
$440$	438.096	0.0474668
$441$	0	0
$442$	−946.157	−0.101819
$443$	7404.68	0.794146	0.397073	−	0.917787i	$-0.370026\pi$
$443$	7404.68	0.794146	0.397073	+	0.917787i	$0.370026\pi$
$444$	0	0
$445$	−13416.6	−1.42923
$446$	−575.857	−0.0611381
$447$	0	0
$448$	10134.7	1.06879
$449$	10822.8	1.13755	0.568774	−	0.822494i	$-0.307418\pi$
$449$	10822.8	1.13755	0.568774	+	0.822494i	$0.307418\pi$
$450$	0	0
$451$	2483.61	0.259309
$452$	16605.9	1.72804
$453$	0	0
$454$	411.623	0.0425516
$455$	−17332.3	−1.78582
$456$	0	0
$457$	8312.90	0.850900	0.425450	−	0.904982i	$-0.360116\pi$
$457$	8312.90	0.850900	0.425450	+	0.904982i	$0.360116\pi$
$458$	725.879	0.0740570
$459$	0	0
$460$	−2783.91	−0.282175
$461$	7234.32	0.730881	0.365440	−	0.930835i	$-0.380918\pi$
$461$	7234.32	0.730881	0.365440	+	0.930835i	$0.380918\pi$
$462$	0	0
$463$	−5295.12	−0.531502	−0.265751	−	0.964042i	$-0.585620\pi$
$463$	−5295.12	−0.531502	−0.265751	+	0.964042i	$0.585620\pi$
$464$	14334.6	1.43420
$465$	0	0
$466$	505.519	0.0502526
$467$	−12565.2	−1.24507	−0.622535	−	0.782592i	$-0.713897\pi$
$467$	−12565.2	−1.24507	−0.622535	+	0.782592i	$0.713897\pi$
$468$	0	0
$469$	8794.23	0.865842
$470$	−1778.75	−0.174570
$471$	0	0
$472$	−1892.38	−0.184542
$473$	−2274.89	−0.221141
$474$	0	0
$475$	−10369.9	−1.00169
$476$	11938.3	1.14956
$477$	0	0
$478$	−1532.07	−0.146601
$479$	−20358.9	−1.94200	−0.971002	−	0.239071i	$-0.923157\pi$
$479$	−20358.9	−1.94200	−0.971002	+	0.239071i	$0.923157\pi$
$480$	0	0
$481$	11738.4	1.11274
$482$	537.074	0.0507532
$483$	0	0
$484$	10110.2	0.949493
$485$	311.302	0.0291454
$486$	0	0
$487$	936.192	0.0871107	0.0435553	−	0.999051i	$-0.486132\pi$
$487$	936.192	0.0871107	0.0435553	+	0.999051i	$0.486132\pi$
$488$	−261.203	−0.0242297
$489$	0	0
$490$	−300.417	−0.0276969
$491$	464.136	0.0426602	0.0213301	−	0.999772i	$-0.493210\pi$
$491$	464.136	0.0426602	0.0213301	+	0.999772i	$0.493210\pi$
$492$	0	0
$493$	16644.6	1.52056
$494$	1258.57	0.114627
$495$	0	0
$496$	21081.6	1.90845
$497$	−2903.81	−0.262080
$498$	0	0
$499$	−10308.5	−0.924796	−0.462398	−	0.886672i	$-0.653011\pi$
$499$	−10308.5	−0.924796	−0.462398	+	0.886672i	$0.653011\pi$
$500$	2162.04	0.193378
$501$	0	0
$502$	−1337.76	−0.118938
$503$	−4196.36	−0.371981	−0.185990	−	0.982552i	$-0.559549\pi$
$503$	−4196.36	−0.371981	−0.185990	+	0.982552i	$0.559549\pi$
$504$	0	0
$505$	625.339	0.0551034
$506$	−41.4782	−0.00364413
$507$	0	0
$508$	4347.37	0.379692
$509$	5925.58	0.516005	0.258003	−	0.966144i	$-0.416936\pi$
$509$	5925.58	0.516005	0.258003	+	0.966144i	$0.416936\pi$
$510$	0	0
$511$	5643.62	0.488570
$512$	−4700.94	−0.405770
$513$	0	0
$514$	1224.25	0.105057
$515$	−2514.95	−0.215189
$516$	0	0
$517$	3777.99	0.321384
$518$	1038.99	0.0881282
$519$	0	0
