LMFDB - Embedded newform 1050.4.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,4,Mod(1,1050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1050.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1050.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,4,-6,8,0,-12,14,16,18,0,12,-24,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$61.9520055060$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1050.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +25.5959 q^{11} -12.0000 q^{12} -89.1918 q^{13} +14.0000 q^{14} +16.0000 q^{16} +18.7878 q^{17} +18.0000 q^{18} -11.7980 q^{19} -21.0000 q^{21} +51.1918 q^{22} -159.373 q^{23} -24.0000 q^{24} -178.384 q^{26} -27.0000 q^{27} +28.0000 q^{28} +19.7980 q^{29} -126.404 q^{31} +32.0000 q^{32} -76.7878 q^{33} +37.5755 q^{34} +36.0000 q^{36} +282.969 q^{37} -23.5959 q^{38} +267.576 q^{39} -144.384 q^{41} -42.0000 q^{42} +326.929 q^{43} +102.384 q^{44} -318.747 q^{46} -117.435 q^{47} -48.0000 q^{48} +49.0000 q^{49} -56.3633 q^{51} -356.767 q^{52} -634.504 q^{53} -54.0000 q^{54} +56.0000 q^{56} +35.3939 q^{57} +39.5959 q^{58} -515.069 q^{59} +395.716 q^{61} -252.808 q^{62} +63.0000 q^{63} +64.0000 q^{64} -153.576 q^{66} -842.120 q^{67} +75.1510 q^{68} +478.120 q^{69} +402.100 q^{71} +72.0000 q^{72} -303.171 q^{73} +565.939 q^{74} -47.1918 q^{76} +179.171 q^{77} +535.151 q^{78} +266.384 q^{79} +81.0000 q^{81} -288.767 q^{82} -753.253 q^{83} -84.0000 q^{84} +653.857 q^{86} -59.3939 q^{87} +204.767 q^{88} -842.967 q^{89} -624.343 q^{91} -637.494 q^{92} +379.212 q^{93} -234.869 q^{94} -96.0000 q^{96} -1201.92 q^{97} +98.0000 q^{98} +230.363 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9} + 12 q^{11} - 24 q^{12} - 100 q^{13} + 28 q^{14} + 32 q^{16} - 80 q^{17} + 36 q^{18} - 4 q^{19} - 42 q^{21} + 24 q^{22} - 64 q^{23}+ \cdots + 108 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 12 * q^6 + 14 * q^7 + 16 * q^8 + 18 * q^9 + 12 * q^11 - 24 * q^12 - 100 * q^13 + 28 * q^14 + 32 * q^16 - 80 * q^17 + 36 * q^18 - 4 * q^19 - 42 * q^21 + 24 * q^22 - 64 * q^23 - 48 * q^24 - 200 * q^26 - 54 * q^27 + 56 * q^28 + 20 * q^29 - 292 * q^31 + 64 * q^32 - 36 * q^33 - 160 * q^34 + 72 * q^36 + 272 * q^37 - 8 * q^38 + 300 * q^39 - 132 * q^41 - 84 * q^42 - 32 * q^43 + 48 * q^44 - 128 * q^46 - 568 * q^47 - 96 * q^48 + 98 * q^49 + 240 * q^51 - 400 * q^52 - 348 * q^53 - 108 * q^54 + 112 * q^56 + 12 * q^57 + 40 * q^58 + 224 * q^59 - 12 * q^61 - 584 * q^62 + 126 * q^63 + 128 * q^64 - 72 * q^66 - 920 * q^67 - 320 * q^68 + 192 * q^69 - 156 * q^71 + 144 * q^72 - 332 * q^73 + 544 * q^74 - 16 * q^76 + 84 * q^77 + 600 * q^78 + 376 * q^79 + 162 * q^81 - 264 * q^82 - 2016 * q^83 - 168 * q^84 - 64 * q^86 - 60 * q^87 + 96 * q^88 + 548 * q^89 - 700 * q^91 - 256 * q^92 + 876 * q^93 - 1136 * q^94 - 192 * q^96 - 1620 * q^97 + 196 * q^98 + 108 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	25.5959	0.701587	0.350794	−	0.936453i	$-0.385912\pi$
$11$	25.5959	0.701587	0.350794	+	0.936453i	$0.385912\pi$
$12$	−12.0000	−0.288675
$13$	−89.1918	−1.90287	−0.951437	−	0.307843i	$-0.900393\pi$
$13$	−89.1918	−1.90287	−0.951437	+	0.307843i	$0.900393\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	18.7878	0.268041	0.134021	−	0.990979i	$-0.457211\pi$
$17$	18.7878	0.268041	0.134021	+	0.990979i	$0.457211\pi$
$18$	18.0000	0.235702
$19$	−11.7980	−0.142455	−0.0712273	−	0.997460i	$-0.522692\pi$
$19$	−11.7980	−0.142455	−0.0712273	+	0.997460i	$0.522692\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	51.1918	0.496097
$23$	−159.373	−1.44486	−0.722428	−	0.691447i	$-0.756974\pi$
$23$	−159.373	−1.44486	−0.722428	+	0.691447i	$0.756974\pi$
$24$	−24.0000	−0.204124
$25$	0	0
$26$	−178.384	−1.34554
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	19.7980	0.126772	0.0633860	−	0.997989i	$-0.479810\pi$
$29$	19.7980	0.126772	0.0633860	+	0.997989i	$0.479810\pi$
$30$	0	0
$31$	−126.404	−0.732350	−0.366175	−	0.930546i	$-0.619333\pi$
$31$	−126.404	−0.732350	−0.366175	+	0.930546i	$0.619333\pi$
$32$	32.0000	0.176777
$33$	−76.7878	−0.405062
$34$	37.5755	0.189534
$35$	0	0
$36$	36.0000	0.166667
$37$	282.969	1.25729	0.628647	−	0.777691i	$-0.283609\pi$
$37$	282.969	1.25729	0.628647	+	0.777691i	$0.283609\pi$
$38$	−23.5959	−0.100731
$39$	267.576	1.09862
$40$	0	0
$41$	−144.384	−0.549974	−0.274987	−	0.961448i	$-0.588674\pi$
$41$	−144.384	−0.549974	−0.274987	+	0.961448i	$0.588674\pi$
$42$	−42.0000	−0.154303
$43$	326.929	1.15945	0.579723	−	0.814814i	$-0.303161\pi$
$43$	326.929	1.15945	0.579723	+	0.814814i	$0.303161\pi$
$44$	102.384	0.350794
$45$	0	0
$46$	−318.747	−1.02167
$47$	−117.435	−0.364460	−0.182230	−	0.983256i	$-0.558332\pi$
$47$	−117.435	−0.364460	−0.182230	+	0.983256i	$0.558332\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	0	0
$51$	−56.3633	−0.154754
$52$	−356.767	−0.951437
$53$	−634.504	−1.64445	−0.822225	−	0.569163i	$-0.807267\pi$
$53$	−634.504	−1.64445	−0.822225	+	0.569163i	$0.807267\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	56.0000	0.133631
$57$	35.3939	0.0822462
$58$	39.5959	0.0896414
$59$	−515.069	−1.13655	−0.568274	−	0.822839i	$-0.692389\pi$
$59$	−515.069	−1.13655	−0.568274	+	0.822839i	$0.692389\pi$
$60$	0	0
$61$	395.716	0.830595	0.415297	−	0.909686i	$-0.363678\pi$
$61$	395.716	0.830595	0.415297	+	0.909686i	$0.363678\pi$
$62$	−252.808	−0.517849
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	0	0
