LMFDB - Embedded newform 2450.4.a.dc.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 214x^{8} + 15801x^{6} - 479776x^{4} + 5017216x^{2} - 1411200 $ x^10 - 214x^8 + 15801x^6 - 479776x^4 + 5017216x^2 - 1411200
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$7.11026$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +7.11026 q^{3} +4.00000 q^{4} +14.2205 q^{6} +8.00000 q^{8} +23.5558 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +7.11026 q^{3} +4.00000 q^{4} +14.2205 q^{6} +8.00000 q^{8} +23.5558 q^{9} -65.2884 q^{11} +28.4410 q^{12} +55.0095 q^{13} +16.0000 q^{16} +116.759 q^{17} +47.1115 q^{18} +98.8324 q^{19} -130.577 q^{22} -87.1142 q^{23} +56.8821 q^{24} +110.019 q^{26} -24.4894 q^{27} -47.4747 q^{29} +165.355 q^{31} +32.0000 q^{32} -464.217 q^{33} +233.518 q^{34} +94.2230 q^{36} +107.127 q^{37} +197.665 q^{38} +391.131 q^{39} +280.281 q^{41} +236.981 q^{43} -261.154 q^{44} -174.228 q^{46} +85.8061 q^{47} +113.764 q^{48} +830.185 q^{51} +220.038 q^{52} -473.771 q^{53} -48.9789 q^{54} +702.724 q^{57} -94.9494 q^{58} +548.294 q^{59} -693.644 q^{61} +330.710 q^{62} +64.0000 q^{64} -928.435 q^{66} +310.152 q^{67} +467.035 q^{68} -619.404 q^{69} -331.372 q^{71} +188.446 q^{72} -277.139 q^{73} +214.253 q^{74} +395.330 q^{76} +782.263 q^{78} +645.756 q^{79} -810.132 q^{81} +560.563 q^{82} -315.047 q^{83} +473.963 q^{86} -337.557 q^{87} -522.307 q^{88} +829.630 q^{89} -348.457 q^{92} +1175.72 q^{93} +171.612 q^{94} +227.528 q^{96} +63.7075 q^{97} -1537.92 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 20 q^{2} + 40 q^{4} + 80 q^{8} + 158 q^{9}+O(q^{10}) $ 10 * q + 20 * q^2 + 40 * q^4 + 80 * q^8 + 158 * q^9	$ 10 q + 20 q^{2} + 40 q^{4} + 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} + 316 q^{18} + 104 q^{22} + 400 q^{23} + 108 q^{29} + 320 q^{32} + 632 q^{36} + 1492 q^{37} + 252 q^{39} + 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} + 968 q^{53} + 3024 q^{57} + 216 q^{58} + 640 q^{64} + 1880 q^{67} - 936 q^{71} + 1264 q^{72} + 2984 q^{74} + 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} + 416 q^{88} + 1600 q^{92} + 304 q^{93} - 312 q^{99}+O(q^{100}) $ 10 * q + 20 * q^2 + 40 * q^4 + 80 * q^8 + 158 * q^9 + 52 * q^11 + 160 * q^16 + 316 * q^18 + 104 * q^22 + 400 * q^23 + 108 * q^29 + 320 * q^32 + 632 * q^36 + 1492 * q^37 + 252 * q^39 + 904 * q^43 + 208 * q^44 + 800 * q^46 - 148 * q^51 + 968 * q^53 + 3024 * q^57 + 216 * q^58 + 640 * q^64 + 1880 * q^67 - 936 * q^71 + 1264 * q^72 + 2984 * q^74 + 504 * q^78 + 3212 * q^79 + 1010 * q^81 + 1808 * q^86 + 416 * q^88 + 1600 * q^92 + 304 * q^93 - 312 * q^99

Display 100 coefficients 10 coefficients

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
$3$	7.11026	1.36837	0.684185	−	0.729309i	$-0.260158\pi$
$3$	7.11026	1.36837	0.684185	+	0.729309i	$0.260158\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	14.2205	0.967583
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	23.5558	0.872436
$10$	0	0
$11$	−65.2884	−1.78956	−0.894781	−	0.446504i	$-0.852669\pi$
$11$	−65.2884	−1.78956	−0.894781	+	0.446504i	$0.852669\pi$
$12$	28.4410	0.684185
$13$	55.0095	1.17361	0.586803	−	0.809730i	$-0.300386\pi$
$13$	55.0095	1.17361	0.586803	+	0.809730i	$0.300386\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	116.759	1.66578	0.832888	−	0.553442i	$-0.186686\pi$
$17$	116.759	1.66578	0.832888	+	0.553442i	$0.186686\pi$
$18$	47.1115	0.616905
$19$	98.8324	1.19335	0.596677	−	0.802482i	$-0.296487\pi$
$19$	98.8324	1.19335	0.596677	+	0.802482i	$0.296487\pi$
$20$	0	0
$21$	0	0
$22$	−130.577	−1.26541
$23$	−87.1142	−0.789764	−0.394882	−	0.918732i	$-0.629214\pi$
$23$	−87.1142	−0.789764	−0.394882	+	0.918732i	$0.629214\pi$
$24$	56.8821	0.483792
$25$	0	0
$26$	110.019	0.829865
$27$	−24.4894	−0.174555
$28$	0	0
$29$	−47.4747	−0.303994	−0.151997	−	0.988381i	$-0.548570\pi$
$29$	−47.4747	−0.303994	−0.151997	+	0.988381i	$0.548570\pi$
$30$	0	0
$31$	165.355	0.958021	0.479010	−	0.877809i	$-0.340996\pi$
$31$	165.355	0.958021	0.479010	+	0.877809i	$0.340996\pi$
$32$	32.0000	0.176777
$33$	−464.217	−2.44878
$34$	233.518	1.17788
$35$	0	0
$36$	94.2230	0.436218
$37$	107.127	0.475986	0.237993	−	0.971267i	$-0.423510\pi$
$37$	107.127	0.475986	0.237993	+	0.971267i	$0.423510\pi$
$38$	197.665	0.843828
$39$	391.131	1.60593
$40$	0	0
$41$	280.281	1.06762	0.533812	−	0.845603i	$-0.320759\pi$
$41$	280.281	1.06762	0.533812	+	0.845603i	$0.320759\pi$
$42$	0	0
$43$	236.981	0.840450	0.420225	−	0.907420i	$-0.361951\pi$
$43$	236.981	0.840450	0.420225	+	0.907420i	$0.361951\pi$
$44$	−261.154	−0.894781
$45$	0	0
$46$	−174.228	−0.558447
$47$	85.8061	0.266300	0.133150	−	0.991096i	$-0.457491\pi$
$47$	85.8061	0.266300	0.133150	+	0.991096i	$0.457491\pi$
$48$	113.764	0.342092
$49$	0	0
$50$	0	0
$51$	830.185	2.27940
$52$	220.038	0.586803
$53$	−473.771	−1.22788	−0.613939	−	0.789354i	$-0.710416\pi$
$53$	−473.771	−1.22788	−0.613939	+	0.789354i	$0.710416\pi$
$54$	−48.9789	−0.123429
$55$	0	0
$56$	0	0
$57$	702.724	1.63295
$58$	−94.9494	−0.214956
$59$	548.294	1.20986	0.604930	−	0.796279i	$-0.293201\pi$
$59$	548.294	1.20986	0.604930	+	0.796279i	$0.293201\pi$
$60$	0	0
$61$	−693.644	−1.45594	−0.727968	−	0.685612i	$-0.759535\pi$