$520$	3158.82	0.266391
$521$	−15787.7	−1.32759	−0.663794	−	0.747915i	$-0.731055\pi$
$521$	−15787.7	−1.32759	−0.663794	+	0.747915i	$0.731055\pi$
$522$	0	0
$523$	−3558.50	−0.297519	−0.148760	−	0.988873i	$-0.547528\pi$
$523$	−3558.50	−0.297519	−0.148760	+	0.988873i	$0.547528\pi$
$524$	−14702.0	−1.22568
$525$	0	0
$526$	−1677.14	−0.139024
$527$	24478.8	2.02336
$528$	0	0
$529$	529.000	0.0434783
$530$	792.735	0.0649702
$531$	0	0
$532$	−15880.3	−1.29417
$533$	17907.6	1.45528
$534$	0	0
$535$	14770.2	1.19359
$536$	−1602.76	−0.129158
$537$	0	0
$538$	1093.69	0.0876438
$539$	638.072	0.0509902
$540$	0	0
$541$	5412.82	0.430158	0.215079	−	0.976597i	$-0.430999\pi$
$541$	5412.82	0.430158	0.215079	+	0.976597i	$0.430999\pi$
$542$	1770.92	0.140346
$543$	0	0
$544$	−3267.45	−0.257520
$545$	21113.3	1.65944
$546$	0	0
$547$	−2315.27	−0.180976	−0.0904881	−	0.995898i	$-0.528843\pi$
$547$	−2315.27	−0.180976	−0.0904881	+	0.995898i	$0.528843\pi$
$548$	−17993.0	−1.40260
$549$	0	0
$550$	−193.212	−0.0149793
$551$	−22140.5	−1.71183
$552$	0	0
$553$	4843.44	0.372448
$554$	1492.98	0.114495
$555$	0	0
$556$	11756.3	0.896721
$557$	−11430.7	−0.869545	−0.434772	−	0.900540i	$-0.643171\pi$
$557$	−11430.7	−0.869545	−0.434772	+	0.900540i	$0.643171\pi$
$558$	0	0
$559$	−16402.7	−1.24108
$560$	−19718.5	−1.48796
$561$	0	0
$562$	626.973	0.0470592
$563$	17414.8	1.30364	0.651819	−	0.758375i	$-0.274006\pi$
$563$	17414.8	1.30364	0.651819	+	0.758375i	$0.274006\pi$
$564$	0	0
$565$	−31847.9	−2.37142
$566$	−468.407	−0.0347856
$567$	0	0
$568$	529.223	0.0390945
$569$	17224.9	1.26908	0.634539	−	0.772891i	$-0.281190\pi$
$569$	17224.9	1.26908	0.634539	+	0.772891i	$0.281190\pi$
$570$	0	0
$571$	−18042.2	−1.32232	−0.661159	−	0.750246i	$-0.729935\pi$
$571$	−18042.2	−1.32232	−0.661159	+	0.750246i	$0.729935\pi$
$572$	−3342.86	−0.244357
$573$	0	0
$574$	1585.03	0.115258
$575$	2464.17	0.178718
$576$	0	0
$577$	16322.9	1.17769	0.588847	−	0.808244i	$-0.299582\pi$
$577$	16322.9	1.17769	0.588847	+	0.808244i	$0.299582\pi$
$578$	−90.0811	−0.00648249
$579$	0	0
$580$	−27687.5	−1.98217
$581$	−4819.41	−0.344136
$582$	0	0
$583$	−1683.73	−0.119611
$584$	−1028.56	−0.0728800
$585$	0	0
$586$	−237.291	−0.0167276
$587$	−13631.1	−0.958460	−0.479230	−	0.877689i	$-0.659084\pi$
$587$	−13631.1	−0.958460	−0.479230	+	0.877689i	$0.659084\pi$
$588$	0	0
$589$	−32561.5	−2.27789
$590$	1808.33	0.126183
$591$	0	0
$592$	13354.6	0.927144
$593$	4990.19	0.345569	0.172785	−	0.984960i	$-0.444724\pi$
$593$	4990.19	0.345569	0.172785	+	0.984960i	$0.444724\pi$
$594$	0	0
$595$	−22896.1	−1.57756
$596$	−2782.08	−0.191205
$597$	0	0
$598$	−299.071	−0.0204514
$599$	−15924.0	−1.08621	−0.543103	−	0.839666i	$-0.682751\pi$
$599$	−15924.0	−1.08621	−0.543103	+	0.839666i	$0.682751\pi$