$66$	−153.576	−0.286422
$67$	−842.120	−1.53554	−0.767772	−	0.640724i	$-0.778634\pi$
$67$	−842.120	−1.53554	−0.767772	+	0.640724i	$0.778634\pi$
$68$	75.1510	0.134021
$69$	478.120	0.834187
$70$	0	0
$71$	402.100	0.672120	0.336060	−	0.941841i	$-0.390906\pi$
$71$	402.100	0.672120	0.336060	+	0.941841i	$0.390906\pi$
$72$	72.0000	0.117851
$73$	−303.171	−0.486076	−0.243038	−	0.970017i	$-0.578144\pi$
$73$	−303.171	−0.486076	−0.243038	+	0.970017i	$0.578144\pi$
$74$	565.939	0.889041
$75$	0	0
$76$	−47.1918	−0.0712273
$77$	179.171	0.265175
$78$	535.151	0.776845
$79$	266.384	0.379373	0.189687	−	0.981845i	$-0.439253\pi$
$79$	266.384	0.379373	0.189687	+	0.981845i	$0.439253\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−288.767	−0.388890
$83$	−753.253	−0.996148	−0.498074	−	0.867135i	$-0.665959\pi$
$83$	−753.253	−0.996148	−0.498074	+	0.867135i	$0.665959\pi$
$84$	−84.0000	−0.109109
$85$	0	0
$86$	653.857	0.819851
$87$	−59.3939	−0.0731919
$88$	204.767	0.248049
$89$	−842.967	−1.00398	−0.501991	−	0.864873i	$-0.667399\pi$
$89$	−842.967	−1.00398	−0.501991	+	0.864873i	$0.667399\pi$
$90$	0	0
$91$	−624.343	−0.719219
$92$	−637.494	−0.722428
$93$	379.212	0.422822
$94$	−234.869	−0.257712
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−1201.92	−1.25811	−0.629053	−	0.777362i	$-0.716557\pi$
$97$	−1201.92	−1.25811	−0.629053	+	0.777362i	$0.716557\pi$
$98$	98.0000	0.101015
$99$	230.363	0.233862
$100$	0	0
$101$	575.371	0.566847	0.283424	−	0.958995i	$-0.408530\pi$
$101$	575.371	0.566847	0.283424	+	0.958995i	$0.408530\pi$
$102$	−112.727	−0.109427
$103$	254.220	0.243195	0.121597	−	0.992579i	$-0.461198\pi$
$103$	254.220	0.243195	0.121597	+	0.992579i	$0.461198\pi$
$104$	−713.535	−0.672768
$105$	0	0
$106$	−1269.01	−1.16280
$107$	−1142.61	−1.03234	−0.516168	−	0.856487i	$-0.672642\pi$
$107$	−1142.61	−1.03234	−0.516168	+	0.856487i	$0.672642\pi$
$108$	−108.000	−0.0962250
$109$	−2054.28	−1.80518	−0.902589	−	0.430502i	$-0.858336\pi$
$109$	−2054.28	−1.80518	−0.902589	+	0.430502i	$0.858336\pi$
$110$	0	0
$111$	−848.908	−0.725899
$112$	112.000	0.0944911
$113$	890.827	0.741610	0.370805	−	0.928711i	$-0.379082\pi$
$113$	890.827	0.741610	0.370805	+	0.928711i	$0.379082\pi$
$114$	70.7878	0.0581568
$115$	0	0
$116$	79.1918	0.0633860
$117$	−802.727	−0.634291
$118$	−1030.14	−0.803661
$119$	131.514	0.101310
$120$	0	0
$121$	−675.849	−0.507775
$122$	791.433	0.587319
$123$	433.151	0.317528
$124$	−505.616	−0.366175
$125$	0	0
$126$	126.000	0.0890871
$127$	−2304.16	−1.60993	−0.804965	−	0.593323i	$-0.797816\pi$
$127$	−2304.16	−1.60993	−0.804965	+	0.593323i	$0.797816\pi$
$128$	128.000	0.0883883
$129$	−980.786	−0.669406
$130$	0	0
$131$	955.192	0.637065	0.318532	−	0.947912i	$-0.396810\pi$
$131$	955.192	0.637065	0.318532	+	0.947912i	$0.396810\pi$
$132$	−307.151	−0.202531
$133$	−82.5857	−0.0538428
$134$	−1684.24	−1.08579
$135$	0	0
$136$	150.302	0.0947669
$137$	1489.53	0.928900	0.464450	−	0.885599i	$-0.346252\pi$
$137$	1489.53	0.928900	0.464450	+	0.885599i	$0.346252\pi$
$138$	956.241	0.589860
$139$	−752.284	−0.459049	−0.229525	−	0.973303i	$-0.573717\pi$
$139$	−752.284	−0.459049	−0.229525	+	0.973303i	$0.573717\pi$
$140$	0	0
$141$	352.304	0.210421
$142$	804.200	0.475260
$143$	−2282.95	−1.33503
$144$	144.000	0.0833333
$145$	0	0
$146$	−606.343	−0.343707
$147$	−147.000	−0.0824786
$148$	1131.88	0.628647
$149$	1709.01	0.939648	0.469824	−	0.882760i	$-0.344317\pi$
$149$	1709.01	0.939648	0.469824	+	0.882760i	$0.344317\pi$
$150$	0	0
$151$	−2773.61	−1.49479	−0.747395	−	0.664380i	$-0.768696\pi$
$151$	−2773.61	−1.49479	−0.747395	+	0.664380i	$0.768696\pi$
$152$	−94.3837	−0.0503653
$153$	169.090	0.0893471
$154$	358.343	0.187507
$155$	0	0
$156$	1070.30	0.549312
$157$	−353.478	−0.179685	−0.0898426	−	0.995956i	$-0.528636\pi$
$157$	−353.478	−0.179685	−0.0898426	+	0.995956i	$0.528636\pi$
$158$	532.767	0.268258
$159$	1903.51	0.949424
$160$	0	0
$161$	−1115.61	−0.546104
$162$	162.000	0.0785674
$163$	−885.231	−0.425378	−0.212689	−	0.977120i	$-0.568222\pi$
$163$	−885.231	−0.425378	−0.212689	+	0.977120i	$0.568222\pi$
$164$	−577.535	−0.274987
$165$	0	0
$166$	−1506.51	−0.704383
$167$	−3220.26	−1.49216	−0.746081	−	0.665855i	$-0.768067\pi$
$167$	−3220.26	−1.49216	−0.746081	+	0.665855i	$0.768067\pi$
$168$	−168.000	−0.0771517
$169$	5758.18	2.62093
$170$	0	0
$171$	−106.182	−0.0474849
$172$	1307.71	0.579723
$173$	−3780.62	−1.66148	−0.830738	−	0.556663i	$-0.812081\pi$
$173$	−3780.62	−1.66148	−0.830738	+	0.556663i	$0.812081\pi$
$174$	−118.788	−0.0517545
$175$	0	0
$176$	409.535	0.175397
$177$	1545.21	0.656186
$178$	−1685.93	−0.709922
$179$	4534.07	1.89325	0.946627	−	0.322330i	$-0.104466\pi$
$179$	4534.07	1.89325	0.946627	+	0.322330i	$0.104466\pi$
$180$	0	0
$181$	1805.49	0.741443	0.370721	−	0.928744i	$-0.379110\pi$
$181$	1805.49	0.741443	0.370721	+	0.928744i	$0.379110\pi$
$182$	−1248.69	−0.508565
$183$	−1187.15	−0.479544
$184$	−1274.99	−0.510833
$185$	0	0
$186$	758.424	0.298981
$187$	480.890	0.188054
$188$	−469.739	−0.182230
$189$	−189.000	−0.0727393
$190$	0	0
$191$	3359.15	1.27256	0.636282	−	0.771456i	$-0.280471\pi$
$191$	3359.15	1.27256	0.636282	+	0.771456i	$0.280471\pi$
$192$	−192.000	−0.0721688
$193$	2800.56	1.04450	0.522251	−	0.852792i	$-0.325092\pi$
$193$	2800.56	1.04450	0.522251	+	0.852792i	$0.325092\pi$
$194$	−2403.84	−0.889616
$195$	0	0
$196$	196.000	0.0714286
$197$	−3252.52	−1.17631	−0.588154	−	0.808749i	$-0.700145\pi$