$61$	−693.644	−1.45594	−0.727968	+	0.685612i	$0.759535\pi$
$62$	330.710	0.677423
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−928.435	−1.73155
$67$	310.152	0.565539	0.282770	−	0.959188i	$-0.408747\pi$
$67$	310.152	0.565539	0.282770	+	0.959188i	$0.408747\pi$
$68$	467.035	0.832888
$69$	−619.404	−1.08069
$70$	0	0
$71$	−331.372	−0.553896	−0.276948	−	0.960885i	$-0.589323\pi$
$71$	−331.372	−0.553896	−0.276948	+	0.960885i	$0.589323\pi$
$72$	188.446	0.308453
$73$	−277.139	−0.444338	−0.222169	−	0.975008i	$-0.571314\pi$
$73$	−277.139	−0.444338	−0.222169	+	0.975008i	$0.571314\pi$
$74$	214.253	0.336573
$75$	0	0
$76$	395.330	0.596677
$77$	0	0
$78$	782.263	1.13556
$79$	645.756	0.919661	0.459830	−	0.888007i	$-0.347910\pi$
$79$	645.756	0.919661	0.459830	+	0.888007i	$0.347910\pi$
$80$	0	0
$81$	−810.132	−1.11129
$82$	560.563	0.754924
$83$	−315.047	−0.416637	−0.208319	−	0.978061i	$-0.566799\pi$
$83$	−315.047	−0.416637	−0.208319	+	0.978061i	$0.566799\pi$
$84$	0	0
$85$	0	0
$86$	473.963	0.594288
$87$	−337.557	−0.415976
$88$	−522.307	−0.632706
$89$	829.630	0.988097	0.494049	−	0.869434i	$-0.335516\pi$
$89$	829.630	0.988097	0.494049	+	0.869434i	$0.335516\pi$
$90$	0	0
$91$	0	0
$92$	−348.457	−0.394882
$93$	1175.72	1.31093
$94$	171.612	0.188303
$95$	0	0
$96$	227.528	0.241896
$97$	63.7075	0.0666858	0.0333429	−	0.999444i	$-0.489385\pi$
$97$	63.7075	0.0666858	0.0333429	+	0.999444i	$0.489385\pi$
$98$	0	0
$99$	−1537.92	−1.56128
$100$	0	0
$101$	857.505	0.844801	0.422400	−	0.906409i	$-0.361188\pi$
$101$	857.505	0.844801	0.422400	+	0.906409i	$0.361188\pi$
$102$	1660.37	1.61178
$103$	1042.61	0.997394	0.498697	−	0.866776i	$-0.333812\pi$
$103$	1042.61	0.997394	0.498697	+	0.866776i	$0.333812\pi$
$104$	440.076	0.414932
$105$	0	0
$106$	−947.543	−0.868241
$107$	−1143.29	−1.03295	−0.516476	−	0.856302i	$-0.672756\pi$
$107$	−1143.29	−1.03295	−0.516476	+	0.856302i	$0.672756\pi$
$108$	−97.9577	−0.0872776
$109$	−151.498	−0.133128	−0.0665638	−	0.997782i	$-0.521204\pi$
$109$	−151.498	−0.133128	−0.0665638	+	0.997782i	$0.521204\pi$
$110$	0	0
$111$	761.697	0.651325
$112$	0	0
$113$	1135.03	0.944912	0.472456	−	0.881354i	$-0.343368\pi$
$113$	1135.03	0.944912	0.472456	+	0.881354i	$0.343368\pi$
$114$	1405.45	1.15467
$115$	0	0
$116$	−189.899	−0.151997
$117$	1295.79	1.02390
$118$	1096.59	0.855500
$119$	0	0
$120$	0	0
$121$	2931.57	2.20254
$122$	−1387.29	−1.02950
$123$	1992.87	1.46090
$124$	661.420	0.479010
$125$	0	0
$126$	0	0
$127$	1638.75	1.14500	0.572501	−	0.819904i	$-0.305973\pi$
$127$	1638.75	1.14500	0.572501	+	0.819904i	$0.305973\pi$
$128$	128.000	0.0883883
$129$	1685.00	1.15005
$130$	0	0
$131$	2320.41	1.54760	0.773799	−	0.633432i	$-0.218354\pi$
$131$	2320.41	1.54760	0.773799	+	0.633432i	$0.218354\pi$
$132$	−1856.87	−1.22439
$133$	0	0
$134$	620.304	0.399896
$135$	0	0
$136$	934.071	0.588941
$137$	−2319.27	−1.44634	−0.723171	−	0.690669i	$-0.757316\pi$
$137$	−2319.27	−1.44634	−0.723171	+	0.690669i	$0.757316\pi$
$138$	−1238.81	−0.764162
$139$	2549.97	1.55601	0.778007	−	0.628256i	$-0.216231\pi$
$139$	2549.97	1.55601	0.778007	+	0.628256i	$0.216231\pi$
$140$	0	0
$141$	610.104	0.364397
$142$	−662.744	−0.391664
$143$	−3591.48	−2.10024
$144$	376.892	0.218109
$145$	0	0
$146$	−554.279	−0.314195
$147$	0	0
$148$	428.506	0.237993
$149$	3286.67	1.80708	0.903538	−	0.428508i	$-0.140961\pi$
$149$	3286.67	1.80708	0.903538	+	0.428508i	$0.140961\pi$
$150$	0	0
$151$	−2501.03	−1.34789	−0.673944	−	0.738782i	$-0.735401\pi$
$151$	−2501.03	−1.34789	−0.673944	+	0.738782i	$0.735401\pi$
$152$	790.660	0.421914
$153$	2750.34	1.45328
$154$	0	0
$155$	0	0
$156$	1564.53	0.802964
$157$	450.666	0.229090	0.114545	−	0.993418i	$-0.463459\pi$
$157$	450.666	0.229090	0.114545	+	0.993418i	$0.463459\pi$
$158$	1291.51	0.650298
$159$	−3368.64	−1.68019
$160$	0	0
$161$	0	0
$162$	−1620.26	−0.785802
$163$	939.794	0.451597	0.225799	−	0.974174i	$-0.427501\pi$
$163$	939.794	0.451597	0.225799	+	0.974174i	$0.427501\pi$
$164$	1121.13	0.533812
$165$	0	0
$166$	−630.094	−0.294607
$167$	1783.15	0.826252	0.413126	−	0.910674i	$-0.364437\pi$
$167$	1783.15	0.826252	0.413126	+	0.910674i	$0.364437\pi$
$168$	0	0
$169$	829.042	0.377352
$170$	0	0
$171$	2328.07	1.04112
$172$	947.926	0.420225
$173$	−3267.58	−1.43601	−0.718004	−	0.696039i	$-0.754944\pi$
$173$	−3267.58	−1.43601	−0.718004	+	0.696039i	$0.754944\pi$
$174$	−675.115	−0.294140
$175$	0	0
$176$	−1044.61	−0.447391
$177$	3898.51	1.65554
$178$	1659.26	0.698690
$179$	1604.72	0.670068	0.335034	−	0.942206i	$-0.391252\pi$
$179$	1604.72	0.670068	0.335034	+	0.942206i	$0.391252\pi$
$180$	0	0
$181$	3709.52	1.52335	0.761676	−	0.647959i	$-0.224377\pi$
$181$	3709.52	1.52335	0.761676	+	0.647959i	$0.224377\pi$
$182$	0	0
$183$	−4931.99	−1.99226
$184$	−696.914	−0.279224
$185$	0	0
$186$	2351.43	0.926965
$187$	−7623.00	−2.98101
$188$	343.225	0.133150
$189$	0	0
$190$	0	0
$191$	−3869.37	−1.46585	−0.732926	−	0.680308i	$-0.761846\pi$
$191$	−3869.37	−1.46585	−0.732926	+	0.680308i	$0.761846\pi$
$192$	455.056	0.171046
$193$	4309.95	1.60744	0.803722	−	0.595005i	$-0.202850\pi$
$193$	4309.95	1.60744	0.803722	+	0.595005i	$0.202850\pi$
$194$	127.415	0.0471540
$195$	0	0
$196$	0	0
