LMFDB - Embedded newform 2450.4.a.bz.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.78233$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -5.78233 q^{3} +4.00000 q^{4} -11.5647 q^{6} +8.00000 q^{8} +6.43534 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -5.78233 q^{3} +4.00000 q^{4} -11.5647 q^{6} +8.00000 q^{8} +6.43534 q^{9} -20.4353 q^{11} -23.1293 q^{12} +53.1293 q^{13} +16.0000 q^{16} -27.7414 q^{17} +12.8707 q^{18} -143.129 q^{19} -40.8707 q^{22} +200.735 q^{23} -46.2586 q^{24} +106.259 q^{26} +118.912 q^{27} +113.082 q^{29} -105.470 q^{31} +32.0000 q^{32} +118.164 q^{33} -55.4827 q^{34} +25.7414 q^{36} +1.38796 q^{37} -286.259 q^{38} -307.211 q^{39} -226.729 q^{41} -268.558 q^{43} -81.7414 q^{44} +401.470 q^{46} -27.7414 q^{47} -92.5173 q^{48} +160.410 q^{51} +212.517 q^{52} -74.2586 q^{53} +237.823 q^{54} +827.621 q^{57} +226.164 q^{58} +665.362 q^{59} +509.845 q^{61} -210.940 q^{62} +64.0000 q^{64} +236.328 q^{66} +981.416 q^{67} -110.965 q^{68} -1160.72 q^{69} -144.707 q^{71} +51.4827 q^{72} -735.552 q^{73} +2.77592 q^{74} -572.517 q^{76} -614.423 q^{78} +146.069 q^{79} -861.341 q^{81} -453.457 q^{82} -712.356 q^{83} -537.116 q^{86} -653.877 q^{87} -163.483 q^{88} -606.457 q^{89} +802.940 q^{92} +609.862 q^{93} -55.4827 q^{94} -185.035 q^{96} -771.552 q^{97} -131.508 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{8} + 40 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^6 + 16 * q^8 + 40 * q^9	$ 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{8} + 40 q^{9} - 68 q^{11} + 8 q^{12} + 52 q^{13} + 32 q^{16} - 164 q^{17} + 80 q^{18} - 232 q^{19} - 136 q^{22} + 198 q^{23} + 16 q^{24} + 104 q^{26} + 170 q^{27} - 18 q^{29} + 196 q^{31} + 64 q^{32} - 252 q^{33} - 328 q^{34} + 160 q^{36} - 160 q^{37} - 464 q^{38} - 316 q^{39} + 62 q^{41} - 198 q^{43} - 272 q^{44} + 396 q^{46} - 164 q^{47} + 32 q^{48} - 900 q^{51} + 208 q^{52} - 40 q^{53} + 340 q^{54} + 136 q^{57} - 36 q^{58} - 80 q^{59} - 174 q^{61} + 392 q^{62} + 128 q^{64} - 504 q^{66} + 1054 q^{67} - 656 q^{68} - 1182 q^{69} - 832 q^{71} + 320 q^{72} - 820 q^{73} - 320 q^{74} - 928 q^{76} - 632 q^{78} - 576 q^{79} - 1370 q^{81} + 124 q^{82} + 298 q^{83} - 396 q^{86} - 1674 q^{87} - 544 q^{88} - 182 q^{89} + 792 q^{92} + 2956 q^{93} - 328 q^{94} + 64 q^{96} - 892 q^{97} - 1728 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^6 + 16 * q^8 + 40 * q^9 - 68 * q^11 + 8 * q^12 + 52 * q^13 + 32 * q^16 - 164 * q^17 + 80 * q^18 - 232 * q^19 - 136 * q^22 + 198 * q^23 + 16 * q^24 + 104 * q^26 + 170 * q^27 - 18 * q^29 + 196 * q^31 + 64 * q^32 - 252 * q^33 - 328 * q^34 + 160 * q^36 - 160 * q^37 - 464 * q^38 - 316 * q^39 + 62 * q^41 - 198 * q^43 - 272 * q^44 + 396 * q^46 - 164 * q^47 + 32 * q^48 - 900 * q^51 + 208 * q^52 - 40 * q^53 + 340 * q^54 + 136 * q^57 - 36 * q^58 - 80 * q^59 - 174 * q^61 + 392 * q^62 + 128 * q^64 - 504 * q^66 + 1054 * q^67 - 656 * q^68 - 1182 * q^69 - 832 * q^71 + 320 * q^72 - 820 * q^73 - 320 * q^74 - 928 * q^76 - 632 * q^78 - 576 * q^79 - 1370 * q^81 + 124 * q^82 + 298 * q^83 - 396 * q^86 - 1674 * q^87 - 544 * q^88 - 182 * q^89 + 792 * q^92 + 2956 * q^93 - 328 * q^94 + 64 * q^96 - 892 * q^97 - 1728 * 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
$3$	−5.78233	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$3$	−5.78233	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−11.5647	−0.786875
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	6.43534	0.238346
$10$	0	0
$11$	−20.4353	−0.560135	−0.280068	−	0.959980i	$-0.590357\pi$
$11$	−20.4353	−0.560135	−0.280068	+	0.959980i	$0.590357\pi$
$12$	−23.1293	−0.556405
$13$	53.1293	1.13349	0.566747	−	0.823892i	$-0.308202\pi$
$13$	53.1293	1.13349	0.566747	+	0.823892i	$0.308202\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−27.7414	−0.395780	−0.197890	−	0.980224i	$-0.563409\pi$
$17$	−27.7414	−0.395780	−0.197890	+	0.980224i	$0.563409\pi$
$18$	12.8707	0.168536
$19$	−143.129	−1.72822	−0.864108	−	0.503306i	$-0.832117\pi$
$19$	−143.129	−1.72822	−0.864108	+	0.503306i	$0.832117\pi$
$20$	0	0
$21$	0	0
$22$	−40.8707	−0.396075
$23$	200.735	1.81983	0.909916	−	0.414793i	$-0.136146\pi$
$23$	200.735	1.81983	0.909916	+	0.414793i	$0.136146\pi$
$24$	−46.2586	−0.393438
$25$	0	0
$26$	106.259	0.801501
$27$	118.912	0.847576
$28$	0	0
$29$	113.082	0.724096	0.362048	−	0.932159i	$-0.382078\pi$
$29$	113.082	0.724096	0.362048	+	0.932159i	$0.382078\pi$
$30$	0	0
$31$	−105.470	−0.611063	−0.305532	−	0.952182i	$-0.598834\pi$
$31$	−105.470	−0.611063	−0.305532	+	0.952182i	$0.598834\pi$
$32$	32.0000	0.176777
$33$	118.164	0.623324
$34$	−55.4827	−0.279859
$35$	0	0
$36$	25.7414	0.119173
$37$	1.38796	0.00616701	0.00308350	−	0.999995i	$-0.499018\pi$
$37$	1.38796	0.00616701	0.00308350	+	0.999995i	$0.499018\pi$
$38$	−286.259	−1.22203