$197$	−3252.52	−1.17631	−0.588154	+	0.808749i	$0.700145\pi$
$198$	460.727	0.165366
$199$	−1525.31	−0.543347	−0.271674	−	0.962389i	$-0.587577\pi$
$199$	−1525.31	−0.543347	−0.271674	+	0.962389i	$0.587577\pi$
$200$	0	0
$201$	2526.36	0.886546
$202$	1150.74	0.400822
$203$	138.586	0.0479153
$204$	−225.453	−0.0773768
$205$	0	0
$206$	508.441	0.171965
$207$	−1434.36	−0.481618
$208$	−1427.07	−0.475719
$209$	−301.980	−0.0999443
$210$	0	0
$211$	−2934.29	−0.957370	−0.478685	−	0.877987i	$-0.658886\pi$
$211$	−2934.29	−0.957370	−0.478685	+	0.877987i	$0.658886\pi$
$212$	−2538.02	−0.822225
$213$	−1206.30	−0.388048
$214$	−2285.21	−0.729971
$215$	0	0
$216$	−216.000	−0.0680414
$217$	−884.829	−0.276802
$218$	−4108.56	−1.27645
$219$	909.514	0.280636
$220$	0	0
$221$	−1675.71	−0.510049
$222$	−1697.82	−0.513288
$223$	3626.30	1.08895	0.544473	−	0.838778i	$-0.316730\pi$
$223$	3626.30	1.08895	0.544473	+	0.838778i	$0.316730\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	1781.65	0.524397
$227$	2189.45	0.640171	0.320086	−	0.947389i	$-0.396288\pi$
$227$	2189.45	0.640171	0.320086	+	0.947389i	$0.396288\pi$
$228$	141.576	0.0411231
$229$	−5501.00	−1.58741	−0.793704	−	0.608304i	$-0.791850\pi$
$229$	−5501.00	−1.58741	−0.793704	+	0.608304i	$0.791850\pi$
$230$	0	0
$231$	−537.514	−0.153099
$232$	158.384	0.0448207
$233$	−3927.72	−1.10435	−0.552174	−	0.833729i	$-0.686202\pi$
$233$	−3927.72	−1.10435	−0.552174	+	0.833729i	$0.686202\pi$
$234$	−1605.45	−0.448512
$235$	0	0
$236$	−2060.28	−0.568274
$237$	−799.151	−0.219031
$238$	263.029	0.0716370
$239$	4435.11	1.20035	0.600174	−	0.799869i	$-0.295098\pi$
$239$	4435.11	1.20035	0.600174	+	0.799869i	$0.295098\pi$
$240$	0	0
$241$	4575.81	1.22304	0.611522	−	0.791227i	$-0.290557\pi$
$241$	4575.81	1.22304	0.611522	+	0.791227i	$0.290557\pi$
$242$	−1351.70	−0.359051
$243$	−243.000	−0.0641500
$244$	1582.87	0.415297
$245$	0	0
$246$	866.302	0.224526
$247$	1052.28	0.271073
$248$	−1011.23	−0.258925
$249$	2259.76	0.575126
$250$	0	0
$251$	1646.96	0.414163	0.207081	−	0.978324i	$-0.433603\pi$
$251$	1646.96	0.414163	0.207081	+	0.978324i	$0.433603\pi$
$252$	252.000	0.0629941
$253$	−4079.31	−1.01369
$254$	−4608.32	−1.13839
$255$	0	0
$256$	256.000	0.0625000
$257$	−2811.64	−0.682432	−0.341216	−	0.939985i	$-0.610839\pi$
$257$	−2811.64	−0.682432	−0.341216	+	0.939985i	$0.610839\pi$
$258$	−1961.57	−0.473341
$259$	1980.79	0.475212
$260$	0	0
$261$	178.182	0.0422573
$262$	1910.38	0.450473
$263$	3103.13	0.727556	0.363778	−	0.931486i	$-0.381487\pi$
$263$	3103.13	0.727556	0.363778	+	0.931486i	$0.381487\pi$
$264$	−614.302	−0.143211
$265$	0	0
$266$	−165.171	−0.0380726
$267$	2528.90	0.579649
$268$	−3368.48	−0.767772
$269$	−61.1877	−0.0138687	−0.00693435	−	0.999976i	$-0.502207\pi$
$269$	−61.1877	−0.0138687	−0.00693435	+	0.999976i	$0.502207\pi$
$270$	0	0
$271$	97.6285	0.0218838	0.0109419	−	0.999940i	$-0.496517\pi$
$271$	97.6285	0.0218838	0.0109419	+	0.999940i	$0.496517\pi$
$272$	300.604	0.0670103
$273$	1873.03	0.415241
$274$	2979.07	0.656832
$275$	0	0
$276$	1912.48	0.417094
$277$	−4719.24	−1.02365	−0.511826	−	0.859089i	$-0.671031\pi$
$277$	−4719.24	−1.02365	−0.511826	+	0.859089i	$0.671031\pi$
$278$	−1504.57	−0.324597
$279$	−1137.64	−0.244117
$280$	0	0
$281$	995.094	0.211254	0.105627	−	0.994406i	$-0.466315\pi$
$281$	995.094	0.211254	0.105627	+	0.994406i	$0.466315\pi$
$282$	704.608	0.148790
$283$	7995.58	1.67946	0.839732	−	0.543002i	$-0.182712\pi$
$283$	7995.58	1.67946	0.839732	+	0.543002i	$0.182712\pi$
$284$	1608.40	0.336060
$285$	0	0
$286$	−4565.89	−0.944010
$287$	−1010.69	−0.207871
$288$	288.000	0.0589256
$289$	−4560.02	−0.928154
$290$	0	0
$291$	3605.76	0.726368
$292$	−1212.69	−0.243038
$293$	8930.79	1.78069	0.890345	−	0.455286i	$-0.150463\pi$
$293$	8930.79	1.78069	0.890345	+	0.455286i	$0.150463\pi$
$294$	−294.000	−0.0583212
$295$	0	0
$296$	2263.76	0.444521
$297$	−691.090	−0.135021
$298$	3418.02	0.664432
$299$	14214.8	2.74938
$300$	0	0
$301$	2288.50	0.438229
$302$	−5547.22	−1.05698
$303$	−1726.11	−0.327270
$304$	−188.767	−0.0356137
$305$	0	0
$306$	338.180	0.0631779
$307$	−2515.79	−0.467700	−0.233850	−	0.972273i	$-0.575132\pi$
$307$	−2515.79	−0.467700	−0.233850	+	0.972273i	$0.575132\pi$
$308$	716.686	0.132588
$309$	−762.661	−0.140409
$310$	0	0
$311$	3759.39	0.685452	0.342726	−	0.939435i	$-0.388650\pi$
$311$	3759.39	0.685452	0.342726	+	0.939435i	$0.388650\pi$
$312$	2140.60	0.388423
$313$	−6825.91	−1.23266	−0.616331	−	0.787487i	$-0.711382\pi$
$313$	−6825.91	−1.23266	−0.616331	+	0.787487i	$0.711382\pi$
$314$	−706.955	−0.127057
$315$	0	0
$316$	1065.53	0.189687
$317$	2464.94	0.436734	0.218367	−	0.975867i	$-0.429927\pi$
$317$	2464.94	0.436734	0.218367	+	0.975867i	$0.429927\pi$
$318$	3807.02	0.671344
$319$	506.747	0.0889416
$320$	0	0
$321$	3427.82	0.596019
$322$	−2231.23	−0.386154
$323$	−221.657	−0.0381837
$324$	324.000	0.0555556
$325$	0	0
$326$	−1770.46	−0.300788
$327$	6162.84	1.04222
$328$	−1155.07	−0.194445
$329$	−822.043	−0.137753
$330$	0	0
$331$	3780.36	0.627756	0.313878	−	0.949463i	$-0.398372\pi$
$331$	3780.36	0.627756	0.313878	+	0.949463i	$0.398372\pi$
$332$	−3013.01	−0.498074
$333$	2546.72	0.419098
$334$	−6440.52	−1.05512
$335$	0	0
$336$	−336.000	−0.0545545
$337$	−1914.39	−0.309446	−0.154723	−	0.987958i	$-0.549449\pi$
$337$	−1914.39	−0.309446	−0.154723	+	0.987958i	$0.549449\pi$
$338$	11516.4	1.85328
$339$	−2672.48	−0.428169
$340$	0	0