$197$	−707.884	−0.256013	−0.128007	−	0.991773i	$-0.540858\pi$
$197$	−707.884	−0.256013	−0.128007	+	0.991773i	$0.540858\pi$
$198$	−3075.84	−1.10399
$199$	−36.9243	−0.0131532	−0.00657662	−	0.999978i	$-0.502093\pi$
$199$	−36.9243	−0.0131532	−0.00657662	+	0.999978i	$0.502093\pi$
$200$	0	0
$201$	2205.26	0.773866
$202$	1715.01	0.597364
$203$	0	0
$204$	3320.74	1.13970
$205$	0	0
$206$	2085.22	0.705264
$207$	−2052.04	−0.689018
$208$	880.152	0.293402
$209$	−6452.61	−2.13558
$210$	0	0
$211$	−4656.12	−1.51915	−0.759575	−	0.650420i	$-0.774593\pi$
$211$	−4656.12	−1.51915	−0.759575	+	0.650420i	$0.774593\pi$
$212$	−1895.09	−0.613939
$213$	−2356.14	−0.757935
$214$	−2286.58	−0.730407
$215$	0	0
$216$	−195.915	−0.0617146
$217$	0	0
$218$	−302.997	−0.0941354
$219$	−1970.53	−0.608019
$220$	0	0
$221$	6422.84	1.95496
$222$	1523.39	0.460556
$223$	−45.2935	−0.0136012	−0.00680062	−	0.999977i	$-0.502165\pi$
$223$	−45.2935	−0.0136012	−0.00680062	+	0.999977i	$0.502165\pi$
$224$	0	0
$225$	0	0
$226$	2270.07	0.668153
$227$	−5323.46	−1.55652	−0.778260	−	0.627942i	$-0.783898\pi$
$227$	−5323.46	−1.55652	−0.778260	+	0.627942i	$0.783898\pi$
$228$	2810.90	0.816474
$229$	−1909.39	−0.550987	−0.275493	−	0.961303i	$-0.588841\pi$
$229$	−1909.39	−0.550987	−0.275493	+	0.961303i	$0.588841\pi$
$230$	0	0
$231$	0	0
$232$	−379.798	−0.107478
$233$	−1505.25	−0.423228	−0.211614	−	0.977353i	$-0.567872\pi$
$233$	−1505.25	−0.423228	−0.211614	+	0.977353i	$0.567872\pi$
$234$	2591.58	0.724004
$235$	0	0
$236$	2193.17	0.604930
$237$	4591.49	1.25844
$238$	0	0
$239$	−1353.19	−0.366238	−0.183119	−	0.983091i	$-0.558619\pi$
$239$	−1353.19	−0.366238	−0.183119	+	0.983091i	$0.558619\pi$
$240$	0	0
$241$	−1230.08	−0.328781	−0.164390	−	0.986395i	$-0.552566\pi$
$241$	−1230.08	−0.328781	−0.164390	+	0.986395i	$0.552566\pi$
$242$	5863.15	1.55743
$243$	−5099.03	−1.34610
$244$	−2774.58	−0.727968
$245$	0	0
$246$	3985.74	1.03302
$247$	5436.72	1.40053
$248$	1322.84	0.338711
$249$	−2240.06	−0.570114
$250$	0	0
$251$	1289.55	0.324286	0.162143	−	0.986767i	$-0.448159\pi$
$251$	1289.55	0.324286	0.162143	+	0.986767i	$0.448159\pi$
$252$	0	0
$253$	5687.55	1.41333
$254$	3277.49	0.809639
$255$	0	0
$256$	256.000	0.0625000
$257$	−3323.54	−0.806679	−0.403339	−	0.915050i	$-0.632151\pi$
$257$	−3323.54	−0.806679	−0.403339	+	0.915050i	$0.632151\pi$
$258$	3370.00	0.813205
$259$	0	0
$260$	0	0
$261$	−1118.30	−0.265215
$262$	4640.82	1.09432
$263$	1918.96	0.449916	0.224958	−	0.974368i	$-0.427776\pi$
$263$	1918.96	0.449916	0.224958	+	0.974368i	$0.427776\pi$
$264$	−3713.74	−0.865776
$265$	0	0
$266$	0	0
$267$	5898.89	1.35208
$268$	1240.61	0.282770
$269$	−5928.46	−1.34373	−0.671867	−	0.740672i	$-0.734507\pi$
$269$	−5928.46	−1.34373	−0.671867	+	0.740672i	$0.734507\pi$
$270$	0	0
$271$	−6970.50	−1.56246	−0.781232	−	0.624240i	$-0.785409\pi$
$271$	−6970.50	−1.56246	−0.781232	+	0.624240i	$0.785409\pi$
$272$	1868.14	0.416444
$273$	0	0
$274$	−4638.55	−1.02272
$275$	0	0
$276$	−2477.62	−0.540344
$277$	−94.1940	−0.0204317	−0.0102158	−	0.999948i	$-0.503252\pi$
$277$	−94.1940	−0.0204317	−0.0102158	+	0.999948i	$0.503252\pi$
$278$	5099.95	1.10027
$279$	3895.06	0.835811
$280$	0	0
$281$	2164.99	0.459618	0.229809	−	0.973236i	$-0.426190\pi$
$281$	2164.99	0.459618	0.229809	+	0.973236i	$0.426190\pi$
$282$	1220.21	0.257668
$283$	629.446	0.132214	0.0661072	−	0.997813i	$-0.478942\pi$
$283$	629.446	0.132214	0.0661072	+	0.997813i	$0.478942\pi$
$284$	−1325.49	−0.276948
$285$	0	0
$286$	−7182.96	−1.48510
$287$	0	0
$288$	753.784	0.154226
$289$	8719.63	1.77481
$290$	0	0
$291$	452.977	0.0912508
$292$	−1108.56	−0.222169
$293$	7714.43	1.53816	0.769081	−	0.639151i	$-0.220714\pi$
$293$	7714.43	1.53816	0.769081	+	0.639151i	$0.220714\pi$
$294$	0	0
$295$	0	0
$296$	857.012	0.168287
$297$	1598.88	0.312378
$298$	6573.33	1.27780
$299$	−4792.11	−0.926872
$300$	0	0
$301$	0	0
$302$	−5002.07	−0.953101
$303$	6097.08	1.15600
$304$	1581.32	0.298338
$305$	0	0
$306$	5500.69	1.02763
$307$	−5840.05	−1.08570	−0.542849	−	0.839830i	$-0.682654\pi$
$307$	−5840.05	−1.08570	−0.542849	+	0.839830i	$0.682654\pi$
$308$	0	0
$309$	7413.24	1.36480
$310$	0	0
$311$	−561.246	−0.102332	−0.0511661	−	0.998690i	$-0.516294\pi$
$311$	−561.246	−0.102332	−0.0511661	+	0.998690i	$0.516294\pi$
$312$	3129.05	0.567781
$313$	−1728.20	−0.312087	−0.156044	−	0.987750i	$-0.549874\pi$
$313$	−1728.20	−0.312087	−0.156044	+	0.987750i	$0.549874\pi$
$314$	901.332	0.161991
$315$	0	0
$316$	2583.02	0.459830
$317$	−10338.4	−1.83174	−0.915868	−	0.401479i	$-0.868496\pi$
$317$	−10338.4	−1.83174	−0.915868	+	0.401479i	$0.868496\pi$
$318$	−6737.27	−1.18807
$319$	3099.55	0.544017
$320$	0	0
$321$	−8129.07	−1.41346
$322$	0	0
$323$	11539.6	1.98786
$324$	−3240.53	−0.555646
$325$	0	0
$326$	1879.59	0.319327
$327$	−1077.19	−0.182168
$328$	2242.25	0.377462
$329$	0	0
$330$	0	0
$331$	−6957.25	−1.15530	−0.577651	−	0.816284i	$-0.696031\pi$
$331$	−6957.25	−1.15530	−0.577651	+	0.816284i	$0.696031\pi$
$332$	−1260.19	−0.208319
$333$	2523.45	0.415267
$334$	3566.30	0.584249
$335$	0	0
$336$	0	0
$337$	4191.83	0.677577	0.338789	−	0.940862i	$-0.389983\pi$
$337$	4191.83	0.677577	0.338789	+	0.940862i	$0.389983\pi$
$338$	1658.08	0.266828