$39$	−307.211	−1.26136
$40$	0	0
$41$	−226.729	−0.863635	−0.431818	−	0.901961i	$-0.642128\pi$
$41$	−226.729	−0.863635	−0.431818	+	0.901961i	$0.642128\pi$
$42$	0	0
$43$	−268.558	−0.952436	−0.476218	−	0.879327i	$-0.657993\pi$
$43$	−268.558	−0.952436	−0.476218	+	0.879327i	$0.657993\pi$
$44$	−81.7414	−0.280068
$45$	0	0
$46$	401.470	1.28682
$47$	−27.7414	−0.0860956	−0.0430478	−	0.999073i	$-0.513707\pi$
$47$	−27.7414	−0.0860956	−0.0430478	+	0.999073i	$0.513707\pi$
$48$	−92.5173	−0.278202
$49$	0	0
$50$	0	0
$51$	160.410	0.440428
$52$	212.517	0.566747
$53$	−74.2586	−0.192457	−0.0962284	−	0.995359i	$-0.530678\pi$
$53$	−74.2586	−0.192457	−0.0962284	+	0.995359i	$0.530678\pi$
$54$	237.823	0.599327
$55$	0	0
$56$	0	0
$57$	827.621	1.92318
$58$	226.164	0.512013
$59$	665.362	1.46818	0.734091	−	0.679051i	$-0.237608\pi$
$59$	665.362	1.46818	0.734091	+	0.679051i	$0.237608\pi$
$60$	0	0
$61$	509.845	1.07015	0.535074	−	0.844806i	$-0.320284\pi$
$61$	509.845	1.07015	0.535074	+	0.844806i	$0.320284\pi$
$62$	−210.940	−0.432087
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	236.328	0.440757
$67$	981.416	1.78954	0.894769	−	0.446529i	$-0.147340\pi$
$67$	981.416	1.78954	0.894769	+	0.446529i	$0.147340\pi$
$68$	−110.965	−0.197890
$69$	−1160.72	−2.02513
$70$	0	0
$71$	−144.707	−0.241881	−0.120940	−	0.992660i	$-0.538591\pi$
$71$	−144.707	−0.241881	−0.120940	+	0.992660i	$0.538591\pi$
$72$	51.4827	0.0842680
$73$	−735.552	−1.17931	−0.589656	−	0.807654i	$-0.700737\pi$
$73$	−735.552	−1.17931	−0.589656	+	0.807654i	$0.700737\pi$
$74$	2.77592	0.00436073
$75$	0	0
$76$	−572.517	−0.864108
$77$	0	0
$78$	−614.423	−0.891919
$79$	146.069	0.208026	0.104013	−	0.994576i	$-0.466832\pi$
$79$	146.069	0.208026	0.104013	+	0.994576i	$0.466832\pi$
$80$	0	0
$81$	−861.341	−1.18154
$82$	−453.457	−0.610682
$83$	−712.356	−0.942063	−0.471031	−	0.882116i	$-0.656118\pi$
$83$	−712.356	−0.942063	−0.471031	+	0.882116i	$0.656118\pi$
$84$	0	0
$85$	0	0
$86$	−537.116	−0.673474
$87$	−653.877	−0.805781
$88$	−163.483	−0.198038
$89$	−606.457	−0.722296	−0.361148	−	0.932509i	$-0.617615\pi$
$89$	−606.457	−0.722296	−0.361148	+	0.932509i	$0.617615\pi$
$90$	0	0
$91$	0	0
$92$	802.940	0.909916
$93$	609.862	0.679997
$94$	−55.4827	−0.0608788
$95$	0	0
$96$	−185.035	−0.196719
$97$	−771.552	−0.807621	−0.403810	−	0.914843i	$-0.632314\pi$
$97$	−771.552	−0.807621	−0.403810	+	0.914843i	$0.632314\pi$
$98$	0	0
$99$	−131.508	−0.133506
$100$	0	0
$101$	−1465.41	−1.44370	−0.721850	−	0.692049i	$-0.756708\pi$
$101$	−1465.41	−1.44370	−0.721850	+	0.692049i	$0.756708\pi$
$102$	320.819	0.311430
$103$	−436.312	−0.417390	−0.208695	−	0.977981i	$-0.566922\pi$
$103$	−436.312	−0.417390	−0.208695	+	0.977981i	$0.566922\pi$
$104$	425.035	0.400751
$105$	0	0
$106$	−148.517	−0.136087
$107$	2184.33	1.97353	0.986763	−	0.162172i	$-0.0518499\pi$
$107$	2184.33	1.97353	0.986763	+	0.162172i	$0.0518499\pi$
$108$	475.647	0.423788
$109$	−65.0256	−0.0571406	−0.0285703	−	0.999592i	$-0.509095\pi$
$109$	−65.0256	−0.0571406	−0.0285703	+	0.999592i	$0.509095\pi$
$110$	0	0
$111$	−8.02564	−0.00686270
$112$	0	0
$113$	613.767	0.510959	0.255479	−	0.966815i	$-0.417767\pi$
$113$	613.767	0.510959	0.255479	+	0.966815i	$0.417767\pi$
$114$	1655.24	1.35989
$115$	0	0
$116$	452.328	0.362048
$117$	341.905	0.270164
$118$	1330.72	1.03816
$119$	0	0
$120$	0	0
$121$	−913.397	−0.686249
$122$	1019.69	0.756708
$123$	1311.02	0.961062
$124$	−421.880	−0.305532
$125$	0	0
$126$	0	0
$127$	2490.47	1.74010	0.870052	−	0.492961i	$-0.164085\pi$
$127$	2490.47	1.74010	0.870052	+	0.492961i	$0.164085\pi$
$128$	128.000	0.0883883
$129$	1552.89	1.05988
$130$	0	0
$131$	−2389.20	−1.59348	−0.796738	−	0.604325i	$-0.793443\pi$
$131$	−2389.20	−1.59348	−0.796738	+	0.604325i	$0.793443\pi$
$132$	472.656	0.311662
$133$	0	0
$134$	1962.83	1.26539
$135$	0	0
$136$	−221.931	−0.139930
$137$	−1738.86	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$137$	−1738.86	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$138$	−2321.43	−1.43198
$139$	697.332	0.425517	0.212759	−	0.977105i	$-0.431755\pi$
$139$	697.332	0.425517	0.212759	+	0.977105i	$0.431755\pi$
$140$	0	0
$141$	160.410	0.0958080
$142$	−289.414	−0.171036
$143$	−1085.72	−0.634910
$144$	102.965	0.0595865
$145$	0	0
$146$	−1471.10	−0.833900
$147$	0	0
$148$	5.55184	0.00308350
$149$	−1639.86	−0.901630	−0.450815	−	0.892618i	$-0.648867\pi$
$149$	−1639.86	−0.901630	−0.450815	+	0.892618i	$0.648867\pi$
$150$	0	0
$151$	2829.31	1.52481	0.762403	−	0.647103i	$-0.224020\pi$
$151$	2829.31	1.52481	0.762403	+	0.647103i	$0.224020\pi$
$152$	−1145.03	−0.611017
$153$	−178.525	−0.0943327
$154$	0	0
$155$	0	0
$156$	−1228.85	−0.630682
$157$	2591.12	1.31716	0.658580	−	0.752511i	$-0.271158\pi$
$157$	2591.12	1.31716	0.658580	+	0.752511i	$0.271158\pi$
$158$	292.138	0.147097
$159$	429.388	0.214168