$341$	−3235.43	−0.513807
$342$	−212.363	−0.0335769
$343$	343.000	0.0539949
$344$	2615.43	0.409926
$345$	0	0
$346$	−7561.24	−1.17484
$347$	9118.93	1.41075	0.705375	−	0.708835i	$-0.250779\pi$
$347$	9118.93	1.41075	0.705375	+	0.708835i	$0.250779\pi$
$348$	−237.576	−0.0365959
$349$	4832.45	0.741189	0.370594	−	0.928795i	$-0.379154\pi$
$349$	4832.45	0.741189	0.370594	+	0.928795i	$0.379154\pi$
$350$	0	0
$351$	2408.18	0.366208
$352$	819.069	0.124024
$353$	−2806.58	−0.423170	−0.211585	−	0.977360i	$-0.567862\pi$
$353$	−2806.58	−0.423170	−0.211585	+	0.977360i	$0.567862\pi$
$354$	3090.42	0.463994
$355$	0	0
$356$	−3371.87	−0.501991
$357$	−394.543	−0.0584914
$358$	9068.15	1.33873
$359$	−1075.06	−0.158049	−0.0790246	−	0.996873i	$-0.525181\pi$
$359$	−1075.06	−0.158049	−0.0790246	+	0.996873i	$0.525181\pi$
$360$	0	0
$361$	−6719.81	−0.979707
$362$	3610.98	0.524279
$363$	2027.55	0.293164
$364$	−2497.37	−0.359609
$365$	0	0
$366$	−2374.30	−0.339089
$367$	−367.020	−0.0522025	−0.0261012	−	0.999659i	$-0.508309\pi$
$367$	−367.020	−0.0522025	−0.0261012	+	0.999659i	$0.508309\pi$
$368$	−2549.98	−0.361214
$369$	−1299.45	−0.183325
$370$	0	0
$371$	−4441.53	−0.621544
$372$	1516.85	0.211411
$373$	1304.56	0.181092	0.0905461	−	0.995892i	$-0.471139\pi$
$373$	1304.56	0.181092	0.0905461	+	0.995892i	$0.471139\pi$
$374$	961.780	0.132974
$375$	0	0
$376$	−939.478	−0.128856
$377$	−1765.82	−0.241231
$378$	−378.000	−0.0514344
$379$	5340.61	0.723822	0.361911	−	0.932213i	$-0.382124\pi$
$379$	5340.61	0.723822	0.361911	+	0.932213i	$0.382124\pi$
$380$	0	0
$381$	6912.48	0.929493
$382$	6718.31	0.899839
$383$	−6662.27	−0.888841	−0.444420	−	0.895818i	$-0.646590\pi$
$383$	−6662.27	−0.888841	−0.444420	+	0.895818i	$0.646590\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	5601.13	0.738575
$387$	2942.36	0.386482
$388$	−4807.67	−0.629053
$389$	−6313.48	−0.822895	−0.411448	−	0.911433i	$-0.634977\pi$
$389$	−6313.48	−0.822895	−0.411448	+	0.911433i	$0.634977\pi$
$390$	0	0
$391$	−2994.27	−0.387281
$392$	392.000	0.0505076
$393$	−2865.58	−0.367810
$394$	−6505.05	−0.831776
$395$	0	0
$396$	921.453	0.116931
$397$	14210.6	1.79650	0.898248	−	0.439490i	$-0.144841\pi$
$397$	14210.6	1.79650	0.898248	+	0.439490i	$0.144841\pi$
$398$	−3050.61	−0.384204
$399$	247.757	0.0310861
$400$	0	0
$401$	5578.59	0.694717	0.347359	−	0.937732i	$-0.387079\pi$
$401$	5578.59	0.694717	0.347359	+	0.937732i	$0.387079\pi$
$402$	5052.72	0.626883
$403$	11274.2	1.39357
$404$	2301.49	0.283424
$405$	0	0
$406$	277.171	0.0338812
$407$	7242.86	0.882101
$408$	−450.906	−0.0547137
$409$	11914.0	1.44037	0.720184	−	0.693783i	$-0.244057\pi$
$409$	11914.0	1.44037	0.720184	+	0.693783i	$0.244057\pi$
$410$	0	0
$411$	−4468.60	−0.536301
$412$	1016.88	0.121597
$413$	−3605.49	−0.429575
$414$	−2868.72	−0.340556
$415$	0	0
$416$	−2854.14	−0.336384
$417$	2256.85	0.265032
$418$	−603.959	−0.0706713
$419$	−12093.0	−1.40998	−0.704990	−	0.709217i	$-0.749049\pi$
$419$	−12093.0	−1.40998	−0.704990	+	0.709217i	$0.749049\pi$
$420$	0	0
$421$	−12188.5	−1.41100	−0.705500	−	0.708710i	$-0.749277\pi$
$421$	−12188.5	−1.41100	−0.705500	+	0.708710i	$0.749277\pi$
$422$	−5868.59	−0.676963
$423$	−1056.91	−0.121487
$424$	−5076.03	−0.581401
$425$	0	0
$426$	−2412.60	−0.274392
$427$	2770.01	0.313935
$428$	−4570.42	−0.516168
$429$	6848.84	0.770781
$430$	0	0
$431$	4968.16	0.555238	0.277619	−	0.960691i	$-0.410455\pi$
$431$	4968.16	0.555238	0.277619	+	0.960691i	$0.410455\pi$
$432$	−432.000	−0.0481125
$433$	−14483.7	−1.60749	−0.803745	−	0.594974i	$-0.797162\pi$
$433$	−14483.7	−1.60749	−0.803745	+	0.594974i	$0.797162\pi$
$434$	−1769.66	−0.195729
$435$	0	0
$436$	−8217.13	−0.902589
$437$	1880.28	0.205826
$438$	1819.03	0.198440
$439$	−9937.64	−1.08040	−0.540202	−	0.841535i	$-0.681652\pi$
$439$	−9937.64	−1.08040	−0.540202	+	0.841535i	$0.681652\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−3351.43	−0.360659
$443$	17139.6	1.83821	0.919105	−	0.394013i	$-0.128913\pi$
$443$	17139.6	1.83821	0.919105	+	0.394013i	$0.128913\pi$
$444$	−3395.63	−0.362950
$445$	0	0
$446$	7252.60	0.770001
$447$	−5127.03	−0.542506
$448$	448.000	0.0472456
$449$	1555.34	0.163477	0.0817385	−	0.996654i	$-0.473953\pi$
$449$	1555.34	0.163477	0.0817385	+	0.996654i	$0.473953\pi$
$450$	0	0
$451$	−3695.63	−0.385855
$452$	3563.31	0.370805
$453$	8320.84	0.863018
$454$	4378.90	0.452669
$455$	0	0
$456$	283.151	0.0290784
$457$	18409.2	1.88435	0.942174	−	0.335124i	$-0.108778\pi$
$457$	18409.2	1.88435	0.942174	+	0.335124i	$0.108778\pi$
$458$	−11002.0	−1.12247
$459$	−507.269	−0.0515845
$460$	0	0
$461$	6616.52	0.668465	0.334232	−	0.942491i	$-0.391523\pi$
$461$	6616.52	0.668465	0.334232	+	0.942491i	$0.391523\pi$
$462$	−1075.03	−0.108257
$463$	−12605.5	−1.26529	−0.632644	−	0.774443i	$-0.718030\pi$
$463$	−12605.5	−1.26529	−0.632644	+	0.774443i	$0.718030\pi$
$464$	316.767	0.0316930
$465$	0	0
$466$	−7855.43	−0.780892
$467$	16291.5	1.61431	0.807155	−	0.590340i	$-0.201006\pi$
$467$	16291.5	1.61431	0.807155	+	0.590340i	$0.201006\pi$
$468$	−3210.91	−0.317146
$469$	−5894.84	−0.580381
$470$	0	0
$471$	1060.43	0.103741
$472$	−4120.55	−0.401830
$473$	8368.04	0.813452
$474$	−1598.30	−0.154879
$475$	0	0
$476$	526.057	0.0506550
$477$	−5710.54	−0.548150
$478$	8870.22	0.848775
$479$	11506.3	1.09757	0.548785	−	0.835963i	$-0.315091\pi$
$479$	11506.3	1.09757	0.548785	+	0.835963i	$0.315091\pi$
$480$	0	0
$481$	−25238.6	−2.39247
$482$	9151.62	0.864823