$339$	8070.38	1.29299
$340$	0	0
$341$	−10795.8	−1.71444
$342$	4656.15	0.736186
$343$	0	0
$344$	1895.85	0.297144
$345$	0	0
$346$	−6535.16	−1.01541
$347$	−4547.62	−0.703542	−0.351771	−	0.936086i	$-0.614420\pi$
$347$	−4547.62	−0.703542	−0.351771	+	0.936086i	$0.614420\pi$
$348$	−1350.23	−0.207988
$349$	4773.11	0.732088	0.366044	−	0.930598i	$-0.380712\pi$
$349$	4773.11	0.732088	0.366044	+	0.930598i	$0.380712\pi$
$350$	0	0
$351$	−1347.15	−0.204859
$352$	−2089.23	−0.316353
$353$	−6826.54	−1.02929	−0.514646	−	0.857403i	$-0.672077\pi$
$353$	−6826.54	−1.02929	−0.514646	+	0.857403i	$0.672077\pi$
$354$	7797.02	1.17064
$355$	0	0
$356$	3318.52	0.494049
$357$	0	0
$358$	3209.43	0.473810
$359$	−8768.61	−1.28911	−0.644554	−	0.764559i	$-0.722957\pi$
$359$	−8768.61	−1.28911	−0.644554	+	0.764559i	$0.722957\pi$
$360$	0	0
$361$	2908.85	0.424093
$362$	7419.04	1.07717
$363$	20844.3	3.01388
$364$	0	0
$365$	0	0
$366$	−9863.98	−1.40874
$367$	−6544.18	−0.930800	−0.465400	−	0.885101i	$-0.654089\pi$
$367$	−6544.18	−0.930800	−0.465400	+	0.885101i	$0.654089\pi$
$368$	−1393.83	−0.197441
$369$	6602.24	0.931433
$370$	0	0
$371$	0	0
$372$	4702.87	0.655463
$373$	−1236.66	−0.171668	−0.0858339	−	0.996309i	$-0.527355\pi$
$373$	−1236.66	−0.171668	−0.0858339	+	0.996309i	$0.527355\pi$
$374$	−15246.0	−2.10789
$375$	0	0
$376$	686.449	0.0941514
$377$	−2611.56	−0.356770
$378$	0	0
$379$	−604.196	−0.0818878	−0.0409439	−	0.999161i	$-0.513036\pi$
$379$	−604.196	−0.0818878	−0.0409439	+	0.999161i	$0.513036\pi$
$380$	0	0
$381$	11651.9	1.56679
$382$	−7738.74	−1.03651
$383$	3319.36	0.442850	0.221425	−	0.975177i	$-0.428929\pi$
$383$	3319.36	0.442850	0.221425	+	0.975177i	$0.428929\pi$
$384$	910.113	0.120948
$385$	0	0
$386$	8619.89	1.13663
$387$	5582.28	0.733238
$388$	254.830	0.0333429
$389$	3702.35	0.482562	0.241281	−	0.970455i	$-0.422432\pi$
$389$	3702.35	0.482562	0.241281	+	0.970455i	$0.422432\pi$
$390$	0	0
$391$	−10171.4	−1.31557
$392$	0	0
$393$	16498.7	2.11769
$394$	−1415.77	−0.181029
$395$	0	0
$396$	−6151.67	−0.780639
$397$	−3654.19	−0.461961	−0.230981	−	0.972958i	$-0.574193\pi$
$397$	−3654.19	−0.461961	−0.230981	+	0.972958i	$0.574193\pi$
$398$	−73.8486	−0.00930075
$399$	0	0
$400$	0	0
$401$	−10069.3	−1.25396	−0.626979	−	0.779036i	$-0.715709\pi$
$401$	−10069.3	−1.25396	−0.626979	+	0.779036i	$0.715709\pi$
$402$	4410.52	0.547206
$403$	9096.09	1.12434
$404$	3430.02	0.422400
$405$	0	0
$406$	0	0
$407$	−6994.12	−0.851807
$408$	6641.48	0.805888
$409$	11596.0	1.40192	0.700959	−	0.713202i	$-0.252756\pi$
$409$	11596.0	1.40192	0.700959	+	0.713202i	$0.252756\pi$
$410$	0	0
$411$	−16490.6	−1.97913
$412$	4170.45	0.498697
$413$	0	0
$414$	−4104.08	−0.487209
$415$	0	0
$416$	1760.30	0.207466
$417$	18131.0	2.12920
$418$	−12905.2	−1.51008
$419$	−11167.4	−1.30206	−0.651032	−	0.759050i	$-0.725664\pi$
$419$	−11167.4	−1.30206	−0.651032	+	0.759050i	$0.725664\pi$
$420$	0	0
$421$	3141.15	0.363635	0.181817	−	0.983332i	$-0.441802\pi$
$421$	3141.15	0.363635	0.181817	+	0.983332i	$0.441802\pi$
$422$	−9312.24	−1.07420
$423$	2021.23	0.232330
$424$	−3790.17	−0.434120
$425$	0	0
$426$	−4712.28	−0.535941
$427$	0	0
$428$	−4573.15	−0.516476
$429$	−25536.3	−2.87391
$430$	0	0
$431$	−13433.2	−1.50128	−0.750640	−	0.660711i	$-0.770255\pi$
$431$	−13433.2	−1.50128	−0.750640	+	0.660711i	$0.770255\pi$
$432$	−391.831	−0.0436388
$433$	−1632.96	−0.181236	−0.0906181	−	0.995886i	$-0.528884\pi$
$433$	−1632.96	−0.181236	−0.0906181	+	0.995886i	$0.528884\pi$
$434$	0	0
$435$	0	0
$436$	−605.993	−0.0665638
$437$	−8609.71	−0.942467
$438$	−3941.06	−0.429934
$439$	4192.02	0.455750	0.227875	−	0.973690i	$-0.426822\pi$
$439$	4192.02	0.455750	0.227875	+	0.973690i	$0.426822\pi$
$440$	0	0
$441$	0	0
$442$	12845.7	1.38237
$443$	−7998.77	−0.857863	−0.428931	−	0.903337i	$-0.641110\pi$
$443$	−7998.77	−0.857863	−0.428931	+	0.903337i	$0.641110\pi$
$444$	3046.79	0.325663
$445$	0	0
$446$	−90.5870	−0.00961754
$447$	23369.1	2.47275
$448$	0	0
$449$	−6912.51	−0.726552	−0.363276	−	0.931682i	$-0.618342\pi$
$449$	−6912.51	−0.726552	−0.363276	+	0.931682i	$0.618342\pi$
$450$	0	0
$451$	−18299.1	−1.91058
$452$	4540.14	0.472456
$453$	−17783.0	−1.84441
$454$	−10646.9	−1.10063
$455$	0	0
$456$	5621.79	0.577335
$457$	−5302.27	−0.542735	−0.271368	−	0.962476i	$-0.587476\pi$
$457$	−5302.27	−0.542735	−0.271368	+	0.962476i	$0.587476\pi$
$458$	−3818.78	−0.389607
$459$	−2859.36	−0.290770
$460$	0	0
$461$	−1050.36	−0.106118	−0.0530589	−	0.998591i	$-0.516897\pi$
$461$	−1050.36	−0.106118	−0.0530589	+	0.998591i	$0.516897\pi$
$462$	0	0
$463$	−14089.9	−1.41428	−0.707142	−	0.707071i	$-0.750016\pi$
$463$	−14089.9	−1.41428	−0.707142	+	0.707071i	$0.750016\pi$
$464$	−759.595	−0.0759986
$465$	0	0
$466$	−3010.50	−0.299268
$467$	2916.25	0.288968	0.144484	−	0.989507i	$-0.453848\pi$
$467$	2916.25	0.288968	0.144484	+	0.989507i	$0.453848\pi$
$468$	5183.16	0.511948
$469$	0	0
$470$	0	0
$471$	3204.35	0.313479
$472$	4386.35	0.427750
$473$	−15472.1	−1.50404
$474$	9182.98	0.889849
$475$	0	0
$476$	0	0
$477$	−11160.0	−1.07124
$478$	−2706.39	−0.258969
$479$	−3078.80	−0.293683	−0.146841	−	0.989160i	$-0.546911\pi$
$479$	−3078.80	−0.293683	−0.146841	+	0.989160i	$0.546911\pi$
$480$	0	0