$160$	0	0
$161$	0	0
$162$	−1722.68	−0.835473
$163$	−3472.54	−1.66865	−0.834325	−	0.551273i	$-0.814142\pi$
$163$	−3472.54	−1.66865	−0.834325	+	0.551273i	$0.814142\pi$
$164$	−906.914	−0.431818
$165$	0	0
$166$	−1424.71	−0.666139
$167$	−798.576	−0.370034	−0.185017	−	0.982735i	$-0.559234\pi$
$167$	−798.576	−0.370034	−0.185017	+	0.982735i	$0.559234\pi$
$168$	0	0
$169$	625.725	0.284809
$170$	0	0
$171$	−921.086	−0.411913
$172$	−1074.23	−0.476218
$173$	−2198.89	−0.966349	−0.483175	−	0.875524i	$-0.660516\pi$
$173$	−2198.89	−0.966349	−0.483175	+	0.875524i	$0.660516\pi$
$174$	−1307.75	−0.569773
$175$	0	0
$176$	−326.965	−0.140034
$177$	−3847.34	−1.63381
$178$	−1212.91	−0.510740
$179$	−2336.57	−0.975663	−0.487832	−	0.872938i	$-0.662212\pi$
$179$	−2336.57	−0.975663	−0.487832	+	0.872938i	$0.662212\pi$
$180$	0	0
$181$	2243.52	0.921324	0.460662	−	0.887576i	$-0.347612\pi$
$181$	2243.52	0.921324	0.460662	+	0.887576i	$0.347612\pi$
$182$	0	0
$183$	−2948.09	−1.19087
$184$	1605.88	0.643408
$185$	0	0
$186$	1219.72	0.480830
$187$	566.904	0.221691
$188$	−110.965	−0.0430478
$189$	0	0
$190$	0	0
$191$	−415.099	−0.157254	−0.0786269	−	0.996904i	$-0.525054\pi$
$191$	−415.099	−0.157254	−0.0786269	+	0.996904i	$0.525054\pi$
$192$	−370.069	−0.139101
$193$	−1820.21	−0.678867	−0.339434	−	0.940630i	$-0.610235\pi$
$193$	−1820.21	−0.678867	−0.339434	+	0.940630i	$0.610235\pi$
$194$	−1543.10	−0.571074
$195$	0	0
$196$	0	0
$197$	−343.337	−0.124171	−0.0620856	−	0.998071i	$-0.519775\pi$
$197$	−343.337	−0.124171	−0.0620856	+	0.998071i	$0.519775\pi$
$198$	−263.017	−0.0944030
$199$	2432.41	0.866479	0.433239	−	0.901279i	$-0.357370\pi$
$199$	2432.41	0.866479	0.433239	+	0.901279i	$0.357370\pi$
$200$	0	0
$201$	−5674.87	−1.99142
$202$	−2930.82	−1.02085
$203$	0	0
$204$	641.639	0.220214
$205$	0	0
$206$	−872.625	−0.295139
$207$	1291.80	0.433749
$208$	850.069	0.283374
$209$	2924.90	0.968035
$210$	0	0
$211$	310.965	0.101459	0.0507293	−	0.998712i	$-0.483845\pi$
$211$	310.965	0.101459	0.0507293	+	0.998712i	$0.483845\pi$
$212$	−297.035	−0.0962284
$213$	836.742	0.269167
$214$	4368.66	1.39549
$215$	0	0
$216$	951.293	0.299663
$217$	0	0
$218$	−130.051	−0.0404045
$219$	4253.20	1.31235
$220$	0	0
$221$	−1473.88	−0.448615
$222$	−16.0513	−0.00485266
$223$	−2999.59	−0.900750	−0.450375	−	0.892840i	$-0.648710\pi$
$223$	−2999.59	−0.900750	−0.450375	+	0.892840i	$0.648710\pi$
$224$	0	0
$225$	0	0
$226$	1227.53	0.361302
$227$	−6145.54	−1.79689	−0.898445	−	0.439086i	$-0.855302\pi$
$227$	−6145.54	−1.79689	−0.898445	+	0.439086i	$0.855302\pi$
$228$	3310.48	0.961588
$229$	−1422.05	−0.410357	−0.205179	−	0.978725i	$-0.565778\pi$
$229$	−1422.05	−0.410357	−0.205179	+	0.978725i	$0.565778\pi$
$230$	0	0
$231$	0	0
$232$	904.656	0.256007
$233$	−3515.64	−0.988486	−0.494243	−	0.869324i	$-0.664555\pi$
$233$	−3515.64	−0.988486	−0.494243	+	0.869324i	$0.664555\pi$
$234$	683.810	0.191035
$235$	0	0
$236$	2661.45	0.734091
$237$	−844.620	−0.231493
$238$	0	0
$239$	−4332.94	−1.17270	−0.586348	−	0.810059i	$-0.699435\pi$
$239$	−4332.94	−1.17270	−0.586348	+	0.810059i	$0.699435\pi$
$240$	0	0
$241$	−6600.26	−1.76415	−0.882075	−	0.471109i	$-0.843854\pi$
$241$	−6600.26	−1.76415	−0.882075	+	0.471109i	$0.843854\pi$
$242$	−1826.79	−0.485251
$243$	1769.94	0.467250
$244$	2039.38	0.535074
$245$	0	0
$246$	2622.04	0.679573
$247$	−7604.36	−1.95892
$248$	−843.759	−0.216043
$249$	4119.08	1.04834
$250$	0	0
$251$	−3659.19	−0.920182	−0.460091	−	0.887872i	$-0.652183\pi$
$251$	−3659.19	−0.920182	−0.460091	+	0.887872i	$0.652183\pi$
$252$	0	0
$253$	−4102.09	−1.01935
$254$	4980.93	1.23044
$255$	0	0
$256$	256.000	0.0625000
$257$	3843.24	0.932820	0.466410	−	0.884569i	$-0.345547\pi$
$257$	3843.24	0.932820	0.466410	+	0.884569i	$0.345547\pi$
$258$	3105.78	0.749449
$259$	0	0
$260$	0	0
$261$	727.721	0.172585
$262$	−4778.40	−1.12676
$263$	462.377	0.108408	0.0542041	−	0.998530i	$-0.482738\pi$
$263$	462.377	0.108408	0.0542041	+	0.998530i	$0.482738\pi$
$264$	945.311	0.220378
$265$	0	0
$266$	0	0
$267$	3506.73	0.803778
$268$	3925.66	0.894769
$269$	−678.880	−0.153874	−0.0769368	−	0.997036i	$-0.524514\pi$
$269$	−678.880	−0.153874	−0.0769368	+	0.997036i	$0.524514\pi$
$270$	0	0
$271$	887.810	0.199006	0.0995030	−	0.995037i	$-0.468275\pi$
$271$	887.810	0.199006	0.0995030	+	0.995037i	$0.468275\pi$
$272$	−443.862	−0.0989451
$273$	0	0
$274$	−3477.73	−0.766778
$275$	0	0
$276$	−4642.86	−1.01256
$277$	5414.86	1.17454	0.587269	−	0.809392i	$-0.300203\pi$
$277$	5414.86	1.17454	0.587269	+	0.809392i	$0.300203\pi$
$278$	1394.66	0.300886
$279$	−678.735	−0.145644
$280$	0	0
$281$	−1424.10	−0.302331	−0.151165	−	0.988508i	$-0.548303\pi$
$281$	−1424.10	−0.302331	−0.151165	+	0.988508i	$0.548303\pi$
$282$	320.819	0.0677465
$283$	−4164.94	−0.874841	−0.437420	−	0.899257i	$-0.644108\pi$