$483$	3346.84	0.315293
$484$	−2703.40	−0.253888
$485$	0	0
$486$	−486.000	−0.0453609
$487$	16502.9	1.53556	0.767780	−	0.640714i	$-0.221361\pi$
$487$	16502.9	1.53556	0.767780	+	0.640714i	$0.221361\pi$
$488$	3165.73	0.293660
$489$	2655.69	0.245592
$490$	0	0
$491$	9269.67	0.852004	0.426002	−	0.904722i	$-0.359922\pi$
$491$	9269.67	0.852004	0.426002	+	0.904722i	$0.359922\pi$
$492$	1732.60	0.158764
$493$	371.959	0.0339801
$494$	2104.56	0.191678
$495$	0	0
$496$	−2022.47	−0.183087
$497$	2814.70	0.254037
$498$	4519.52	0.406676
$499$	13167.3	1.18126	0.590631	−	0.806941i	$-0.298879\pi$
$499$	13167.3	1.18126	0.590631	+	0.806941i	$0.298879\pi$
$500$	0	0
$501$	9660.78	0.861500
$502$	3293.91	0.292857
$503$	−16299.1	−1.44481	−0.722406	−	0.691469i	$-0.756964\pi$
$503$	−16299.1	−1.44481	−0.722406	+	0.691469i	$0.756964\pi$
$504$	504.000	0.0445435
$505$	0	0
$506$	−8158.62	−0.716788
$507$	−17274.6	−1.51319
$508$	−9216.64	−0.804965
$509$	−10595.4	−0.922656	−0.461328	−	0.887230i	$-0.652627\pi$
$509$	−10595.4	−0.922656	−0.461328	+	0.887230i	$0.652627\pi$
$510$	0	0
$511$	−2122.20	−0.183719
$512$	512.000	0.0441942
$513$	318.545	0.0274154
$514$	−5623.27	−0.482552
$515$	0	0
$516$	−3923.14	−0.334703
$517$	−3005.85	−0.255700
$518$	3961.57	0.336026
$519$	11341.9	0.959254
$520$	0	0
$521$	−3930.10	−0.330481	−0.165241	−	0.986253i	$-0.552840\pi$
$521$	−3930.10	−0.330481	−0.165241	+	0.986253i	$0.552840\pi$
$522$	356.363	0.0298805
$523$	15430.7	1.29013	0.645064	−	0.764129i	$-0.276831\pi$
$523$	15430.7	1.29013	0.645064	+	0.764129i	$0.276831\pi$
$524$	3820.77	0.318532
$525$	0	0
$526$	6206.26	0.514459
$527$	−2374.85	−0.196300
$528$	−1228.60	−0.101265
$529$	13232.9	1.08761
$530$	0	0
$531$	−4635.62	−0.378849
$532$	−330.343	−0.0269214
$533$	12877.8	1.04653
$534$	5057.80	0.409874
$535$	0	0
$536$	−6736.96	−0.542896
$537$	−13602.2	−1.09307
$538$	−122.375	−0.00980665
$539$	1254.20	0.100227
$540$	0	0
$541$	−24132.0	−1.91777	−0.958885	−	0.283794i	$-0.908407\pi$
$541$	−24132.0	−1.91777	−0.958885	+	0.283794i	$0.908407\pi$
$542$	195.257	0.0154742
$543$	−5416.48	−0.428072
$544$	601.208	0.0473834
$545$	0	0
$546$	3746.06	0.293620
$547$	−3505.64	−0.274022	−0.137011	−	0.990569i	$-0.543750\pi$
$547$	−3505.64	−0.274022	−0.137011	+	0.990569i	$0.543750\pi$
$548$	5958.13	0.464450
$549$	3561.45	0.276865
$550$	0	0
$551$	−233.576	−0.0180593
$552$	3824.96	0.294930
$553$	1864.69	0.143390
$554$	−9438.48	−0.723831
$555$	0	0
$556$	−3009.13	−0.229525
$557$	14427.0	1.09747	0.548737	−	0.835995i	$-0.315109\pi$
$557$	14427.0	1.09747	0.548737	+	0.835995i	$0.315109\pi$
$558$	−2275.27	−0.172616
$559$	−29159.4	−2.20628
$560$	0	0
$561$	−1442.67	−0.108573
$562$	1990.19	0.149379
$563$	9972.47	0.746518	0.373259	−	0.927727i	$-0.378240\pi$
$563$	9972.47	0.746518	0.373259	+	0.927727i	$0.378240\pi$
$564$	1409.22	0.105210
$565$	0	0
$566$	15991.2	1.18756
$567$	567.000	0.0419961
$568$	3216.80	0.237630
$569$	−25180.3	−1.85521	−0.927605	−	0.373563i	$-0.878136\pi$
$569$	−25180.3	−1.85521	−0.927605	+	0.373563i	$0.878136\pi$
$570$	0	0
$571$	6146.38	0.450470	0.225235	−	0.974305i	$-0.427685\pi$
$571$	6146.38	0.450470	0.225235	+	0.974305i	$0.427685\pi$
$572$	−9131.79	−0.667516
$573$	−10077.5	−0.734715
$574$	−2021.37	−0.146987
$575$	0	0
$576$	576.000	0.0416667
$577$	−12735.9	−0.918895	−0.459447	−	0.888205i	$-0.651952\pi$
$577$	−12735.9	−0.918895	−0.459447	+	0.888205i	$0.651952\pi$
$578$	−9120.04	−0.656304
$579$	−8401.69	−0.603044
$580$	0	0
$581$	−5272.77	−0.376508
$582$	7211.51	0.513620
$583$	−16240.7	−1.15372
$584$	−2425.37	−0.171854
$585$	0	0
$586$	17861.6	1.25914
$587$	−13223.9	−0.929829	−0.464915	−	0.885355i	$-0.653915\pi$
$587$	−13223.9	−0.929829	−0.464915	+	0.885355i	$0.653915\pi$
$588$	−588.000	−0.0412393
$589$	1491.31	0.104327
$590$	0	0
$591$	9757.57	0.679142
$592$	4527.51	0.314324
$593$	2498.96	0.173052	0.0865261	−	0.996250i	$-0.472423\pi$
$593$	2498.96	0.173052	0.0865261	+	0.996250i	$0.472423\pi$
$594$	−1382.18	−0.0954739
$595$	0	0
$596$	6836.04	0.469824
$597$	4575.92	0.313702
$598$	28429.6	1.94410
$599$	−9468.29	−0.645850	−0.322925	−	0.946425i	$-0.604666\pi$
$599$	−9468.29	−0.645850	−0.322925	+	0.946425i	$0.604666\pi$
$600$	0	0
$601$	−12334.1	−0.837137	−0.418569	−	0.908185i	$-0.637468\pi$
$601$	−12334.1	−0.837137	−0.418569	+	0.908185i	$0.637468\pi$
$602$	4577.00	0.309875
$603$	−7579.08	−0.511848
$604$	−11094.4	−0.747395
$605$	0	0
$606$	−3452.23	−0.231415
$607$	−25245.6	−1.68812	−0.844060	−	0.536248i	$-0.819841\pi$
$607$	−25245.6	−1.68812	−0.844060	+	0.536248i	$0.819841\pi$
$608$	−377.535	−0.0251827
$609$	−415.757	−0.0276639
$610$	0	0
$611$	10474.2	0.693521
$612$	676.359	0.0446735
$613$	4007.86	0.264071	0.132036	−	0.991245i	$-0.457849\pi$
$613$	4007.86	0.264071	0.132036	+	0.991245i	$0.457849\pi$
$614$	−5031.58	−0.330714
$615$	0	0
$616$	1433.37	0.0937535
$617$	14349.2	0.936266	0.468133	−	0.883658i	$-0.344927\pi$
$617$	14349.2	0.936266	0.468133	+	0.883658i	$0.344927\pi$
$618$	−1525.32	−0.0992839
$619$	−6004.19	−0.389869	−0.194934	−	0.980816i	$-0.562449\pi$
$619$	−6004.19	−0.389869	−0.194934	+	0.980816i	$0.562449\pi$
$620$	0	0
$621$	4303.08	0.278062
$622$	7518.78	0.484688
$623$	−5900.77	−0.379469
$624$	4281.21	0.274656
$625$	0	0
$626$	−13651.8	−0.871624
$627$	905.939	0.0577029
$628$	−1413.91	−0.0898426
$629$	5316.36	0.337007
$630$	0	0
$631$	17130.6	1.08076	0.540378	−	0.841422i	$-0.318281\pi$