$600$	0	0
$601$	−1599.09	−0.108533	−0.0542666	−	0.998526i	$-0.517282\pi$
$601$	−1599.09	−0.108533	−0.0542666	+	0.998526i	$0.517282\pi$
$602$	−1451.83	−0.0982925
$603$	0	0
$604$	−24618.0	−1.65843
$605$	−19390.0	−1.30300
$606$	0	0
$607$	15008.9	1.00361	0.501805	−	0.864981i	$-0.332670\pi$
$607$	15008.9	1.00361	0.501805	+	0.864981i	$0.332670\pi$
$608$	4346.35	0.289914
$609$	0	0
$610$	249.601	0.0165673
$611$	27240.5	1.80366
$612$	0	0
$613$	25798.6	1.69983	0.849914	−	0.526921i	$-0.176654\pi$
$613$	25798.6	1.69983	0.849914	+	0.526921i	$0.176654\pi$
$614$	1773.48	0.116566
$615$	0	0
$616$	−593.839	−0.0388416
$617$	14138.0	0.922487	0.461244	−	0.887274i	$-0.347403\pi$
$617$	14138.0	0.922487	0.461244	+	0.887274i	$0.347403\pi$
$618$	0	0
$619$	−14928.1	−0.969320	−0.484660	−	0.874703i	$-0.661057\pi$
$619$	−14928.1	−0.969320	−0.484660	+	0.874703i	$0.661057\pi$
$620$	−40719.3	−2.63762
$621$	0	0
$622$	−140.344	−0.00904709
$623$	18186.2	1.16952
$624$	0	0
$625$	−17538.7	−1.12248
$626$	−302.206	−0.0192949
$627$	0	0
$628$	−28368.7	−1.80260
$629$	15506.6	0.982971
$630$	0	0
$631$	−15845.5	−0.999683	−0.499842	−	0.866117i	$-0.666608\pi$
$631$	−15845.5	−0.999683	−0.499842	+	0.866117i	$0.666608\pi$
$632$	−882.721	−0.0555581
$633$	0	0
$634$	−1616.97	−0.101291
$635$	−8337.69	−0.521057
$636$	0	0
$637$	4600.71	0.286165
$638$	−412.523	−0.0255987
$639$	0	0
$640$	7238.39	0.447067
$641$	−18989.4	−1.17010	−0.585052	−	0.810996i	$-0.698926\pi$
$641$	−18989.4	−1.17010	−0.585052	+	0.810996i	$0.698926\pi$
$642$	0	0
$643$	17937.4	1.10013	0.550063	−	0.835123i	$-0.314604\pi$
$643$	17937.4	1.10013	0.550063	+	0.835123i	$0.314604\pi$
$644$	3773.58	0.230901
$645$	0	0
$646$	1662.59	0.101260
$647$	8350.82	0.507426	0.253713	−	0.967280i	$-0.418348\pi$
$647$	8350.82	0.507426	0.253713	+	0.967280i	$0.418348\pi$
$648$	0	0
$649$	−3840.80	−0.232303
$650$	−1393.12	−0.0840659
$651$	0	0
$652$	6301.30	0.378494
$653$	−222.430	−0.0133298	−0.00666491	−	0.999978i	$-0.502122\pi$
$653$	−222.430	−0.0133298	−0.00666491	+	0.999978i	$0.502122\pi$
$654$	0	0
$655$	28196.4	1.68202
$656$	20373.1	1.21256
$657$	0	0
$658$	2411.10	0.142849
$659$	−8736.62	−0.516435	−0.258217	−	0.966087i	$-0.583135\pi$
$659$	−8736.62	−0.516435	−0.258217	+	0.966087i	$0.583135\pi$
$660$	0	0
$661$	5051.38	0.297241	0.148620	−	0.988894i	$-0.452517\pi$
$661$	5051.38	0.297241	0.148620	+	0.988894i	$0.452517\pi$
$662$	830.237	0.0487433
$663$	0	0
$664$	878.342	0.0513348
$665$	30456.3	1.77600
$666$	0	0
$667$	5261.20	0.305419
$668$	22759.1	1.31823
$669$	0	0
$670$	1531.57	0.0883128
$671$	−530.141	−0.0305005
$672$	0	0
$673$	−741.691	−0.0424816	−0.0212408	−	0.999774i	$-0.506762\pi$
$673$	−741.691	−0.0424816	−0.0212408	+	0.999774i	$0.506762\pi$