$283$	−4164.94	−0.874841	−0.437420	+	0.899257i	$0.644108\pi$
$284$	−578.827	−0.120940
$285$	0	0
$286$	−2171.43	−0.448949
$287$	0	0
$288$	205.931	0.0421340
$289$	−4143.42	−0.843358
$290$	0	0
$291$	4461.37	0.898728
$292$	−2942.21	−0.589656
$293$	−6486.84	−1.29340	−0.646698	−	0.762746i	$-0.723851\pi$
$293$	−6486.84	−1.29340	−0.646698	+	0.762746i	$0.723851\pi$
$294$	0	0
$295$	0	0
$296$	11.1037	0.00218037
$297$	−2430.00	−0.474757
$298$	−3279.73	−0.637548
$299$	10664.9	2.06277
$300$	0	0
$301$	0	0
$302$	5658.61	1.07820
$303$	8473.48	1.60656
$304$	−2290.07	−0.432054
$305$	0	0
$306$	−357.050	−0.0667033
$307$	−4307.13	−0.800719	−0.400359	−	0.916358i	$-0.631115\pi$
$307$	−4307.13	−0.800719	−0.400359	+	0.916358i	$0.631115\pi$
$308$	0	0
$309$	2522.90	0.464475
$310$	0	0
$311$	−2646.85	−0.482602	−0.241301	−	0.970450i	$-0.577574\pi$
$311$	−2646.85	−0.482602	−0.241301	+	0.970450i	$0.577574\pi$
$312$	−2457.69	−0.445959
$313$	−634.766	−0.114630	−0.0573148	−	0.998356i	$-0.518254\pi$
$313$	−634.766	−0.114630	−0.0573148	+	0.998356i	$0.518254\pi$
$314$	5182.24	0.931372
$315$	0	0
$316$	584.276	0.104013
$317$	−7219.04	−1.27906	−0.639530	−	0.768766i	$-0.720871\pi$
$317$	−7219.04	−1.27906	−0.639530	+	0.768766i	$0.720871\pi$
$318$	858.776	0.151440
$319$	−2310.87	−0.405592
$320$	0	0
$321$	−12630.5	−2.19616
$322$	0	0
$323$	3970.60	0.683994
$324$	−3445.36	−0.590769
$325$	0	0
$326$	−6945.07	−1.17991
$327$	376.000	0.0635866
$328$	−1813.83	−0.305341
$329$	0	0
$330$	0	0
$331$	−8398.10	−1.39457	−0.697283	−	0.716796i	$-0.745608\pi$
$331$	−8398.10	−1.39457	−0.697283	+	0.716796i	$0.745608\pi$
$332$	−2849.42	−0.471031
$333$	8.93199	0.00146988
$334$	−1597.15	−0.261654
$335$	0	0
$336$	0	0
$337$	−11964.8	−1.93401	−0.967007	−	0.254748i	$-0.918007\pi$
$337$	−11964.8	−1.93401	−0.967007	+	0.254748i	$0.918007\pi$
$338$	1251.45	0.201390
$339$	−3549.00	−0.568600
$340$	0	0
$341$	2155.31	0.342278
$342$	−1842.17	−0.291267
$343$	0	0
$344$	−2148.47	−0.336737
$345$	0	0
$346$	−4397.78	−0.683312
$347$	9014.24	1.39455	0.697276	−	0.716803i	$-0.254395\pi$
$347$	9014.24	1.39455	0.697276	+	0.716803i	$0.254395\pi$
$348$	−2615.51	−0.402891
$349$	4439.73	0.680955	0.340477	−	0.940253i	$-0.389411\pi$
$349$	4439.73	0.680955	0.340477	+	0.940253i	$0.389411\pi$
$350$	0	0
$351$	6317.70	0.960723
$352$	−653.931	−0.0990188
$353$	8903.85	1.34250	0.671252	−	0.741229i	$-0.265757\pi$
$353$	8903.85	1.34250	0.671252	+	0.741229i	$0.265757\pi$
$354$	−7694.69	−1.15528
$355$	0	0
$356$	−2425.83	−0.361148
$357$	0	0
$358$	−4673.15	−0.689898
$359$	−10567.2	−1.55352	−0.776760	−	0.629797i	$-0.783138\pi$
$359$	−10567.2	−1.55352	−0.776760	+	0.629797i	$0.783138\pi$
$360$	0	0
$361$	13627.0	1.98673
$362$	4487.04	0.651475
$363$	5281.56	0.763664
$364$	0	0
$365$	0	0
$366$	−5896.18	−0.842072
$367$	3593.28	0.511084	0.255542	−	0.966798i	$-0.417746\pi$
$367$	3593.28	0.511084	0.255542	+	0.966798i	$0.417746\pi$
$368$	3211.76	0.454958
$369$	−1459.08	−0.205844
$370$	0	0
$371$	0	0
$372$	2439.45	0.339998
$373$	−8762.21	−1.21633	−0.608163	−	0.793812i	$-0.708093\pi$
$373$	−8762.21	−1.21633	−0.608163	+	0.793812i	$0.708093\pi$
$374$	1133.81	0.156759
$375$	0	0
$376$	−221.931	−0.0304394
$377$	6007.97	0.820759
$378$	0	0
$379$	−13221.6	−1.79195	−0.895974	−	0.444106i	$-0.853521\pi$
$379$	−13221.6	−1.79195	−0.895974	+	0.444106i	$0.853521\pi$
$380$	0	0
$381$	−14400.7	−1.93640
$382$	−830.197	−0.111195
$383$	7109.63	0.948525	0.474263	−	0.880383i	$-0.342715\pi$
$383$	7109.63	0.948525	0.474263	+	0.880383i	$0.342715\pi$
$384$	−740.138	−0.0983594
$385$	0	0
$386$	−3640.41	−0.480032
$387$	−1728.26	−0.227009
$388$	−3086.21	−0.403810
$389$	−5282.31	−0.688493	−0.344247	−	0.938879i	$-0.611866\pi$
$389$	−5282.31	−0.688493	−0.344247	+	0.938879i	$0.611866\pi$
$390$	0	0
$391$	−5568.66	−0.720254
$392$	0	0
$393$	13815.1	1.77323
$394$	−686.673	−0.0878023
$395$	0	0
$396$	−526.033	−0.0667530
$397$	−11937.1	−1.50908	−0.754538	−	0.656256i	$-0.772139\pi$
$397$	−11937.1	−1.50908	−0.754538	+	0.656256i	$0.772139\pi$
$398$	4864.83	0.612693
$399$	0	0
$400$	0	0
$401$	−4144.05	−0.516070	−0.258035	−	0.966136i	$-0.583075\pi$
$401$	−4144.05	−0.516070	−0.258035	+	0.966136i	$0.583075\pi$
$402$	−11349.7	−1.40814
$403$	−5603.54	−0.692636
$404$	−5861.64	−0.721850
$405$	0	0
$406$	0	0
$407$	−28.3634	−0.00345436
$408$	1283.28	0.155715
$409$	8288.13	1.00201	0.501004	−	0.865445i	$-0.332964\pi$
$409$	8288.13	1.00201	0.501004	+	0.865445i	$0.332964\pi$
$410$	0	0
$411$	10054.7	1.20672
$412$	−1745.25	−0.208695
$413$	0	0
$414$	2583.60	0.306707
$415$	0	0
$416$	1700.14	0.200375
$417$	−4032.20	−0.473520
$418$	5849.79	0.684504
$419$	−404.482	−0.0471604	−0.0235802	−	0.999722i	$-0.507507\pi$
$419$	−404.482	−0.0471604	−0.0235802	+	0.999722i	$0.507507\pi$
$420$	0	0
$421$	−1345.62	−0.155776	−0.0778879	−	0.996962i	$-0.524818\pi$