$631$	17130.6	1.08076	0.540378	+	0.841422i	$0.318281\pi$
$632$	2131.07	0.134129
$633$	8802.88	0.552738
$634$	4929.87	0.308817
$635$	0	0
$636$	7614.05	0.474712
$637$	−4370.40	−0.271839
$638$	1013.49	0.0628912
$639$	3618.90	0.224040
$640$	0	0
$641$	13484.7	0.830911	0.415456	−	0.909613i	$-0.363622\pi$
$641$	13484.7	0.830911	0.415456	+	0.909613i	$0.363622\pi$
$642$	6855.64	0.421449
$643$	18209.9	1.11684	0.558419	−	0.829559i	$-0.311408\pi$
$643$	18209.9	1.11684	0.558419	+	0.829559i	$0.311408\pi$
$644$	−4462.46	−0.273052
$645$	0	0
$646$	−443.314	−0.0270000
$647$	13402.9	0.814406	0.407203	−	0.913338i	$-0.366504\pi$
$647$	13402.9	0.814406	0.407203	+	0.913338i	$0.366504\pi$
$648$	648.000	0.0392837
$649$	−13183.7	−0.797387
$650$	0	0
$651$	2654.49	0.159812
$652$	−3540.92	−0.212689
$653$	1922.32	0.115201	0.0576003	−	0.998340i	$-0.481655\pi$
$653$	1922.32	0.115201	0.0576003	+	0.998340i	$0.481655\pi$
$654$	12325.7	0.736961
$655$	0	0
$656$	−2310.14	−0.137494
$657$	−2728.54	−0.162025
$658$	−1644.09	−0.0974060
$659$	22722.3	1.34315	0.671574	−	0.740937i	$-0.265618\pi$
$659$	22722.3	1.34315	0.671574	+	0.740937i	$0.265618\pi$
$660$	0	0
$661$	−16049.3	−0.944393	−0.472196	−	0.881493i	$-0.656539\pi$
$661$	−16049.3	−0.944393	−0.472196	+	0.881493i	$0.656539\pi$
$662$	7560.72	0.443891
$663$	5027.14	0.294477
$664$	−6026.02	−0.352191
$665$	0	0
$666$	5093.45	0.296347
$667$	−3155.27	−0.183167
$668$	−12881.0	−0.746081
$669$	−10878.9	−0.628703
$670$	0	0
$671$	10128.7	0.582735
$672$	−672.000	−0.0385758
$673$	−16507.8	−0.945513	−0.472757	−	0.881193i	$-0.656741\pi$
$673$	−16507.8	−0.945513	−0.472757	+	0.881193i	$0.656741\pi$
$674$	−3828.78	−0.218811
$675$	0	0
$676$	23032.7	1.31047
$677$	15118.9	0.858298	0.429149	−	0.903234i	$-0.358813\pi$
$677$	15118.9	0.858298	0.429149	+	0.903234i	$0.358813\pi$
$678$	−5344.96	−0.302761
$679$	−8413.43	−0.475520
$680$	0	0
$681$	−6568.35	−0.369603
$682$	−6470.86	−0.363317
$683$	68.6595	0.00384654	0.00192327	−	0.999998i	$-0.499388\pi$
$683$	68.6595	0.00384654	0.00192327	+	0.999998i	$0.499388\pi$
$684$	−424.727	−0.0237424
$685$	0	0
$686$	686.000	0.0381802
$687$	16503.0	0.916490
$688$	5230.86	0.289861
$689$	56592.6	3.12918
$690$	0	0
$691$	−3167.95	−0.174406	−0.0872030	−	0.996191i	$-0.527793\pi$
$691$	−3167.95	−0.174406	−0.0872030	+	0.996191i	$0.527793\pi$
$692$	−15122.5	−0.830738
$693$	1612.54	0.0883917
$694$	18237.9	0.997550
$695$	0	0
$696$	−475.151	−0.0258772
$697$	−2712.64	−0.147416
$698$	9664.89	0.524100
$699$	11783.1	0.637596
$700$	0	0
$701$	6930.44	0.373408	0.186704	−	0.982416i	$-0.440219\pi$
$701$	6930.44	0.373408	0.186704	+	0.982416i	$0.440219\pi$
$702$	4816.36	0.258948
$703$	−3338.46	−0.179107
$704$	1638.14	0.0876984
$705$	0	0
$706$	−5613.15	−0.299226
$707$	4027.60	0.214248
$708$	6180.83	0.328093
$709$	20689.6	1.09593	0.547966	−	0.836501i	$-0.315402\pi$
$709$	20689.6	1.09593	0.547966	+	0.836501i	$0.315402\pi$
$710$	0	0
$711$	2397.45	0.126458
$712$	−6743.74	−0.354961
$713$	20145.5	1.05814
$714$	−789.086	−0.0413596
$715$	0	0
$716$	18136.3	0.946627
$717$	−13305.3	−0.693022
$718$	−2150.13	−0.111758
$719$	−2807.47	−0.145620	−0.0728101	−	0.997346i	$-0.523197\pi$
$719$	−2807.47	−0.145620	−0.0728101	+	0.997346i	$0.523197\pi$
$720$	0	0
$721$	1779.54	0.0919190
$722$	−13439.6	−0.692757
$723$	−13727.4	−0.706125
$724$	7221.97	0.370721
$725$	0	0
$726$	4055.09	0.207298
$727$	3059.84	0.156098	0.0780489	−	0.996950i	$-0.475131\pi$
$727$	3059.84	0.156098	0.0780489	+	0.996950i	$0.475131\pi$
$728$	−4994.74	−0.254282
$729$	729.000	0.0370370
$730$	0	0
$731$	6142.25	0.310779
$732$	−4748.60	−0.239772
$733$	483.122	0.0243445	0.0121723	−	0.999926i	$-0.496125\pi$
$733$	483.122	0.0243445	0.0121723	+	0.999926i	$0.496125\pi$
$734$	−734.041	−0.0369127
$735$	0	0
$736$	−5099.95	−0.255417
$737$	−21554.8	−1.07732
$738$	−2598.91	−0.129630
$739$	15972.4	0.795066	0.397533	−	0.917588i	$-0.369866\pi$
$739$	15972.4	0.795066	0.397533	+	0.917588i	$0.369866\pi$
$740$	0	0
$741$	−3156.84	−0.156504
$742$	−8883.06	−0.439498
$743$	1310.03	0.0646840	0.0323420	−	0.999477i	$-0.489703\pi$
$743$	1310.03	0.0646840	0.0323420	+	0.999477i	$0.489703\pi$
$744$	3033.70	0.149490
$745$	0	0
$746$	2609.11	0.128052
$747$	−6779.28	−0.332049
$748$	1923.56	0.0940271
$749$	−7998.24	−0.390186
$750$	0	0
$751$	−14752.9	−0.716830	−0.358415	−	0.933562i	$-0.616683\pi$
$751$	−14752.9	−0.716830	−0.358415	+	0.933562i	$0.616683\pi$
$752$	−1878.96	−0.0911150
$753$	−4940.87	−0.239117
$754$	−3531.63	−0.170576
$755$	0	0
$756$	−756.000	−0.0363696
$757$	−3934.60	−0.188911	−0.0944555	−	0.995529i	$-0.530111\pi$
$757$	−3934.60	−0.188911	−0.0944555	+	0.995529i	$0.530111\pi$
$758$	10681.2	0.511820
$759$	12237.9	0.585255
$760$	0	0
$761$	17102.3	0.814662	0.407331	−	0.913281i	$-0.366460\pi$
$761$	17102.3	0.814662	0.407331	+	0.913281i	$0.366460\pi$
$762$	13825.0	0.657251
$763$	−14380.0	−0.682293
$764$	13436.6	0.636282
$765$	0	0
$766$	−13324.5	−0.628505
$767$	45940.0	2.16271
$768$	−768.000	−0.0360844
$769$	−32173.5	−1.50872	−0.754361	−	0.656460i	$-0.772053\pi$
$769$	−32173.5	−1.50872	−0.754361	+	0.656460i	$0.772053\pi$
$770$	0	0
$771$	8434.91	0.394002
$772$	11202.3	0.522251
$773$	19329.5	0.899398	0.449699	−	0.893180i	$-0.351531\pi$
$773$	19329.5	0.899398	0.449699	+	0.893180i	$0.351531\pi$
$774$	5884.71	0.273284
$775$	0	0
$776$	−9615.35	−0.444808
$777$	−5942.36	−0.274364
$778$	−12627.0	−0.581875
$779$	1703.43	0.0783464