$674$	74.8171	0.00427574
$675$	0	0
$676$	−6649.60	−0.378334
$677$	−28146.0	−1.59784	−0.798919	−	0.601438i	$-0.794595\pi$
$677$	−28146.0	−1.59784	−0.798919	+	0.601438i	$0.794595\pi$
$678$	0	0
$679$	−421.970	−0.0238494
$680$	4172.83	0.235325
$681$	0	0
$682$	−606.687	−0.0340634
$683$	15890.9	0.890261	0.445130	−	0.895466i	$-0.353157\pi$
$683$	15890.9	0.890261	0.445130	+	0.895466i	$0.353157\pi$
$684$	0	0
$685$	34508.2	1.92480
$686$	−1265.04	−0.0704075
$687$	0	0
$688$	−18661.0	−1.03408
$689$	−12140.3	−0.671273
$690$	0	0
$691$	−10267.0	−0.565231	−0.282616	−	0.959233i	$-0.591202\pi$
$691$	−10267.0	−0.565231	−0.282616	+	0.959233i	$0.591202\pi$
$692$	−5364.17	−0.294675
$693$	0	0
$694$	−1756.18	−0.0960572
$695$	−22547.0	−1.23058
$696$	0	0
$697$	23656.2	1.28557
$698$	−1878.46	−0.101863
$699$	0	0
$700$	17578.0	0.949121
$701$	2254.95	0.121495	0.0607476	−	0.998153i	$-0.480652\pi$
$701$	2254.95	0.121495	0.0607476	+	0.998153i	$0.480652\pi$
$702$	0	0
$703$	−20626.8	−1.10662
$704$	−3748.80	−0.200694
$705$	0	0
$706$	1564.52	0.0834017
$707$	−847.646	−0.0450905
$708$	0	0
$709$	−20509.7	−1.08640	−0.543201	−	0.839603i	$-0.682788\pi$
$709$	−20509.7	−1.08640	−0.543201	+	0.839603i	$0.682788\pi$
$710$	−505.716	−0.0267312
$711$	0	0
$712$	−3314.45	−0.174458
$713$	7737.51	0.406412
$714$	0	0
$715$	6411.17	0.335335
$716$	−17253.7	−0.900560
$717$	0	0
$718$	542.320	0.0281883
$719$	−16691.7	−0.865782	−0.432891	−	0.901446i	$-0.642506\pi$
$719$	−16691.7	−0.865782	−0.432891	+	0.901446i	$0.642506\pi$
$720$	0	0
$721$	3409.02	0.176087
$722$	−592.377	−0.0305346
$723$	0	0
$724$	24572.6	1.26137
$725$	24507.5	1.25543
$726$	0	0
$727$	23099.2	1.17841	0.589203	−	0.807985i	$-0.299442\pi$
$727$	23099.2	1.17841	0.589203	+	0.807985i	$0.299442\pi$
$728$	−4281.78	−0.217985
$729$	0	0
$730$	982.870	0.0498324
$731$	−21668.2	−1.09634
$732$	0	0
$733$	12216.5	0.615590	0.307795	−	0.951453i	$-0.400409\pi$
$733$	12216.5	0.615590	0.307795	+	0.951453i	$0.400409\pi$
$734$	−931.602	−0.0468475
$735$	0	0
$736$	−1032.81	−0.0517254
$737$	−3252.97	−0.162585
$738$	0	0
$739$	−3094.18	−0.154020	−0.0770102	−	0.997030i	$-0.524537\pi$
$739$	−3094.18	−0.154020	−0.0770102	+	0.997030i	$0.524537\pi$
$740$	−25794.5	−1.28139
$741$	0	0
$742$	−1074.55	−0.0531645
$743$	1452.57	0.0717223	0.0358611	−	0.999357i	$-0.488583\pi$
$743$	1452.57	0.0717223	0.0358611	+	0.999357i	$0.488583\pi$
$744$	0	0
$745$	5335.66	0.262394
$746$	1907.41	0.0936131
$747$	0	0
$748$	−4415.96	−0.215860
$749$	−20021.0	−0.976703
$750$	0	0
$751$	30754.7	1.49435	0.747174	−	0.664629i	$-0.231410\pi$
$751$	30754.7	1.49435	0.747174	+	0.664629i	$0.231410\pi$
$752$	30991.0	1.50282
$753$	0	0
$754$	−2974.43	−0.143664
$755$	47214.1	2.27589
$756$	0	0