$481$	5892.97	0.558620
$482$	−2460.15	−0.232483
$483$	0	0
$484$	11726.3	1.10127
$485$	0	0
$486$	−10198.1	−0.951838
$487$	8829.97	0.821610	0.410805	−	0.911723i	$-0.365248\pi$
$487$	8829.97	0.821610	0.410805	+	0.911723i	$0.365248\pi$
$488$	−5549.15	−0.514751
$489$	6682.18	0.617952
$490$	0	0
$491$	10006.5	0.919727	0.459863	−	0.887990i	$-0.347898\pi$
$491$	10006.5	0.919727	0.459863	+	0.887990i	$0.347898\pi$
$492$	7971.49	0.730452
$493$	−5543.09	−0.506386
$494$	10873.4	0.990322
$495$	0	0
$496$	2645.68	0.239505
$497$	0	0
$498$	−4480.13	−0.403131
$499$	−9161.11	−0.821859	−0.410930	−	0.911667i	$-0.634796\pi$
$499$	−9161.11	−0.821859	−0.410930	+	0.911667i	$0.634796\pi$
$500$	0	0
$501$	12678.6	1.13062
$502$	2579.10	0.229305
$503$	17579.1	1.55828	0.779139	−	0.626852i	$-0.215657\pi$
$503$	17579.1	1.55828	0.779139	+	0.626852i	$0.215657\pi$
$504$	0	0
$505$	0	0
$506$	11375.1	0.999376
$507$	5894.70	0.516357
$508$	6554.99	0.572501
$509$	−12943.5	−1.12713	−0.563567	−	0.826070i	$-0.690571\pi$
$509$	−12943.5	−1.12713	−0.563567	+	0.826070i	$0.690571\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−2420.35	−0.208306
$514$	−6647.07	−0.570408
$515$	0	0
$516$	6740.00	0.575023
$517$	−5602.14	−0.476561
$518$	0	0
$519$	−23233.3	−1.96499
$520$	0	0
$521$	1154.83	0.0971091	0.0485546	−	0.998821i	$-0.484539\pi$
$521$	1154.83	0.0971091	0.0485546	+	0.998821i	$0.484539\pi$
$522$	−2236.61	−0.187536
$523$	9561.01	0.799376	0.399688	−	0.916651i	$-0.369118\pi$
$523$	9561.01	0.799376	0.399688	+	0.916651i	$0.369118\pi$
$524$	9281.64	0.773799
$525$	0	0
$526$	3837.91	0.318139
$527$	19306.7	1.59585
$528$	−7427.48	−0.612196
$529$	−4578.12	−0.376273
$530$	0	0
$531$	12915.5	1.05552
$532$	0	0
$533$	15418.1	1.25297
$534$	11797.8	0.956067
$535$	0	0
$536$	2481.22	0.199948
$537$	11410.0	0.916901
$538$	−11856.9	−0.950163
$539$	0	0
$540$	0	0
$541$	3152.36	0.250519	0.125259	−	0.992124i	$-0.460024\pi$
$541$	3152.36	0.250519	0.125259	+	0.992124i	$0.460024\pi$
$542$	−13941.0	−1.10483
$543$	26375.7	2.08451
$544$	3736.28	0.294470
$545$	0	0
$546$	0	0
$547$	2010.30	0.157138	0.0785688	−	0.996909i	$-0.474965\pi$
$547$	2010.30	0.157138	0.0785688	+	0.996909i	$0.474965\pi$
$548$	−9277.09	−0.723171
$549$	−16339.3	−1.27021
$550$	0	0
$551$	−4692.04	−0.362773
$552$	−4955.23	−0.382081
$553$	0	0
$554$	−188.388	−0.0144474
$555$	0	0
$556$	10199.9	0.778007
$557$	17816.1	1.35528	0.677641	−	0.735393i	$-0.263002\pi$
$557$	17816.1	1.35528	0.677641	+	0.735393i	$0.263002\pi$
$558$	7790.13	0.591008
$559$	13036.2	0.986357
$560$	0	0
$561$	−54201.5	−4.07912
$562$	4329.99	0.324999
$563$	8438.73	0.631706	0.315853	−	0.948808i	$-0.397709\pi$
$563$	8438.73	0.631706	0.315853	+	0.948808i	$0.397709\pi$
$564$	2440.41	0.182199
$565$	0	0
$566$	1258.89	0.0934896
$567$	0	0
$568$	−2650.98	−0.195832
$569$	19603.9	1.44436	0.722178	−	0.691707i	$-0.243141\pi$
$569$	19603.9	1.44436	0.722178	+	0.691707i	$0.243141\pi$
$570$	0	0
$571$	−7487.34	−0.548748	−0.274374	−	0.961623i	$-0.588471\pi$
$571$	−7487.34	−0.548748	−0.274374	+	0.961623i	$0.588471\pi$
$572$	−14365.9	−1.05012
$573$	−27512.2	−2.00583
$574$	0	0
$575$	0	0
$576$	1507.57	0.109054
$577$	5396.08	0.389327	0.194663	−	0.980870i	$-0.437639\pi$
$577$	5396.08	0.389327	0.194663	+	0.980870i	$0.437639\pi$
$578$	17439.3	1.25498
$579$	30644.8	2.19958
$580$	0	0
$581$	0	0
$582$	905.954	0.0645241
$583$	30931.8	2.19736
$584$	−2217.11	−0.157097
$585$	0	0
$586$	15428.9	1.08765
$587$	−7274.14	−0.511475	−0.255738	−	0.966746i	$-0.582318\pi$
$587$	−7274.14	−0.511475	−0.255738	+	0.966746i	$0.582318\pi$
$588$	0	0
$589$	16342.4	1.14326
$590$	0	0
$591$	−5033.24	−0.350321
$592$	1714.02	0.118997
$593$	−21335.9	−1.47750	−0.738752	−	0.673977i	$-0.764585\pi$
$593$	−21335.9	−1.47750	−0.738752	+	0.673977i	$0.764585\pi$
$594$	3197.75	0.220884
$595$	0	0
$596$	13146.7	0.903538
$597$	−262.541	−0.0179985
$598$	−9584.21	−0.655397
$599$	−24465.1	−1.66881	−0.834405	−	0.551152i	$-0.814188\pi$
$599$	−24465.1	−1.66881	−0.834405	+	0.551152i	$0.814188\pi$
$600$	0	0
$601$	7109.15	0.482510	0.241255	−	0.970462i	$-0.422441\pi$
$601$	7109.15	0.482510	0.241255	+	0.970462i	$0.422441\pi$
$602$	0	0
$603$	7305.87	0.493396
$604$	−10004.1	−0.673944
$605$	0	0
$606$	12194.2	0.817415
$607$	9130.66	0.610547	0.305274	−	0.952265i	$-0.401252\pi$
$607$	9130.66	0.610547	0.305274	+	0.952265i	$0.401252\pi$
$608$	3162.64	0.210957
$609$	0	0
$610$	0	0
$611$	4720.15	0.312532
$612$	11001.4	0.726641
$613$	20367.8	1.34200	0.671002	−	0.741455i	$-0.265864\pi$
$613$	20367.8	1.34200	0.671002	+	0.741455i	$0.265864\pi$
$614$	−11680.1	−0.767705
$615$	0	0
$616$	0	0
$617$	14687.5	0.958343	0.479171	−	0.877721i	$-0.340937\pi$
$617$	14687.5	0.958343	0.479171	+	0.877721i	$0.340937\pi$
$618$	14826.5	0.965062
$619$	−12586.5	−0.817275	−0.408637	−	0.912697i	$-0.633996\pi$
$619$	−12586.5	−0.817275	−0.408637	+	0.912697i	$0.633996\pi$
$620$	0	0
$621$	2133.38	0.137857
$622$	−1122.49	−0.0723598
$623$	0	0
$624$	6258.10	0.401482
$625$	0	0
$626$	−3456.39	−0.220679
$627$	−45879.7	−2.92226
$628$	1802.66	0.114545
$629$	12508.0	0.792886
$630$	0	0
$631$	−15855.9	−1.00034	−0.500168	−	0.865928i	$-0.666729\pi$
$631$	−15855.9	−1.00034	−0.500168	+	0.865928i	$0.666729\pi$