$421$	−1345.62	−0.155776	−0.0778879	+	0.996962i	$0.524818\pi$
$422$	621.931	0.0717420
$423$	−178.525	−0.0205205
$424$	−594.069	−0.0680437
$425$	0	0
$426$	1673.48	0.190330
$427$	0	0
$428$	8737.32	0.986763
$429$	6277.97	0.706534
$430$	0	0
$431$	−2877.79	−0.321620	−0.160810	−	0.986985i	$-0.551411\pi$
$431$	−2877.79	−0.321620	−0.160810	+	0.986985i	$0.551411\pi$
$432$	1902.59	0.211894
$433$	−702.525	−0.0779704	−0.0389852	−	0.999240i	$-0.512413\pi$
$433$	−702.525	−0.0779704	−0.0389852	+	0.999240i	$0.512413\pi$
$434$	0	0
$435$	0	0
$436$	−260.103	−0.0285703
$437$	−28731.1	−3.14506
$438$	8506.41	0.927972
$439$	11009.5	1.19694	0.598468	−	0.801146i	$-0.295776\pi$
$439$	11009.5	1.19694	0.598468	+	0.801146i	$0.295776\pi$
$440$	0	0
$441$	0	0
$442$	−2947.76	−0.317219
$443$	16642.9	1.78494	0.892471	−	0.451105i	$-0.148970\pi$
$443$	16642.9	1.78494	0.892471	+	0.451105i	$0.148970\pi$
$444$	−32.1026	−0.00343135
$445$	0	0
$446$	−5999.18	−0.636926
$447$	9482.23	1.00334
$448$	0	0
$449$	−8698.08	−0.914226	−0.457113	−	0.889409i	$-0.651117\pi$
$449$	−8698.08	−0.914226	−0.457113	+	0.889409i	$0.651117\pi$
$450$	0	0
$451$	4633.27	0.483753
$452$	2455.07	0.255479
$453$	−16360.0	−1.69682
$454$	−12291.1	−1.27059
$455$	0	0
$456$	6620.97	0.679946
$457$	−6488.46	−0.664152	−0.332076	−	0.943253i	$-0.607749\pi$
$457$	−6488.46	−0.664152	−0.332076	+	0.943253i	$0.607749\pi$
$458$	−2844.10	−0.290166
$459$	−3298.77	−0.335454
$460$	0	0
$461$	8797.74	0.888832	0.444416	−	0.895820i	$-0.353411\pi$
$461$	8797.74	0.888832	0.444416	+	0.895820i	$0.353411\pi$
$462$	0	0
$463$	538.084	0.0540106	0.0270053	−	0.999635i	$-0.491403\pi$
$463$	538.084	0.0540106	0.0270053	+	0.999635i	$0.491403\pi$
$464$	1809.31	0.181024
$465$	0	0
$466$	−7031.28	−0.698965
$467$	9566.10	0.947893	0.473947	−	0.880554i	$-0.342829\pi$
$467$	9566.10	0.947893	0.473947	+	0.880554i	$0.342829\pi$
$468$	1367.62	0.135082
$469$	0	0
$470$	0	0
$471$	−14982.7	−1.46575
$472$	5322.90	0.519081
$473$	5488.08	0.533493
$474$	−1689.24	−0.163691
$475$	0	0
$476$	0	0
$477$	−477.880	−0.0458713
$478$	−8665.87	−0.829221
$479$	4111.74	0.392214	0.196107	−	0.980583i	$-0.437170\pi$
$479$	4111.74	0.392214	0.196107	+	0.980583i	$0.437170\pi$
$480$	0	0
$481$	73.7414	0.00699026
$482$	−13200.5	−1.24744
$483$	0	0
$484$	−3653.59	−0.343124
$485$	0	0
$486$	3539.88	0.330396
$487$	3571.73	0.332342	0.166171	−	0.986097i	$-0.446860\pi$
$487$	3571.73	0.332342	0.166171	+	0.986097i	$0.446860\pi$
$488$	4078.76	0.378354
$489$	20079.3	1.85689
$490$	0	0
$491$	21626.1	1.98772	0.993861	−	0.110634i	$-0.0352881\pi$
$491$	21626.1	1.98772	0.993861	+	0.110634i	$0.0352881\pi$
$492$	5244.08	0.480531
$493$	−3137.05	−0.286583
$494$	−15208.7	−1.38517
$495$	0	0
$496$	−1687.52	−0.152766
$497$	0	0
$498$	8238.15	0.741286
$499$	170.407	0.0152875	0.00764377	−	0.999971i	$-0.497567\pi$
$499$	170.407	0.0152875	0.00764377	+	0.999971i	$0.497567\pi$
$500$	0	0
$501$	4617.63	0.411777
$502$	−7318.37	−0.650667
$503$	57.5343	0.00510005	0.00255003	−	0.999997i	$-0.499188\pi$
$503$	57.5343	0.00510005	0.00255003	+	0.999997i	$0.499188\pi$
$504$	0	0
$505$	0	0
$506$	−8204.17	−0.720791
$507$	−3618.15	−0.316938
$508$	9961.86	0.870052
$509$	−2127.18	−0.185237	−0.0926187	−	0.995702i	$-0.529524\pi$
$509$	−2127.18	−0.185237	−0.0926187	+	0.995702i	$0.529524\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−17019.7	−1.46480
$514$	7686.48	0.659604
$515$	0	0
$516$	6211.57	0.529940
$517$	566.904	0.0482252
$518$	0	0
$519$	12714.7	1.07536
$520$	0	0
$521$	6823.57	0.573793	0.286896	−	0.957962i	$-0.407376\pi$
$521$	6823.57	0.573793	0.286896	+	0.957962i	$0.407376\pi$
$522$	1455.44	0.122036
$523$	−10746.7	−0.898513	−0.449257	−	0.893403i	$-0.648311\pi$
$523$	−10746.7	−0.898513	−0.449257	+	0.893403i	$0.648311\pi$
$524$	−9556.79	−0.796738
$525$	0	0
$526$	924.753	0.0766562
$527$	2925.88	0.241847
$528$	1890.62	0.155831
$529$	28127.5	2.31179
$530$	0	0
$531$	4281.83	0.349935
$532$	0	0
$533$	−12045.9	−0.978925
$534$	7013.47	0.568357
$535$	0	0
$536$	7851.33	0.632697
$537$	13510.8	1.08573
$538$	−1357.76	−0.108805
$539$	0	0
$540$	0	0
$541$	−14896.8	−1.18385	−0.591926	−	0.805992i	$-0.701632\pi$
$541$	−14896.8	−1.18385	−0.591926	+	0.805992i	$0.701632\pi$
$542$	1775.62	0.140719
$543$	−12972.8	−1.02526
$544$	−887.724	−0.0699648
$545$	0	0
$546$	0	0
$547$	−7150.91	−0.558960	−0.279480	−	0.960152i	$-0.590162\pi$
$547$	−7150.91	−0.558960	−0.279480	+	0.960152i	$0.590162\pi$
$548$	−6955.45	−0.542194
$549$	3281.03	0.255065
$550$	0	0
$551$	−16185.3	−1.25140
$552$	−9285.73	−0.715990
$553$	0	0
$554$	10829.7	0.830524
$555$	0	0
$556$	2789.33	0.212759
$557$	12197.3	0.927856	0.463928	−	0.885873i	$-0.346440\pi$
$557$	12197.3	0.927856	0.463928	+	0.885873i	$0.346440\pi$
$558$	−1357.47	−0.102986
$559$	−14268.3	−1.07958
$560$	0	0
$561$	−3278.03	−0.246699
$562$	−2848.21	−0.213780