$757$	14257.0	0.684516	0.342258	−	0.939606i	$-0.388808\pi$
$757$	14257.0	0.684516	0.342258	+	0.939606i	$0.388808\pi$
$758$	3322.19	0.159192
$759$	0	0
$760$	−5550.68	−0.264927
$761$	−31092.0	−1.48106	−0.740528	−	0.672025i	$-0.765425\pi$
$761$	−31092.0	−1.48106	−0.740528	+	0.672025i	$0.765425\pi$
$762$	0	0
$763$	−28619.1	−1.35790
$764$	−6791.28	−0.321597
$765$	0	0
$766$	87.0876	0.00410783
$767$	−27693.5	−1.30372
$768$	0	0
$769$	−1825.91	−0.0856230	−0.0428115	−	0.999083i	$-0.513631\pi$
$769$	−1825.91	−0.0856230	−0.0428115	+	0.999083i	$0.513631\pi$
$770$	567.462	0.0265583
$771$	0	0
$772$	−33395.8	−1.55692
$773$	7810.87	0.363438	0.181719	−	0.983351i	$-0.441834\pi$
$773$	7810.87	0.363438	0.181719	+	0.983351i	$0.441834\pi$
$774$	0	0
$775$	36042.6	1.67056
$776$	76.9044	0.00355761
$777$	0	0
$778$	−3114.19	−0.143508
$779$	−31467.3	−1.44728
$780$	0	0
$781$	1074.12	0.0492124
$782$	−395.076	−0.0180664
$783$	0	0
$784$	5234.13	0.238435
$785$	54407.4	2.47374
$786$	0	0
$787$	6847.58	0.310152	0.155076	−	0.987903i	$-0.450438\pi$
$787$	6847.58	0.310152	0.155076	+	0.987903i	$0.450438\pi$
$788$	30321.8	1.37077
$789$	0	0
$790$	843.513	0.0379884
$791$	43169.9	1.94051
$792$	0	0
$793$	−3822.49	−0.171174
$794$	−2126.03	−0.0950253
$795$	0	0
$796$	13888.8	0.618437
$797$	12059.9	0.535991	0.267995	−	0.963420i	$-0.413639\pi$
$797$	12059.9	0.535991	0.267995	+	0.963420i	$0.413639\pi$
$798$	0	0
$799$	35985.0	1.59332
$800$	−4811.00	−0.212618
$801$	0	0
$802$	1772.91	0.0780592
$803$	−2087.57	−0.0917418
$804$	0	0
$805$	−7237.24	−0.316868
$806$	−4374.42	−0.191169
$807$	0	0
$808$	154.484	0.00672616
$809$	17144.1	0.745061	0.372531	−	0.928020i	$-0.378490\pi$
$809$	17144.1	0.745061	0.372531	+	0.928020i	$0.378490\pi$
$810$	0	0
$811$	5857.32	0.253611	0.126805	−	0.991928i	$-0.459528\pi$
$811$	5857.32	0.253611	0.126805	+	0.991928i	$0.459528\pi$
$812$	37530.4	1.62199
$813$	0	0
$814$	−384.319	−0.0165484
$815$	−12085.1	−0.519412
$816$	0	0
$817$	28822.9	1.23425
$818$	−1099.49	−0.0469958
$819$	0	0
$820$	−39350.9	−1.67585
$821$	9358.29	0.397816	0.198908	−	0.980018i	$-0.436261\pi$
$821$	9358.29	0.397816	0.198908	+	0.980018i	$0.436261\pi$
$822$	0	0
$823$	9014.93	0.381823	0.190912	−	0.981607i	$-0.438856\pi$
$823$	9014.93	0.381823	0.190912	+	0.981607i	$0.438856\pi$
$824$	−621.296	−0.0262669
$825$	0	0
$826$	−2451.19	−0.103254
$827$	2953.43	0.124185	0.0620924	−	0.998070i	$-0.480223\pi$
$827$	2953.43	0.124185	0.0620924	+	0.998070i	$0.480223\pi$
$828$	0	0
$829$	−37652.3	−1.57747	−0.788733	−	0.614736i	$-0.789262\pi$
$829$	−37652.3	−1.57747	−0.788733	+	0.614736i	$0.789262\pi$
$830$	−839.328	−0.0351006
$831$	0	0
$832$	−27030.1	−1.12632
$833$	6077.59	0.252792
$834$	0	0
$835$	−43648.9	−1.80902