$632$	5166.05	0.325149
$633$	−33106.2	−2.07876
$634$	−20676.7	−1.29523
$635$	0	0
$636$	−13474.5	−0.840095
$637$	0	0
$638$	6199.10	0.384678
$639$	−7805.72	−0.483239
$640$	0	0
$641$	−31817.0	−1.96052	−0.980262	−	0.197702i	$-0.936652\pi$
$641$	−31817.0	−1.96052	−0.980262	+	0.197702i	$0.936652\pi$
$642$	−16258.1	−0.999467
$643$	25566.4	1.56803	0.784013	−	0.620745i	$-0.213170\pi$
$643$	25566.4	1.56803	0.784013	+	0.620745i	$0.213170\pi$
$644$	0	0
$645$	0	0
$646$	23079.1	1.40563
$647$	−2994.06	−0.181930	−0.0909648	−	0.995854i	$-0.528995\pi$
$647$	−2994.06	−0.181930	−0.0909648	+	0.995854i	$0.528995\pi$
$648$	−6481.05	−0.392901
$649$	−35797.2	−2.16512
$650$	0	0
$651$	0	0
$652$	3759.18	0.225799
$653$	−5621.73	−0.336899	−0.168450	−	0.985710i	$-0.553876\pi$
$653$	−5621.73	−0.336899	−0.168450	+	0.985710i	$0.553876\pi$
$654$	−2154.38	−0.128812
$655$	0	0
$656$	4484.50	0.266906
$657$	−6528.23	−0.387657
$658$	0	0
$659$	8074.77	0.477312	0.238656	−	0.971104i	$-0.423293\pi$
$659$	8074.77	0.477312	0.238656	+	0.971104i	$0.423293\pi$
$660$	0	0
$661$	8738.07	0.514178	0.257089	−	0.966388i	$-0.417237\pi$
$661$	8738.07	0.514178	0.257089	+	0.966388i	$0.417237\pi$
$662$	−13914.5	−0.816922
$663$	45668.1	2.67511
$664$	−2520.38	−0.147304
$665$	0	0
$666$	5046.89	0.293638
$667$	4135.72	0.240084
$668$	7132.59	0.413126
$669$	−322.049	−0.0186115
$670$	0	0
$671$	45286.9	2.60549
$672$	0	0
$673$	−16097.3	−0.922002	−0.461001	−	0.887400i	$-0.652509\pi$
$673$	−16097.3	−0.922002	−0.461001	+	0.887400i	$0.652509\pi$
$674$	8383.66	0.479120
$675$	0	0
$676$	3316.17	0.188676
$677$	1358.88	0.0771430	0.0385715	−	0.999256i	$-0.487719\pi$
$677$	1358.88	0.0771430	0.0385715	+	0.999256i	$0.487719\pi$
$678$	16140.8	0.914281
$679$	0	0
$680$	0	0
$681$	−37851.1	−2.12990
$682$	−21591.5	−1.21229
$683$	−13584.6	−0.761056	−0.380528	−	0.924769i	$-0.624258\pi$
$683$	−13584.6	−0.761056	−0.380528	+	0.924769i	$0.624258\pi$
$684$	9312.29	0.520562
$685$	0	0
$686$	0	0
$687$	−13576.3	−0.753954
$688$	3791.70	0.210112
$689$	−26061.9	−1.44104
$690$	0	0
$691$	1143.06	0.0629290	0.0314645	−	0.999505i	$-0.489983\pi$
$691$	1143.06	0.0629290	0.0314645	+	0.999505i	$0.489983\pi$
$692$	−13070.3	−0.718004
$693$	0	0
$694$	−9095.24	−0.497479
$695$	0	0
$696$	−2700.46	−0.147070
$697$	32725.3	1.77842
$698$	9546.22	0.517664
$699$	−10702.7	−0.579133
$700$	0	0
$701$	15804.4	0.851534	0.425767	−	0.904833i	$-0.360004\pi$
$701$	15804.4	0.851534	0.425767	+	0.904833i	$0.360004\pi$
$702$	−2694.30	−0.144857
$703$	10587.6	0.568020
$704$	−4178.46	−0.223695
$705$	0	0
$706$	−13653.1	−0.727820
$707$	0	0
$708$	15594.0	0.827768
$709$	−6434.79	−0.340851	−0.170426	−	0.985371i	$-0.554514\pi$
$709$	−6434.79	−0.340851	−0.170426	+	0.985371i	$0.554514\pi$
$710$	0	0
$711$	15211.3	0.802345
$712$	6637.04	0.349345
$713$	−14404.8	−0.756610
$714$	0	0
$715$	0	0
$716$	6418.87	0.335034
$717$	−9621.55	−0.501148
$718$	−17537.2	−0.911537
$719$	2572.58	0.133437	0.0667185	−	0.997772i	$-0.478747\pi$
$719$	2572.58	0.133437	0.0667185	+	0.997772i	$0.478747\pi$
$720$	0	0
$721$	0	0
$722$	5817.71	0.299879
$723$	−8746.16	−0.449894
$724$	14838.1	0.761676
$725$	0	0
$726$	41688.5	2.13114
$727$	−20925.5	−1.06752	−0.533759	−	0.845637i	$-0.679221\pi$
$727$	−20925.5	−1.06752	−0.533759	+	0.845637i	$0.679221\pi$
$728$	0	0
$729$	−14381.9	−0.730674
$730$	0	0
$731$	27669.7	1.40000
$732$	−19728.0	−0.996129
$733$	−6344.82	−0.319715	−0.159858	−	0.987140i	$-0.551103\pi$
$733$	−6344.82	−0.319715	−0.159858	+	0.987140i	$0.551103\pi$
$734$	−13088.4	−0.658175
$735$	0	0
$736$	−2787.65	−0.139612
$737$	−20249.3	−1.01207
$738$	13204.5	0.658623
$739$	−9871.30	−0.491369	−0.245685	−	0.969350i	$-0.579013\pi$
$739$	−9871.30	−0.491369	−0.245685	+	0.969350i	$0.579013\pi$
$740$	0	0
$741$	38656.5	1.91644
$742$	0	0
$743$	−20268.1	−1.00076	−0.500380	−	0.865806i	$-0.666807\pi$
$743$	−20268.1	−1.00076	−0.500380	+	0.865806i	$0.666807\pi$
$744$	9405.74	0.463482
$745$	0	0
$746$	−2473.33	−0.121387
$747$	−7421.17	−0.363489
$748$	−30492.0	−1.49050
$749$	0	0
$750$	0	0
$751$	−10226.5	−0.496897	−0.248449	−	0.968645i	$-0.579921\pi$
$751$	−10226.5	−0.496897	−0.248449	+	0.968645i	$0.579921\pi$
$752$	1372.90	0.0665751
$753$	9169.05	0.443743
$754$	−5223.12	−0.252274
$755$	0	0
$756$	0	0
$757$	18533.0	0.889819	0.444910	−	0.895575i	$-0.353236\pi$
$757$	18533.0	0.889819	0.444910	+	0.895575i	$0.353236\pi$
$758$	−1208.39	−0.0579034
$759$	40439.9	1.93396
$760$	0	0
$761$	16527.6	0.787288	0.393644	−	0.919263i	$-0.371214\pi$
$761$	16527.6	0.787288	0.393644	+	0.919263i	$0.371214\pi$
$762$	23303.8	1.10789
$763$	0	0
$764$	−15477.5	−0.732926
$765$	0	0
$766$	6638.73	0.313142
$767$	30161.3	1.41990
$768$	1820.23	0.0855231
$769$	31787.3	1.49061	0.745304	−	0.666724i	$-0.232304\pi$
$769$	31787.3	1.49061	0.745304	+	0.666724i	$0.232304\pi$
$770$	0	0
$771$	−23631.2	−1.10383
$772$	17239.8	0.803722
$773$	35515.3	1.65252	0.826259	−	0.563291i	$-0.190465\pi$
$773$	35515.3	1.65252	0.826259	+	0.563291i	$0.190465\pi$
$774$	11164.6	0.518478
$775$	0	0
$776$	509.660	0.0235770
$777$	0	0
$778$	7404.71	0.341223
$779$	27700.9	1.27405
$780$	0	0
$781$	21634.8	0.991232
$782$	−20342.7	−0.930248
$783$	1162.63	0.0530638
$784$	0	0