$563$	6040.23	0.452159	0.226079	−	0.974109i	$-0.427409\pi$
$563$	6040.23	0.452159	0.226079	+	0.974109i	$0.427409\pi$
$564$	641.639	0.0479040
$565$	0	0
$566$	−8329.88	−0.618606
$567$	0	0
$568$	−1157.65	−0.0855178
$569$	5156.95	0.379948	0.189974	−	0.981789i	$-0.439160\pi$
$569$	5156.95	0.379948	0.189974	+	0.981789i	$0.439160\pi$
$570$	0	0
$571$	−13464.2	−0.986795	−0.493398	−	0.869804i	$-0.664245\pi$
$571$	−13464.2	−0.986795	−0.493398	+	0.869804i	$0.664245\pi$
$572$	−4342.86	−0.317455
$573$	2400.24	0.174994
$574$	0	0
$575$	0	0
$576$	411.862	0.0297932
$577$	5180.09	0.373744	0.186872	−	0.982384i	$-0.440165\pi$
$577$	5180.09	0.373744	0.186872	+	0.982384i	$0.440165\pi$
$578$	−8286.83	−0.596344
$579$	10525.0	0.755450
$580$	0	0
$581$	0	0
$582$	8922.73	0.635497
$583$	1517.50	0.107802
$584$	−5884.41	−0.416950
$585$	0	0
$586$	−12973.7	−0.914569
$587$	1803.06	0.126781	0.0633905	−	0.997989i	$-0.479809\pi$
$587$	1803.06	0.126781	0.0633905	+	0.997989i	$0.479809\pi$
$588$	0	0
$589$	15095.8	1.05605
$590$	0	0
$591$	1985.29	0.138179
$592$	22.2074	0.00154175
$593$	17231.6	1.19328	0.596642	−	0.802507i	$-0.296501\pi$
$593$	17231.6	1.19328	0.596642	+	0.802507i	$0.296501\pi$
$594$	−4860.00	−0.335704
$595$	0	0
$596$	−6559.45	−0.450815
$597$	−14065.0	−0.964226
$598$	21329.8	1.45860
$599$	−2931.67	−0.199975	−0.0999874	−	0.994989i	$-0.531880\pi$
$599$	−2931.67	−0.199975	−0.0999874	+	0.994989i	$0.531880\pi$
$600$	0	0
$601$	−5877.40	−0.398909	−0.199454	−	0.979907i	$-0.563917\pi$
$601$	−5877.40	−0.398909	−0.199454	+	0.979907i	$0.563917\pi$
$602$	0	0
$603$	6315.75	0.426529
$604$	11317.2	0.762403
$605$	0	0
$606$	16947.0	1.13601
$607$	−22425.3	−1.49953	−0.749765	−	0.661705i	$-0.769833\pi$
$607$	−22425.3	−1.49953	−0.749765	+	0.661705i	$0.769833\pi$
$608$	−4580.14	−0.305508
$609$	0	0
$610$	0	0
$611$	−1473.88	−0.0975888
$612$	−714.100	−0.0471663
$613$	−26204.0	−1.72654	−0.863272	−	0.504739i	$-0.831589\pi$
$613$	−26204.0	−1.72654	−0.863272	+	0.504739i	$0.831589\pi$
$614$	−8614.25	−0.566194
$615$	0	0
$616$	0	0
$617$	9424.07	0.614909	0.307454	−	0.951563i	$-0.400523\pi$
$617$	9424.07	0.614909	0.307454	+	0.951563i	$0.400523\pi$
$618$	5045.80	0.328434
$619$	−10551.8	−0.685157	−0.342578	−	0.939489i	$-0.611300\pi$
$619$	−10551.8	−0.685157	−0.342578	+	0.939489i	$0.611300\pi$
$620$	0	0
$621$	23869.7	1.54245
$622$	−5293.71	−0.341251
$623$	0	0
$624$	−4915.38	−0.315341
$625$	0	0
$626$	−1269.53	−0.0810554
$627$	−16912.7	−1.07724
$628$	10364.5	0.658580
$629$	−38.5039	−0.00244078
$630$	0	0
$631$	4238.56	0.267408	0.133704	−	0.991021i	$-0.457313\pi$
$631$	4238.56	0.267408	0.133704	+	0.991021i	$0.457313\pi$
$632$	1168.55	0.0735483
$633$	−1798.10	−0.112904
$634$	−14438.1	−0.904432
$635$	0	0
$636$	1717.55	0.107084
$637$	0	0
$638$	−4621.74	−0.286797
$639$	−931.237	−0.0576513
$640$	0	0
$641$	−18319.7	−1.12883	−0.564417	−	0.825490i	$-0.690899\pi$
$641$	−18319.7	−1.12883	−0.564417	+	0.825490i	$0.690899\pi$
$642$	−25261.0	−1.55292
$643$	3583.89	0.219806	0.109903	−	0.993942i	$-0.464946\pi$
$643$	3583.89	0.219806	0.109903	+	0.993942i	$0.464946\pi$
$644$	0	0
$645$	0	0
$646$	7941.20	0.483657
$647$	7974.68	0.484570	0.242285	−	0.970205i	$-0.422103\pi$
$647$	7974.68	0.484570	0.242285	+	0.970205i	$0.422103\pi$
$648$	−6890.72	−0.417736
$649$	−13596.9	−0.822381
$650$	0	0
$651$	0	0
$652$	−13890.1	−0.834325
$653$	−20010.0	−1.19916	−0.599579	−	0.800315i	$-0.704665\pi$
$653$	−20010.0	−1.19916	−0.599579	+	0.800315i	$0.704665\pi$
$654$	751.999	0.0449625
$655$	0	0
$656$	−3627.66	−0.215909
$657$	−4733.53	−0.281084
$658$	0	0
$659$	−12398.6	−0.732897	−0.366448	−	0.930438i	$-0.619426\pi$
$659$	−12398.6	−0.732897	−0.366448	+	0.930438i	$0.619426\pi$
$660$	0	0
$661$	−25974.6	−1.52844	−0.764218	−	0.644958i	$-0.776875\pi$
$661$	−25974.6	−1.52844	−0.764218	+	0.644958i	$0.776875\pi$
$662$	−16796.2	−0.986107
$663$	8522.46	0.499223
$664$	−5698.85	−0.333070
$665$	0	0
$666$	17.8640	0.00103936
$667$	22699.5	1.31773
$668$	−3194.30	−0.185017
$669$	17344.6	1.00236
$670$	0	0
$671$	−10418.9	−0.599427
$672$	0	0
$673$	23005.5	1.31768	0.658839	−	0.752284i	$-0.271048\pi$
$673$	23005.5	1.31768	0.658839	+	0.752284i	$0.271048\pi$
$674$	−23929.6	−1.36755
$675$	0	0
$676$	2502.90	0.142404
$677$	12121.1	0.688111	0.344055	−	0.938949i	$-0.388199\pi$
$677$	12121.1	0.688111	0.344055	+	0.938949i	$0.388199\pi$
$678$	−7098.01	−0.402061
$679$	0	0
$680$	0	0
$681$	35535.6	1.99960
$682$	4310.63	0.242027
$683$	14223.1	0.796823	0.398412	−	0.917207i	$-0.369562\pi$
$683$	14223.1	0.796823	0.398412	+	0.917207i	$0.369562\pi$
$684$	−3684.34	−0.205957
$685$	0	0
$686$	0	0
$687$	8222.77	0.456650
$688$	−4296.93	−0.238109
$689$	−3945.31	−0.218149
$690$	0	0
$691$	2442.82	0.134485	0.0672425	−	0.997737i	$-0.478580\pi$
$691$	2442.82	0.134485	0.0672425	+	0.997737i	$0.478580\pi$