$836$	5874.09	0.243014
$837$	0	0
$838$	3885.97	0.160189
$839$	−4167.68	−0.171495	−0.0857475	−	0.996317i	$-0.527328\pi$
$839$	−4167.68	−0.171495	−0.0857475	+	0.996317i	$0.527328\pi$
$840$	0	0
$841$	27936.5	1.14545
$842$	3453.56	0.141351
$843$	0	0
$844$	−6383.71	−0.260351
$845$	12753.1	0.519194
$846$	0	0
$847$	26283.2	1.06623
$848$	−13811.7	−0.559311
$849$	0	0
$850$	−1840.33	−0.0742621
$851$	4901.49	0.197439
$852$	0	0
$853$	−30540.2	−1.22588	−0.612941	−	0.790129i	$-0.710014\pi$
$853$	−30540.2	−1.22588	−0.612941	+	0.790129i	$0.710014\pi$
$854$	−338.334	−0.0135568
$855$	0	0
$856$	3648.84	0.145695
$857$	−42576.3	−1.69706	−0.848529	−	0.529148i	$-0.822512\pi$
$857$	−42576.3	−1.69706	−0.848529	+	0.529148i	$0.822512\pi$
$858$	0	0
$859$	−3130.47	−0.124343	−0.0621714	−	0.998065i	$-0.519803\pi$
$859$	−3130.47	−0.124343	−0.0621714	+	0.998065i	$0.519803\pi$
$860$	36044.0	1.42917
$861$	0	0
$862$	1227.21	0.0484907
$863$	−1198.11	−0.0472587	−0.0236294	−	0.999721i	$-0.507522\pi$
$863$	−1198.11	−0.0472587	−0.0236294	+	0.999721i	$0.507522\pi$
$864$	0	0
$865$	10287.8	0.404387
$866$	822.274	0.0322656
$867$	0	0
$868$	55195.0	2.15834
$869$	−1791.58	−0.0699369
$870$	0	0
$871$	−23455.0	−0.912449
$872$	5215.85	0.202558
$873$	0	0
$874$	525.529	0.0203390
$875$	5620.58	0.217154
$876$	0	0
$877$	3725.02	0.143427	0.0717133	−	0.997425i	$-0.477153\pi$
$877$	3725.02	0.143427	0.0717133	+	0.997425i	$0.477153\pi$
$878$	2419.26	0.0929911
$879$	0	0
$880$	7293.85	0.279404
$881$	−20793.4	−0.795172	−0.397586	−	0.917565i	$-0.630152\pi$
$881$	−20793.4	−0.795172	−0.397586	+	0.917565i	$0.630152\pi$
$882$	0	0
$883$	−4745.60	−0.180863	−0.0904316	−	0.995903i	$-0.528825\pi$
$883$	−4745.60	−0.180863	−0.0904316	+	0.995903i	$0.528825\pi$
$884$	−31840.5	−1.21144
$885$	0	0
$886$	−1748.01	−0.0662815
$887$	18391.5	0.696198	0.348099	−	0.937458i	$-0.386827\pi$
$887$	18391.5	0.696198	0.348099	+	0.937458i	$0.386827\pi$
$888$	0	0
$889$	11301.7	0.426376
$890$	3167.23	0.119287
$891$	0	0
$892$	−19379.0	−0.727418
$893$	−47867.1	−1.79374
$894$	0	0
$895$	33090.3	1.23585
$896$	−9811.64	−0.365830
$897$	0	0
$898$	−2554.91	−0.0949428
$899$	76953.8	2.85490
$900$	0	0
$901$	−16037.4	−0.592989
$902$	−586.300	−0.0216426
$903$	0	0
$904$	−7867.75	−0.289466
$905$	−47127.0	−1.73100
$906$	0	0
$907$	−8492.10	−0.310888	−0.155444	−	0.987845i	$-0.549681\pi$
$907$	−8492.10	−0.310888	−0.155444	+	0.987845i	$0.549681\pi$
$908$	13852.1	0.506276
$909$	0	0
$910$	4091.59	0.149049
$911$	−29585.7	−1.07598	−0.537990	−	0.842951i	$-0.680816\pi$
$911$	−29585.7	−1.07598	−0.537990	+	0.842951i	$0.680816\pi$
$912$	0	0
$913$	1782.69	0.0646205
$914$	−1962.41	−0.0710183
$915$	0	0
$916$	24427.6	0.881127
$917$	−38220.2	−1.37638
$918$	0	0