$780$	0	0
$781$	10292.1	0.471551
$782$	−5988.54	−0.273849
$783$	−534.545	−0.0243973
$784$	784.000	0.0357143
$785$	0	0
$786$	−5731.15	−0.260081
$787$	−4050.20	−0.183449	−0.0917244	−	0.995784i	$-0.529238\pi$
$787$	−4050.20	−0.183449	−0.0917244	+	0.995784i	$0.529238\pi$
$788$	−13010.1	−0.588154
$789$	−9309.39	−0.420054
$790$	0	0
$791$	6235.79	0.280302
$792$	1842.91	0.0826828
$793$	−35294.7	−1.58052
$794$	28421.2	1.27031
$795$	0	0
$796$	−6101.22	−0.271674
$797$	−30864.7	−1.37175	−0.685874	−	0.727720i	$-0.740580\pi$
$797$	−30864.7	−1.37175	−0.685874	+	0.727720i	$0.740580\pi$
$798$	495.514	0.0219812
$799$	−2206.33	−0.0976902
$800$	0	0
$801$	−7586.71	−0.334661
$802$	11157.2	0.491239
$803$	−7759.95	−0.341025
$804$	10105.4	0.443273
$805$	0	0
$806$	22548.4	0.985402
$807$	183.563	0.00800710
$808$	4602.97	0.200411
$809$	14213.1	0.617682	0.308841	−	0.951114i	$-0.400059\pi$
$809$	14213.1	0.617682	0.308841	+	0.951114i	$0.400059\pi$
$810$	0	0
$811$	1619.57	0.0701242	0.0350621	−	0.999385i	$-0.488837\pi$
$811$	1619.57	0.0701242	0.0350621	+	0.999385i	$0.488837\pi$
$812$	554.343	0.0239577
$813$	−292.885	−0.0126346
$814$	14485.7	0.623740
$815$	0	0
$816$	−901.812	−0.0386884
$817$	−3857.09	−0.165168
$818$	23828.0	1.01849
$819$	−5619.09	−0.239740
$820$	0	0
$821$	−20628.1	−0.876888	−0.438444	−	0.898758i	$-0.644470\pi$
$821$	−20628.1	−0.876888	−0.438444	+	0.898758i	$0.644470\pi$
$822$	−8937.20	−0.379222
$823$	11868.2	0.502671	0.251335	−	0.967900i	$-0.419130\pi$
$823$	11868.2	0.502671	0.251335	+	0.967900i	$0.419130\pi$
$824$	2033.76	0.0859824
$825$	0	0
$826$	−7210.97	−0.303755
$827$	−9149.35	−0.384709	−0.192354	−	0.981326i	$-0.561612\pi$
$827$	−9149.35	−0.384709	−0.192354	+	0.981326i	$0.561612\pi$
$828$	−5737.44	−0.240809
$829$	−11693.7	−0.489916	−0.244958	−	0.969534i	$-0.578774\pi$
$829$	−11693.7	−0.489916	−0.244958	+	0.969534i	$0.578774\pi$
$830$	0	0
$831$	14157.7	0.591006
$832$	−5708.28	−0.237859
$833$	920.600	0.0382916
$834$	4513.70	0.187406
$835$	0	0
$836$	−1207.92	−0.0499722
$837$	3412.91	0.140941
$838$	−24186.0	−0.997007
$839$	−41472.3	−1.70654	−0.853268	−	0.521473i	$-0.825383\pi$
$839$	−41472.3	−1.70654	−0.853268	+	0.521473i	$0.825383\pi$
$840$	0	0
$841$	−23997.0	−0.983929
$842$	−24377.0	−0.997727
$843$	−2985.28	−0.121967
$844$	−11737.2	−0.478685
$845$	0	0
$846$	−2113.82	−0.0859040
$847$	−4730.94	−0.191921
$848$	−10152.1	−0.411112
$849$	−23986.8	−0.969639
$850$	0	0
$851$	−45097.8	−1.81661
$852$	−4825.20	−0.194024
$853$	10326.4	0.414501	0.207251	−	0.978288i	$-0.433548\pi$
$853$	10326.4	0.414501	0.207251	+	0.978288i	$0.433548\pi$
$854$	5540.03	0.221986
$855$	0	0
$856$	−9140.85	−0.364986
$857$	−17375.4	−0.692571	−0.346285	−	0.938129i	$-0.612557\pi$
$857$	−17375.4	−0.692571	−0.346285	+	0.938129i	$0.612557\pi$
$858$	13697.7	0.545025
$859$	10624.9	0.422022	0.211011	−	0.977484i	$-0.432324\pi$
$859$	10624.9	0.422022	0.211011	+	0.977484i	$0.432324\pi$
$860$	0	0
$861$	3032.06	0.120014
$862$	9936.31	0.392613
$863$	22377.1	0.882648	0.441324	−	0.897348i	$-0.354509\pi$
$863$	22377.1	0.882648	0.441324	+	0.897348i	$0.354509\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	−28967.4	−1.13667
$867$	13680.1	0.535870
$868$	−3539.31	−0.138401
$869$	6818.33	0.266164
$870$	0	0
$871$	75110.3	2.92195
$872$	−16434.3	−0.638227
$873$	−10817.3	−0.419369
$874$	3760.56	0.145541
$875$	0	0
$876$	3638.06	0.140318
$877$	−22267.1	−0.857362	−0.428681	−	0.903456i	$-0.641021\pi$
$877$	−22267.1	−0.857362	−0.428681	+	0.903456i	$0.641021\pi$
$878$	−19875.3	−0.763961
$879$	−26792.4	−1.02808
$880$	0	0
$881$	−7273.13	−0.278136	−0.139068	−	0.990283i	$-0.544411\pi$
$881$	−7273.13	−0.278136	−0.139068	+	0.990283i	$0.544411\pi$
$882$	882.000	0.0336718
$883$	−43452.0	−1.65603	−0.828016	−	0.560705i	$-0.810530\pi$
$883$	−43452.0	−1.65603	−0.828016	+	0.560705i	$0.810530\pi$
$884$	−6702.86	−0.255024
$885$	0	0
$886$	34279.2	1.29981
$887$	34206.2	1.29485	0.647424	−	0.762130i	$-0.275846\pi$
$887$	34206.2	1.29485	0.647424	+	0.762130i	$0.275846\pi$
$888$	−6791.27	−0.256644
$889$	−16129.1	−0.608496
$890$	0	0
$891$	2073.27	0.0779541
$892$	14505.2	0.544473
$893$	1385.49	0.0519190
$894$	−10254.1	−0.383610
$895$	0	0
$896$	896.000	0.0334077
$897$	−42644.4	−1.58735
$898$	3110.69	0.115596
$899$	−2502.54	−0.0928415
$900$	0	0
$901$	−11920.9	−0.440780
$902$	−7391.27	−0.272841
$903$	−6865.50	−0.253012
$904$	7126.61	0.262199
$905$	0	0
$906$	16641.7	0.610246
$907$	−24922.5	−0.912391	−0.456196	−	0.889880i	$-0.650788\pi$
$907$	−24922.5	−0.912391	−0.456196	+	0.889880i	$0.650788\pi$
$908$	8757.80	0.320086
$909$	5178.34	0.188949
$910$	0	0
$911$	29200.7	1.06198	0.530989	−	0.847379i	$-0.321821\pi$
$911$	29200.7	1.06198	0.530989	+	0.847379i	$0.321821\pi$
$912$	566.302	0.0205616
$913$	−19280.2	−0.698885
$914$	36818.4	1.33244
$915$	0	0
$916$	−22004.0	−0.793704
$917$	6686.34	0.240788
$918$	−1014.54	−0.0364758
$919$	−31961.7	−1.14725	−0.573623	−	0.819119i	$-0.694463\pi$
$919$	−31961.7	−1.14725	−0.573623	+	0.819119i	$0.694463\pi$
$920$	0	0
$921$	7547.38	0.270027
$922$	13233.0	0.472676
$923$	−35864.0	−1.27896
$924$	−2150.06	−0.0765494
$925$	0	0
$926$	−25211.0	−0.894693
$927$	2287.98	0.0810650
$928$	633.535	0.0224103
$929$	−21349.4	−0.753985	−0.376993	−	0.926216i	$-0.623042\pi$
$929$	−21349.4	−0.753985	−0.376993	+	0.926216i	$0.623042\pi$
$930$	0	0
$931$	−578.100	−0.0203507
$932$	−15710.9	−0.552174
$933$	−11278.2	−0.395746