$692$	−8795.55	−0.483175
$693$	0	0
$694$	18028.5	0.986097
$695$	0	0
$696$	−5231.02	−0.284887
$697$	6289.76	0.341810
$698$	8879.46	0.481508
$699$	20328.6	1.10000
$700$	0	0
$701$	5484.01	0.295475	0.147738	−	0.989027i	$-0.452801\pi$
$701$	5484.01	0.295475	0.147738	+	0.989027i	$0.452801\pi$
$702$	12635.4	0.679333
$703$	−198.658	−0.0106579
$704$	−1307.86	−0.0700169
$705$	0	0
$706$	17807.7	0.949294
$707$	0	0
$708$	−15389.4	−0.816904
$709$	10856.6	0.575074	0.287537	−	0.957770i	$-0.407164\pi$
$709$	10856.6	0.575074	0.287537	+	0.957770i	$0.407164\pi$
$710$	0	0
$711$	940.004	0.0495822
$712$	−4851.66	−0.255370
$713$	−21171.5	−1.11203
$714$	0	0
$715$	0	0
$716$	−9346.29	−0.487832
$717$	25054.5	1.30499
$718$	−21134.3	−1.09850
$719$	−10679.7	−0.553943	−0.276971	−	0.960878i	$-0.589331\pi$
$719$	−10679.7	−0.553943	−0.276971	+	0.960878i	$0.589331\pi$
$720$	0	0
$721$	0	0
$722$	27254.0	1.40483
$723$	38164.9	1.96316
$724$	8974.09	0.460662
$725$	0	0
$726$	10563.1	0.539992
$727$	18540.2	0.945831	0.472916	−	0.881108i	$-0.343202\pi$
$727$	18540.2	0.945831	0.472916	+	0.881108i	$0.343202\pi$
$728$	0	0
$729$	13021.8	0.661577
$730$	0	0
$731$	7450.17	0.376956
$732$	−11792.4	−0.595435
$733$	14239.0	0.717501	0.358750	−	0.933434i	$-0.383203\pi$
$733$	14239.0	0.717501	0.358750	+	0.933434i	$0.383203\pi$
$734$	7186.57	0.361391
$735$	0	0
$736$	6423.52	0.321704
$737$	−20055.6	−1.00238
$738$	−2918.15	−0.145554
$739$	4765.10	0.237195	0.118597	−	0.992942i	$-0.462160\pi$
$739$	4765.10	0.237195	0.118597	+	0.992942i	$0.462160\pi$
$740$	0	0
$741$	43970.9	2.17991
$742$	0	0
$743$	−13720.9	−0.677485	−0.338743	−	0.940879i	$-0.610002\pi$
$743$	−13720.9	−0.677485	−0.338743	+	0.940879i	$0.610002\pi$
$744$	4878.89	0.240415
$745$	0	0
$746$	−17524.4	−0.860073
$747$	−4584.25	−0.224537
$748$	2267.62	0.110845
$749$	0	0
$750$	0	0
$751$	25957.9	1.26127	0.630636	−	0.776079i	$-0.282794\pi$
$751$	25957.9	1.26127	0.630636	+	0.776079i	$0.282794\pi$
$752$	−443.862	−0.0215239
$753$	21158.6	1.02399
$754$	12015.9	0.580364
$755$	0	0
$756$	0	0
$757$	7731.70	0.371220	0.185610	−	0.982624i	$-0.440574\pi$
$757$	7731.70	0.371220	0.185610	+	0.982624i	$0.440574\pi$
$758$	−26443.2	−1.26710
$759$	23719.6	1.13434
$760$	0	0
$761$	24447.7	1.16456	0.582278	−	0.812989i	$-0.302161\pi$
$761$	24447.7	1.16456	0.582278	+	0.812989i	$0.302161\pi$
$762$	−28801.4	−1.36924
$763$	0	0
$764$	−1660.39	−0.0786269
$765$	0	0
$766$	14219.3	0.670709
$767$	35350.2	1.66418
$768$	−1480.28	−0.0695506
$769$	38807.0	1.81979	0.909893	−	0.414843i	$-0.136164\pi$
$769$	38807.0	1.81979	0.909893	+	0.414843i	$0.136164\pi$
$770$	0	0
$771$	−22222.9	−1.03805
$772$	−7280.83	−0.339434
$773$	−35375.7	−1.64602	−0.823012	−	0.568023i	$-0.807708\pi$
$773$	−35375.7	−1.64602	−0.823012	+	0.568023i	$0.807708\pi$
$774$	−3456.53	−0.160520
$775$	0	0
$776$	−6172.41	−0.285537
$777$	0	0
$778$	−10564.6	−0.486838
$779$	32451.5	1.49255
$780$	0	0
$781$	2957.13	0.135486
$782$	−11137.3	−0.509296
$783$	13446.8	0.613727
$784$	0	0
$785$	0	0
$786$	27630.3	1.25387
$787$	20972.1	0.949904	0.474952	−	0.880012i	$-0.342465\pi$
$787$	20972.1	0.949904	0.474952	+	0.880012i	$0.342465\pi$
$788$	−1373.35	−0.0620856
$789$	−2673.61	−0.120638
$790$	0	0
$791$	0	0
$792$	−1052.07	−0.0472015
$793$	27087.7	1.21301
$794$	−23874.1	−1.06708
$795$	0	0
$796$	9729.66	0.433239
$797$	873.157	0.0388065	0.0194033	−	0.999812i	$-0.493823\pi$
$797$	873.157	0.0388065	0.0194033	+	0.999812i	$0.493823\pi$
$798$	0	0
$799$	769.583	0.0340750
$800$	0	0
$801$	−3902.76	−0.172156
$802$	−8288.10	−0.364917
$803$	15031.3	0.660575
$804$	−22699.5	−0.995708
$805$	0	0
$806$	−11207.1	−0.489768
$807$	3925.51	0.171232
$808$	−11723.3	−0.510425
$809$	1871.57	0.0813359	0.0406679	−	0.999173i	$-0.487051\pi$
$809$	1871.57	0.0813359	0.0406679	+	0.999173i	$0.487051\pi$
$810$	0	0
$811$	−39990.1	−1.73150	−0.865748	−	0.500480i	$-0.833157\pi$
$811$	−39990.1	−1.73150	−0.865748	+	0.500480i	$0.833157\pi$
$812$	0	0
$813$	−5133.61	−0.221456
$814$	−56.7269	−0.00244260
$815$	0	0
$816$	2566.56	0.110107
$817$	38438.6	1.64602
$818$	16576.3	0.708527
$819$	0	0
$820$	0	0
$821$	−27502.1	−1.16910	−0.584550	−	0.811357i	$-0.698729\pi$
$821$	−27502.1	−1.16910	−0.584550	+	0.811357i	$0.698729\pi$
$822$	20109.4	0.853278
$823$	1303.62	0.0552142	0.0276071	−	0.999619i	$-0.491211\pi$
$823$	1303.62	0.0552142	0.0276071	+	0.999619i	$0.491211\pi$
$824$	−3490.50	−0.147570
$825$	0	0
$826$	0	0
$827$	36754.0	1.54542	0.772709	−	0.634760i	$-0.218901\pi$
$827$	36754.0	1.54542	0.772709	+	0.634760i	$0.218901\pi$
$828$	5167.19	0.216875
$829$	21461.2	0.899128	0.449564	−	0.893248i	$-0.351579\pi$
$829$	21461.2	0.899128	0.449564	+	0.893248i	$0.351579\pi$
$830$	0	0
$831$	−31310.5	−1.30704
$832$	3400.28	0.141687
$833$	0	0
$834$	−8064.40	−0.334829
$835$	0	0
$836$	11699.6	0.484017
$837$	−12541.6	−0.517922