$919$	−12182.1	−0.437268	−0.218634	−	0.975807i	$-0.570160\pi$
$919$	−12182.1	−0.437268	−0.218634	+	0.975807i	$0.570160\pi$
$920$	1318.99	0.0472673
$921$	0	0
$922$	−1707.79	−0.0610012
$923$	7744.74	0.276188
$924$	0	0
$925$	22832.0	0.811578
$926$	1250.01	0.0443605
$927$	0	0
$928$	−10271.9	−0.363352
$929$	−4720.51	−0.166712	−0.0833558	−	0.996520i	$-0.526564\pi$
$929$	−4720.51	−0.166712	−0.0833558	+	0.996520i	$0.526564\pi$
$930$	0	0
$931$	−8084.38	−0.284592
$932$	17012.0	0.597903
$933$	0	0
$934$	2966.24	0.103917
$935$	8469.23	0.296228
$936$	0	0
$937$	−14115.7	−0.492144	−0.246072	−	0.969252i	$-0.579140\pi$
$937$	−14115.7	−0.492144	−0.246072	+	0.969252i	$0.579140\pi$
$938$	−2076.04	−0.0722654
$939$	0	0
$940$	−59859.4	−2.07702
$941$	49060.5	1.69960	0.849802	−	0.527101i	$-0.176721\pi$
$941$	49060.5	1.69960	0.849802	+	0.527101i	$0.176721\pi$
$942$	0	0
$943$	7477.49	0.258219
$944$	−31506.3	−1.08627
$945$	0	0
$946$	537.029	0.0184570
$947$	−6322.87	−0.216965	−0.108482	−	0.994098i	$-0.534599\pi$
$947$	−6322.87	−0.216965	−0.108482	+	0.994098i	$0.534599\pi$
$948$	0	0
$949$	−15052.1	−0.514869
$950$	2448.00	0.0836038
$951$	0	0
$952$	−5656.27	−0.192564
$953$	−44749.8	−1.52108	−0.760540	−	0.649291i	$-0.775066\pi$
$953$	−44749.8	−1.52108	−0.760540	+	0.649291i	$0.775066\pi$
$954$	0	0
$955$	13024.8	0.441332
$956$	−51558.0	−1.74425
$957$	0	0
$958$	4806.07	0.162085
$959$	−46775.8	−1.57505
$960$	0	0
$961$	83382.9	2.79893
$962$	−2771.07	−0.0928720
$963$	0	0
$964$	18073.9	0.603859
$965$	64048.7	2.13658
$966$	0	0
$967$	−13589.8	−0.451934	−0.225967	−	0.974135i	$-0.572554\pi$
$967$	−13589.8	−0.451934	−0.225967	+	0.974135i	$0.572554\pi$
$968$	−4790.13	−0.159050
$969$	0	0
$970$	−73.4885	−0.00243255
$971$	32341.7	1.06889	0.534447	−	0.845202i	$-0.320520\pi$
$971$	32341.7	1.06889	0.534447	+	0.845202i	$0.320520\pi$
$972$	0	0
$973$	30562.4	1.00697
$974$	−221.005	−0.00727048
$975$	0	0
$976$	−4348.76	−0.142623
$977$	−50596.6	−1.65684	−0.828418	−	0.560111i	$-0.810758\pi$
$977$	−50596.6	−1.65684	−0.828418	+	0.560111i	$0.810758\pi$
$978$	0	0
$979$	−6727.04	−0.219609
$980$	−10109.8	−0.329536
$981$	0	0
$982$	−109.568	−0.00356053
$983$	50185.4	1.62835	0.814174	−	0.580621i	$-0.197190\pi$
$983$	50185.4	1.62835	0.814174	+	0.580621i	$0.197190\pi$
$984$	0	0
$985$	−58153.2	−1.88113
$986$	−3929.25	−0.126910
$987$	0	0
$988$	42354.1	1.36383
$989$	−6849.10	−0.220211
$990$	0	0
$991$	−4158.30	−0.133292	−0.0666462	−	0.997777i	$-0.521230\pi$
$991$	−4158.30	−0.133292	−0.0666462	+	0.997777i	$0.521230\pi$
$992$	−15106.6	−0.483502
$993$	0	0
$994$	685.498	0.0218739
$995$	−26636.9	−0.848690
$996$	0	0
$997$	−6534.65	−0.207577	−0.103789	−	0.994599i	$-0.533097\pi$
$997$	−6534.65	−0.207577	−0.103789	+	0.994599i	$0.533097\pi$