$785$	0	0
$786$	32997.4	1.49743
$787$	12720.5	0.576157	0.288079	−	0.957607i	$-0.406984\pi$
$787$	12720.5	0.576157	0.288079	+	0.957607i	$0.406984\pi$
$788$	−2831.54	−0.128007
$789$	13644.3	0.615651
$790$	0	0
$791$	0	0
$792$	−12303.3	−0.551995
$793$	−38157.0	−1.70869
$794$	−7308.38	−0.326656
$795$	0	0
$796$	−147.697	−0.00657662
$797$	19581.7	0.870287	0.435143	−	0.900361i	$-0.356698\pi$
$797$	19581.7	0.870287	0.435143	+	0.900361i	$0.356698\pi$
$798$	0	0
$799$	10018.6	0.443596
$800$	0	0
$801$	19542.6	0.862051
$802$	−20138.6	−0.886682
$803$	18094.0	0.795171
$804$	8821.05	0.386933
$805$	0	0
$806$	18192.2	0.795028
$807$	−42152.9	−1.83872
$808$	6860.04	0.298682
$809$	−9825.05	−0.426984	−0.213492	−	0.976945i	$-0.568484\pi$
$809$	−9825.05	−0.426984	−0.213492	+	0.976945i	$0.568484\pi$
$810$	0	0
$811$	−4742.79	−0.205354	−0.102677	−	0.994715i	$-0.532741\pi$
$811$	−4742.79	−0.205354	−0.102677	+	0.994715i	$0.532741\pi$
$812$	0	0
$813$	−49562.1	−2.13803
$814$	−13988.2	−0.602319
$815$	0	0
$816$	13283.0	0.569849
$817$	23421.5	1.00295
$818$	23192.0	0.991305
$819$	0	0
$820$	0	0
$821$	29300.8	1.24556	0.622781	−	0.782396i	$-0.286003\pi$
$821$	29300.8	1.24556	0.622781	+	0.782396i	$0.286003\pi$
$822$	−32981.3	−1.39946
$823$	−27537.3	−1.16633	−0.583166	−	0.812353i	$-0.698186\pi$
$823$	−27537.3	−1.16633	−0.583166	+	0.812353i	$0.698186\pi$
$824$	8340.90	0.352632
$825$	0	0
$826$	0	0
$827$	−844.574	−0.0355124	−0.0177562	−	0.999842i	$-0.505652\pi$
$827$	−844.574	−0.0355124	−0.0177562	+	0.999842i	$0.505652\pi$
$828$	−8208.16	−0.344509
$829$	−16243.7	−0.680538	−0.340269	−	0.940328i	$-0.610518\pi$
$829$	−16243.7	−0.680538	−0.340269	+	0.940328i	$0.610518\pi$
$830$	0	0
$831$	−669.744	−0.0279581
$832$	3520.61	0.146701
$833$	0	0
$834$	36261.9	1.50557
$835$	0	0
$836$	−25810.4	−1.06779
$837$	−4049.45	−0.167228
$838$	−22334.9	−0.920699
$839$	−13135.2	−0.540499	−0.270250	−	0.962790i	$-0.587106\pi$
$839$	−13135.2	−0.540499	−0.270250	+	0.962790i	$0.587106\pi$
$840$	0	0
$841$	−22135.2	−0.907588
$842$	6282.30	0.257129
$843$	15393.7	0.628928
$844$	−18624.5	−0.759575
$845$	0	0
$846$	4042.46	0.164282
$847$	0	0
$848$	−7580.34	−0.306969
$849$	4475.52	0.180918
$850$	0	0
$851$	−9332.24	−0.375917
$852$	−9424.56	−0.378967
$853$	−24870.1	−0.998282	−0.499141	−	0.866521i	$-0.666351\pi$
$853$	−24870.1	−0.998282	−0.499141	+	0.866521i	$0.666351\pi$
$854$	0	0
$855$	0	0
$856$	−9146.30	−0.365203
$857$	48576.3	1.93622	0.968108	−	0.250535i	$-0.0806065\pi$
$857$	48576.3	1.93622	0.968108	+	0.250535i	$0.0806065\pi$
$858$	−51072.7	−2.03216
$859$	−4535.56	−0.180153	−0.0900765	−	0.995935i	$-0.528711\pi$
$859$	−4535.56	−0.180153	−0.0900765	+	0.995935i	$0.528711\pi$
$860$	0	0
$861$	0	0
$862$	−26866.3	−1.06157
$863$	−19685.6	−0.776484	−0.388242	−	0.921557i	$-0.626918\pi$
$863$	−19685.6	−0.776484	−0.388242	+	0.921557i	$0.626918\pi$
$864$	−783.662	−0.0308573
$865$	0	0
$866$	−3265.93	−0.128153
$867$	61998.8	2.42859
$868$	0	0
$869$	−42160.4	−1.64579
$870$	0	0
$871$	17061.3	0.663720
$872$	−1211.99	−0.0470677
$873$	1500.68	0.0581791
$874$	−17219.4	−0.666425
$875$	0	0
$876$	−7882.13	−0.304010
$877$	−47634.9	−1.83411	−0.917056	−	0.398759i	$-0.869441\pi$
$877$	−47634.9	−1.83411	−0.917056	+	0.398759i	$0.869441\pi$
$878$	8384.04	0.322264
$879$	54851.6	2.10478
$880$	0	0
$881$	26891.9	1.02839	0.514195	−	0.857674i	$-0.328091\pi$
$881$	26891.9	1.02839	0.514195	+	0.857674i	$0.328091\pi$
$882$	0	0
$883$	8927.28	0.340234	0.170117	−	0.985424i	$-0.445585\pi$
$883$	8927.28	0.340234	0.170117	+	0.985424i	$0.445585\pi$
$884$	25691.4	0.977482
$885$	0	0
$886$	−15997.5	−0.606601
$887$	−20711.5	−0.784018	−0.392009	−	0.919961i	$-0.628220\pi$
$887$	−20711.5	−0.784018	−0.392009	+	0.919961i	$0.628220\pi$
$888$	6093.58	0.230278
$889$	0	0
$890$	0	0
$891$	52892.2	1.98873
$892$	−181.174	−0.00680062
$893$	8480.43	0.317790
$894$	46738.1	1.74850
$895$	0	0
$896$	0	0
$897$	−34073.1	−1.26830
$898$	−13825.0	−0.513750
$899$	−7850.18	−0.291233
$900$	0	0
$901$	−55317.0	−2.04537
$902$	−36598.2	−1.35098
$903$	0	0
$904$	9080.27	0.334077
$905$	0	0
$906$	−35566.0	−1.30419
$907$	−47137.0	−1.72564	−0.862821	−	0.505509i	$-0.831305\pi$
$907$	−47137.0	−1.72564	−0.862821	+	0.505509i	$0.831305\pi$
$908$	−21293.8	−0.778260
$909$	20199.2	0.737034
$910$	0	0
$911$	6096.61	0.221723	0.110862	−	0.993836i	$-0.464639\pi$
$911$	6096.61	0.221723	0.110862	+	0.993836i	$0.464639\pi$
$912$	11243.6	0.408237
$913$	20568.9	0.745599
$914$	−10604.5	−0.383772
$915$	0	0
$916$	−7637.56	−0.275493
$917$	0	0
$918$	−5718.71	−0.205605
$919$	23012.9	0.826033	0.413017	−	0.910723i	$-0.364475\pi$
$919$	23012.9	0.826033	0.413017	+	0.910723i	$0.364475\pi$
$920$	0	0
$921$	−41524.3	−1.48564
$922$	−2100.73	−0.0750366
$923$	−18228.6	−0.650056
$924$	0	0
$925$	0	0
$926$	−28179.8	−1.00005
$927$	24559.5	0.870162
$928$	−1519.19	−0.0537391
$929$	−50361.0	−1.77857	−0.889284	−	0.457356i	$-0.848797\pi$
$929$	−50361.0	−1.77857	−0.889284	+	0.457356i	$0.848797\pi$
$930$	0	0
$931$	0	0
$932$	−6021.00	−0.211614
$933$	−3990.60	−0.140028
$934$	5832.50	0.204331
$935$	0	0
$936$	10366.3	0.362002
$937$	−8167.89	−0.284774	−0.142387	−	0.989811i	$-0.545478\pi$
$937$	−8167.89	−0.284774	−0.142387	+	0.989811i	$0.545478\pi$