$934$	32583.1	1.14149
$935$	0	0
$936$	−6421.81	−0.224256
$937$	3802.95	0.132590	0.0662951	−	0.997800i	$-0.478882\pi$
$937$	3802.95	0.132590	0.0662951	+	0.997800i	$0.478882\pi$
$938$	−11789.7	−0.410391
$939$	20477.7	0.711678
$940$	0	0
$941$	−42690.8	−1.47894	−0.739470	−	0.673190i	$-0.764924\pi$
$941$	−42690.8	−1.47894	−0.739470	+	0.673190i	$0.764924\pi$
$942$	2120.87	0.0733562
$943$	23010.9	0.794633
$944$	−8241.11	−0.284137
$945$	0	0
$946$	16736.1	0.575197
$947$	−40333.0	−1.38400	−0.691998	−	0.721899i	$-0.743270\pi$
$947$	−40333.0	−1.38400	−0.691998	+	0.721899i	$0.743270\pi$
$948$	−3196.60	−0.109516
$949$	27040.4	0.924941
$950$	0	0
$951$	−7394.81	−0.252148
$952$	1052.11	0.0358185
$953$	39603.5	1.34615	0.673077	−	0.739573i	$-0.264972\pi$
$953$	39603.5	1.34615	0.673077	+	0.739573i	$0.264972\pi$
$954$	−11421.1	−0.387601
$955$	0	0
$956$	17740.4	0.600174
$957$	−1520.24	−0.0513505
$958$	23012.6	0.776099
$959$	10426.7	0.351091
$960$	0	0
$961$	−13813.0	−0.463664
$962$	−50477.1	−1.69173
$963$	−10283.5	−0.344112
$964$	18303.2	0.611522
$965$	0	0
$966$	6693.69	0.222946
$967$	−4139.69	−0.137666	−0.0688332	−	0.997628i	$-0.521928\pi$
$967$	−4139.69	−0.137666	−0.0688332	+	0.997628i	$0.521928\pi$
$968$	−5406.79	−0.179526
$969$	664.971	0.0220454
$970$	0	0
$971$	42137.2	1.39263	0.696316	−	0.717735i	$-0.254821\pi$
$971$	42137.2	1.39263	0.696316	+	0.717735i	$0.254821\pi$
$972$	−972.000	−0.0320750
$973$	−5265.99	−0.173504
$974$	33005.8	1.08580
$975$	0	0
$976$	6331.46	0.207649
$977$	−39782.4	−1.30271	−0.651357	−	0.758772i	$-0.725800\pi$
$977$	−39782.4	−1.30271	−0.651357	+	0.758772i	$0.725800\pi$
$978$	5311.38	0.173660
$979$	−21576.5	−0.704381
$980$	0	0
$981$	−18488.5	−0.601726
$982$	18539.3	0.602458
$983$	3799.91	0.123294	0.0616471	−	0.998098i	$-0.480365\pi$
$983$	3799.91	0.123294	0.0616471	+	0.998098i	$0.480365\pi$
$984$	3465.21	0.112263
$985$	0	0
$986$	743.918	0.0240276
$987$	2466.13	0.0795317
$988$	4209.13	0.135537
$989$	−52103.7	−1.67523
$990$	0	0
$991$	20113.7	0.644735	0.322367	−	0.946615i	$-0.395521\pi$
$991$	20113.7	0.644735	0.322367	+	0.946615i	$0.395521\pi$
$992$	−4044.93	−0.129462
$993$	−11341.1	−0.362435
$994$	5629.40	0.179632
$995$	0	0
$996$	9039.04	0.287563
$997$	40814.2	1.29649	0.648244	−	0.761432i	$-0.275504\pi$
$997$	40814.2	1.29649	0.648244	+	0.761432i	$0.275504\pi$
$998$	26334.6	0.835279
$999$	−7640.17	−0.241966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.bg.1.2		2
5.2	odd	4		210.4.g.a.169.4	yes	4
5.3	odd	4		210.4.g.a.169.2	✓	4
5.4	even	2		1050.4.a.bc.1.2		2
15.2	even	4		630.4.g.e.379.1		4
15.8	even	4		630.4.g.e.379.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.g.a.169.2	✓	4	5.3	odd	4
210.4.g.a.169.4	yes	4	5.2	odd	4
630.4.g.e.379.1		4	15.2	even	4
630.4.g.e.379.3		4	15.8	even	4
1050.4.a.bc.1.2		2	5.4	even	2
1050.4.a.bg.1.2		2	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +25.5959 q^{11} -12.0000 q^{12} -89.1918 q^{13} +14.0000 q^{14} +16.0000 q^{16} +18.7878 q^{17} +18.0000 q^{18} -11.7980 q^{19} -21.0000 q^{21} +51.1918 q^{22} -159.373 q^{23} -24.0000 q^{24} -178.384 q^{26} -27.0000 q^{27} +28.0000 q^{28} +19.7980 q^{29} -126.404 q^{31} +32.0000 q^{32} -76.7878 q^{33} +37.5755 q^{34} +36.0000 q^{36} +282.969 q^{37} -23.5959 q^{38} +267.576 q^{39} -144.384 q^{41} -42.0000 q^{42} +326.929 q^{43} +102.384 q^{44} -318.747 q^{46} -117.435 q^{47} -48.0000 q^{48} +49.0000 q^{49} -56.3633 q^{51} -356.767 q^{52} -634.504 q^{53} -54.0000 q^{54} +56.0000 q^{56} +35.3939 q^{57} +39.5959 q^{58} -515.069 q^{59} +395.716 q^{61} -252.808 q^{62} +63.0000 q^{63} +64.0000 q^{64} -153.576 q^{66} -842.120 q^{67} +75.1510 q^{68} +478.120 q^{69} +402.100 q^{71} +72.0000 q^{72} -303.171 q^{73} +565.939 q^{74} -47.1918 q^{76} +179.171 q^{77} +535.151 q^{78} +266.384 q^{79} +81.0000 q^{81} -288.767 q^{82} -753.253 q^{83} -84.0000 q^{84} +653.857 q^{86} -59.3939 q^{87} +204.767 q^{88} -842.967 q^{89} -624.343 q^{91} -637.494 q^{92} +379.212 q^{93} -234.869 q^{94} -96.0000 q^{96} -1201.92 q^{97} +98.0000 q^{98} +230.363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 12 q^{6} + 14 q^{7} + 16 q^{8} + 18 q^{9} + 12 q^{11} - 24 q^{12} - 100 q^{13} + 28 q^{14} + 32 q^{16} - 80 q^{17} + 36 q^{18} - 4 q^{19} - 42 q^{21} + 24 q^{22} - 64 q^{23}+ \cdots + 108 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 12 * q^6 + 14 * q^7 + 16 * q^8 + 18 * q^9 + 12 * q^11 - 24 * q^12 - 100 * q^13 + 28 * q^14 + 32 * q^16 - 80 * q^17 + 36 * q^18 - 4 * q^19 - 42 * q^21 + 24 * q^22 - 64 * q^23 - 48 * q^24 - 200 * q^26 - 54 * q^27 + 56 * q^28 + 20 * q^29 - 292 * q^31 + 64 * q^32 - 36 * q^33 - 160 * q^34 + 72 * q^36 + 272 * q^37 - 8 * q^38 + 300 * q^39 - 132 * q^41 - 84 * q^42 - 32 * q^43 + 48 * q^44 - 128 * q^46 - 568 * q^47 - 96 * q^48 + 98 * q^49 + 240 * q^51 - 400 * q^52 - 348 * q^53 - 108 * q^54 + 112 * q^56 + 12 * q^57 + 40 * q^58 + 224 * q^59 - 12 * q^61 - 584 * q^62 + 126 * q^63 + 128 * q^64 - 72 * q^66 - 920 * q^67 - 320 * q^68 + 192 * q^69 - 156 * q^71 + 144 * q^72 - 332 * q^73 + 544 * q^74 - 16 * q^76 + 84 * q^77 + 600 * q^78 + 376 * q^79 + 162 * q^81 - 264 * q^82 - 2016 * q^83 - 168 * q^84 - 64 * q^86 - 60 * q^87 + 96 * q^88 + 548 * q^89 - 700 * q^91 - 256 * q^92 + 876 * q^93 - 1136 * q^94 - 192 * q^96 - 1620 * q^97 + 196 * q^98 + 108 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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