$998$	2433.51	0.0771859
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.c.1.1		2
3.2	odd	2		69.4.a.a.1.2	✓	2
12.11	even	2		1104.4.a.h.1.1		2
15.14	odd	2		1725.4.a.n.1.1		2
69.68	even	2		1587.4.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.a.1.2	✓	2	3.2	odd	2
207.4.a.c.1.1		2	1.1	even	1	trivial
1104.4.a.h.1.1		2	12.11	even	2
1587.4.a.b.1.2		2	69.68	even	2
1725.4.a.n.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-0.236068 q^{2} -7.94427 q^{4} +15.2361 q^{5} -20.6525 q^{7} +3.76393 q^{8} -3.59675 q^{10} +7.63932 q^{11} +55.0820 q^{13} +4.87539 q^{14} +62.6656 q^{16} +72.7639 q^{17} -96.7902 q^{19} -121.039 q^{20} -1.80340 q^{22} +23.0000 q^{23} +107.138 q^{25} -13.0031 q^{26} +164.069 q^{28} +228.748 q^{29} +336.413 q^{31} -44.9048 q^{32} -17.1772 q^{34} -314.663 q^{35} +213.108 q^{37} +22.8491 q^{38} +57.3475 q^{40} +325.108 q^{41} -297.787 q^{43} -60.6888 q^{44} -5.42956 q^{46} +494.545 q^{47} +83.5248 q^{49} -25.2918 q^{50} -437.587 q^{52} -220.403 q^{53} +116.393 q^{55} -77.7345 q^{56} -54.0000 q^{58} -502.768 q^{59} -69.3963 q^{61} -79.4164 q^{62} -490.724 q^{64} +839.234 q^{65} -425.820 q^{67} -578.056 q^{68} +74.2817 q^{70} +140.604 q^{71} -273.266 q^{73} -50.3081 q^{74} +768.928 q^{76} -157.771 q^{77} -234.521 q^{79} +954.778 q^{80} -76.7477 q^{82} +233.358 q^{83} +1108.64 q^{85} +70.2980 q^{86} +28.7539 q^{88} -880.581 q^{89} -1137.58 q^{91} -182.718 q^{92} -116.746 q^{94} -1474.70 q^{95} +20.4319 q^{97} -19.7175 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{4} + 26 q^{5} - 10 q^{7} + 12 q^{8} + 42 q^{10} + 60 q^{11} - 24 q^{13} + 50 q^{14} + 18 q^{16} + 150 q^{17} - 46 q^{19} - 14 q^{20} + 220 q^{22} + 46 q^{23} + 98 q^{25} - 348 q^{26}+ \cdots - 992 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^4 + 26 * q^5 - 10 * q^7 + 12 * q^8 + 42 * q^10 + 60 * q^11 - 24 * q^13 + 50 * q^14 + 18 * q^16 + 150 * q^17 - 46 * q^19 - 14 * q^20 + 220 * q^22 + 46 * q^23 + 98 * q^25 - 348 * q^26 + 270 * q^28 + 216 * q^29 + 324 * q^31 - 300 * q^32 + 310 * q^34 - 200 * q^35 + 140 * q^37 + 238 * q^38 + 146 * q^40 + 364 * q^41 - 126 * q^43 + 460 * q^44 + 92 * q^46 - 120 * q^47 - 146 * q^49 - 64 * q^50 - 1224 * q^52 - 490 * q^53 + 680 * q^55 + 10 * q^56 - 108 * q^58 + 32 * q^59 - 908 * q^61 - 132 * q^62 - 1214 * q^64 - 12 * q^65 - 874 * q^67 + 190 * q^68 + 560 * q^70 - 488 * q^71 + 652 * q^73 - 360 * q^74 + 1274 * q^76 + 400 * q^77 - 1198 * q^79 + 474 * q^80 + 88 * q^82 + 100 * q^83 + 1940 * q^85 + 798 * q^86 + 460 * q^88 - 102 * q^89 - 1980 * q^91 + 46 * q^92 - 2720 * q^94 - 928 * q^95 + 1624 * q^97 - 992 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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