$938$	0	0
$939$	−12287.9	−0.427051
$940$	0	0
$941$	−22954.2	−0.795204	−0.397602	−	0.917558i	$-0.630157\pi$
$941$	−22954.2	−0.795204	−0.397602	+	0.917558i	$0.630157\pi$
$942$	6408.70	0.221663
$943$	−24416.5	−0.843171
$944$	8772.70	0.302465
$945$	0	0
$946$	−30944.3	−1.06352
$947$	46759.1	1.60451	0.802253	−	0.596984i	$-0.203634\pi$
$947$	46759.1	1.60451	0.802253	+	0.596984i	$0.203634\pi$
$948$	18366.0	0.629218
$949$	−15245.3	−0.521478
$950$	0	0
$951$	−73508.4	−2.50649
$952$	0	0
$953$	44798.2	1.52273	0.761363	−	0.648326i	$-0.224531\pi$
$953$	44798.2	1.52273	0.761363	+	0.648326i	$0.224531\pi$
$954$	−22320.1	−0.757484
$955$	0	0
$956$	−5412.77	−0.183119
$957$	22038.6	0.744416
$958$	−6157.61	−0.207665
$959$	0	0
$960$	0	0
$961$	−2448.71	−0.0821963
$962$	11785.9	0.395004
$963$	−26931.0	−0.901184
$964$	−4920.31	−0.164390
$965$	0	0
$966$	0	0
$967$	30697.4	1.02085	0.510425	−	0.859923i	$-0.329488\pi$
$967$	30697.4	1.02085	0.510425	+	0.859923i	$0.329488\pi$
$968$	23452.6	0.778714
$969$	82049.3	2.72013
$970$	0	0
$971$	−12380.9	−0.409187	−0.204593	−	0.978847i	$-0.565587\pi$
$971$	−12380.9	−0.409187	−0.204593	+	0.978847i	$0.565587\pi$
$972$	−20396.1	−0.673051
$973$	0	0
$974$	17659.9	0.580966
$975$	0	0
$976$	−11098.3	−0.363984
$977$	−38163.6	−1.24970	−0.624852	−	0.780743i	$-0.714841\pi$
$977$	−38163.6	−1.24970	−0.624852	+	0.780743i	$0.714841\pi$
$978$	13364.4	0.436958
$979$	−54165.2	−1.76826
$980$	0	0
$981$	−3568.66	−0.116145
$982$	20013.0	0.650345
$983$	−3079.77	−0.0999283	−0.0499641	−	0.998751i	$-0.515911\pi$
$983$	−3079.77	−0.0999283	−0.0499641	+	0.998751i	$0.515911\pi$
$984$	15943.0	0.516508
$985$	0	0
$986$	−11086.2	−0.358069
$987$	0	0
$988$	21746.9	0.700264
$989$	−20644.4	−0.663757
$990$	0	0
$991$	−11201.4	−0.359056	−0.179528	−	0.983753i	$-0.557457\pi$
$991$	−11201.4	−0.359056	−0.179528	+	0.983753i	$0.557457\pi$
$992$	5291.36	0.169356
$993$	−49467.8	−1.58088
$994$	0	0
$995$	0	0
$996$	−8960.26	−0.285057
$997$	25637.7	0.814396	0.407198	−	0.913340i	$-0.366506\pi$
$997$	25637.7	0.814396	0.407198	+	0.913340i	$0.366506\pi$
$998$	−18322.2	−0.581142
$999$	−2623.47	−0.0830859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.dc.1.9		10
5.2	odd	4		490.4.c.g.99.12	yes	20
5.3	odd	4		490.4.c.g.99.9	yes	20
5.4	even	2		2450.4.a.db.1.2		10
7.6	odd	2	inner	2450.4.a.dc.1.2		10
35.13	even	4		490.4.c.g.99.2	✓	20
35.27	even	4		490.4.c.g.99.19	yes	20
35.34	odd	2		2450.4.a.db.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.4.c.g.99.2	✓	20	35.13	even	4
490.4.c.g.99.9	yes	20	5.3	odd	4
490.4.c.g.99.12	yes	20	5.2	odd	4
490.4.c.g.99.19	yes	20	35.27	even	4
2450.4.a.db.1.2		10	5.4	even	2
2450.4.a.db.1.9		10	35.34	odd	2
2450.4.a.dc.1.2		10	7.6	odd	2	inner
2450.4.a.dc.1.9		10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +7.11026 q^{3} +4.00000 q^{4} +14.2205 q^{6} +8.00000 q^{8} +23.5558 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +7.11026 q^{3} +4.00000 q^{4} +14.2205 q^{6} +8.00000 q^{8} +23.5558 q^{9} -65.2884 q^{11} +28.4410 q^{12} +55.0095 q^{13} +16.0000 q^{16} +116.759 q^{17} +47.1115 q^{18} +98.8324 q^{19} -130.577 q^{22} -87.1142 q^{23} +56.8821 q^{24} +110.019 q^{26} -24.4894 q^{27} -47.4747 q^{29} +165.355 q^{31} +32.0000 q^{32} -464.217 q^{33} +233.518 q^{34} +94.2230 q^{36} +107.127 q^{37} +197.665 q^{38} +391.131 q^{39} +280.281 q^{41} +236.981 q^{43} -261.154 q^{44} -174.228 q^{46} +85.8061 q^{47} +113.764 q^{48} +830.185 q^{51} +220.038 q^{52} -473.771 q^{53} -48.9789 q^{54} +702.724 q^{57} -94.9494 q^{58} +548.294 q^{59} -693.644 q^{61} +330.710 q^{62} +64.0000 q^{64} -928.435 q^{66} +310.152 q^{67} +467.035 q^{68} -619.404 q^{69} -331.372 q^{71} +188.446 q^{72} -277.139 q^{73} +214.253 q^{74} +395.330 q^{76} +782.263 q^{78} +645.756 q^{79} -810.132 q^{81} +560.563 q^{82} -315.047 q^{83} +473.963 q^{86} -337.557 q^{87} -522.307 q^{88} +829.630 q^{89} -348.457 q^{92} +1175.72 q^{93} +171.612 q^{94} +227.528 q^{96} +63.7075 q^{97} -1537.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 20 q^{2} + 40 q^{4} + 80 q^{8} + 158 q^{9}+O(q^{10}) \) 10 * q + 20 * q^2 + 40 * q^4 + 80 * q^8 + 158 * q^9	\( 10 q + 20 q^{2} + 40 q^{4} + 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} + 316 q^{18} + 104 q^{22} + 400 q^{23} + 108 q^{29} + 320 q^{32} + 632 q^{36} + 1492 q^{37} + 252 q^{39} + 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} + 968 q^{53} + 3024 q^{57} + 216 q^{58} + 640 q^{64} + 1880 q^{67} - 936 q^{71} + 1264 q^{72} + 2984 q^{74} + 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} + 416 q^{88} + 1600 q^{92} + 304 q^{93} - 312 q^{99}+O(q^{100}) \) 10 * q + 20 * q^2 + 40 * q^4 + 80 * q^8 + 158 * q^9 + 52 * q^11 + 160 * q^16 + 316 * q^18 + 104 * q^22 + 400 * q^23 + 108 * q^29 + 320 * q^32 + 632 * q^36 + 1492 * q^37 + 252 * q^39 + 904 * q^43 + 208 * q^44 + 800 * q^46 - 148 * q^51 + 968 * q^53 + 3024 * q^57 + 216 * q^58 + 640 * q^64 + 1880 * q^67 - 936 * q^71 + 1264 * q^72 + 2984 * q^74 + 504 * q^78 + 3212 * q^79 + 1010 * q^81 + 1808 * q^86 + 416 * q^88 + 1600 * q^92 + 304 * q^93 - 312 * q^99

Introduction

L-functions

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