$838$	−808.963	−0.0333475
$839$	11299.8	0.464973	0.232486	−	0.972600i	$-0.425314\pi$
$839$	11299.8	0.464973	0.232486	+	0.972600i	$0.425314\pi$
$840$	0	0
$841$	−11601.5	−0.475685
$842$	−2691.24	−0.110150
$843$	8234.64	0.336437
$844$	1243.86	0.0507293
$845$	0	0
$846$	−357.050	−0.0145102
$847$	0	0
$848$	−1188.14	−0.0481142
$849$	24083.1	0.973532
$850$	0	0
$851$	278.612	0.0112229
$852$	3346.97	0.134584
$853$	−13188.1	−0.529370	−0.264685	−	0.964335i	$-0.585268\pi$
$853$	−13188.1	−0.529370	−0.264685	+	0.964335i	$0.585268\pi$
$854$	0	0
$855$	0	0
$856$	17474.6	0.697746
$857$	13279.5	0.529309	0.264655	−	0.964343i	$-0.414742\pi$
$857$	13279.5	0.529309	0.264655	+	0.964343i	$0.414742\pi$
$858$	12555.9	0.499595
$859$	4066.00	0.161502	0.0807509	−	0.996734i	$-0.474268\pi$
$859$	4066.00	0.161502	0.0807509	+	0.996734i	$0.474268\pi$
$860$	0	0
$861$	0	0
$862$	−5755.58	−0.227420
$863$	−18299.4	−0.721807	−0.360903	−	0.932603i	$-0.617532\pi$
$863$	−18299.4	−0.721807	−0.360903	+	0.932603i	$0.617532\pi$
$864$	3805.17	0.149832
$865$	0	0
$866$	−1405.05	−0.0551334
$867$	23958.6	0.938497
$868$	0	0
$869$	−2984.97	−0.116523
$870$	0	0
$871$	52142.0	2.02843
$872$	−520.205	−0.0202023
$873$	−4965.20	−0.192493
$874$	−57462.1	−2.22390
$875$	0	0
$876$	17012.8	0.656175
$877$	14113.3	0.543411	0.271705	−	0.962381i	$-0.412412\pi$
$877$	14113.3	0.543411	0.271705	+	0.962381i	$0.412412\pi$
$878$	22019.0	0.846362
$879$	37509.0	1.43930
$880$	0	0
$881$	14794.7	0.565772	0.282886	−	0.959154i	$-0.408708\pi$
$881$	14794.7	0.565772	0.282886	+	0.959154i	$0.408708\pi$
$882$	0	0
$883$	10477.5	0.399314	0.199657	−	0.979866i	$-0.436017\pi$
$883$	10477.5	0.399314	0.199657	+	0.979866i	$0.436017\pi$
$884$	−5895.52	−0.224307
$885$	0	0
$886$	33285.8	1.26214
$887$	−857.232	−0.0324499	−0.0162249	−	0.999868i	$-0.505165\pi$
$887$	−857.232	−0.0324499	−0.0162249	+	0.999868i	$0.505165\pi$
$888$	−64.2051	−0.00242633
$889$	0	0
$890$	0	0
$891$	17601.8	0.661821
$892$	−11998.4	−0.450375
$893$	3970.60	0.148792
$894$	18964.5	0.709470
$895$	0	0
$896$	0	0
$897$	−61668.0	−2.29547
$898$	−17396.2	−0.646456
$899$	−11926.7	−0.442468
$900$	0	0
$901$	2060.04	0.0761706
$902$	9266.55	0.342065
$903$	0	0
$904$	4910.14	0.180651
$905$	0	0
$906$	−32720.0	−1.19983
$907$	52444.7	1.91995	0.959977	−	0.280078i	$-0.0903602\pi$
$907$	52444.7	1.91995	0.959977	+	0.280078i	$0.0903602\pi$
$908$	−24582.2	−0.898445
$909$	−9430.41	−0.344100
$910$	0	0
$911$	−2743.07	−0.0997606	−0.0498803	−	0.998755i	$-0.515884\pi$
$911$	−2743.07	−0.0997606	−0.0498803	+	0.998755i	$0.515884\pi$
$912$	13241.9	0.480794
$913$	14557.2	0.527683
$914$	−12976.9	−0.469626
$915$	0	0
$916$	−5688.21	−0.205179
$917$	0	0
$918$	−6597.54	−0.237202
$919$	−53455.6	−1.91876	−0.959378	−	0.282124i	$-0.908961\pi$
$919$	−53455.6	−1.91876	−0.959378	+	0.282124i	$0.908961\pi$
$920$	0	0
$921$	24905.2	0.891048
$922$	17595.5	0.628499
$923$	−7688.17	−0.274170
$924$	0	0
$925$	0	0
$926$	1076.17	0.0381913
$927$	−2807.82	−0.0994831
$928$	3618.62	0.128003
$929$	8558.80	0.302266	0.151133	−	0.988513i	$-0.451708\pi$
$929$	8558.80	0.302266	0.151133	+	0.988513i	$0.451708\pi$
$930$	0	0
$931$	0	0
$932$	−14062.6	−0.494243
$933$	15305.0	0.537044
$934$	19132.2	0.670262
$935$	0	0
$936$	2735.24	0.0955173
$937$	13653.3	0.476022	0.238011	−	0.971262i	$-0.423505\pi$
$937$	13653.3	0.476022	0.238011	+	0.971262i	$0.423505\pi$
$938$	0	0
$939$	3670.43	0.127561
$940$	0	0
$941$	5269.56	0.182553	0.0912767	−	0.995826i	$-0.470905\pi$
$941$	5269.56	0.182553	0.0912767	+	0.995826i	$0.470905\pi$
$942$	−29965.4	−1.03644
$943$	−45512.3	−1.57167
$944$	10645.8	0.367046
$945$	0	0
$946$	10976.2	0.377236
$947$	39151.7	1.34346	0.671731	−	0.740795i	$-0.265551\pi$
$947$	39151.7	1.34346	0.671731	+	0.740795i	$0.265551\pi$
$948$	−3378.48	−0.115747
$949$	−39079.4	−1.33674
$950$	0	0
$951$	41742.9	1.42335
$952$	0	0
$953$	−20474.4	−0.695941	−0.347971	−	0.937505i	$-0.613129\pi$
$953$	−20474.4	−0.695941	−0.347971	+	0.937505i	$0.613129\pi$
$954$	−955.759	−0.0324359
$955$	0	0
$956$	−17331.7	−0.586348
$957$	13362.2	0.451347
$958$	8223.49	0.277337
$959$	0	0
$960$	0	0
$961$	−18667.1	−0.626602
$962$	147.483	0.00494286
$963$	14056.9	0.470382
$964$	−26401.0	−0.882075
$965$	0	0
$966$	0	0
$967$	−2823.17	−0.0938852	−0.0469426	−	0.998898i	$-0.514948\pi$
$967$	−2823.17	−0.0938852	−0.0469426	+	0.998898i	$0.514948\pi$
$968$	−7307.18	−0.242626
$969$	−22959.3	−0.761156
$970$	0	0
$971$	−8367.21	−0.276536	−0.138268	−	0.990395i	$-0.544154\pi$
$971$	−8367.21	−0.276536	−0.138268	+	0.990395i	$0.544154\pi$
$972$	7079.76	0.233625
$973$	0	0
$974$	7143.45	0.235001
$975$	0	0
$976$	8157.52	0.267537
$977$	20786.2	0.680665	0.340332	−	0.940305i	$-0.389460\pi$
$977$	20786.2	0.680665	0.340332	+	0.940305i	$0.389460\pi$
$978$	40158.7	1.31302
$979$	12393.2	0.404583
$980$	0	0
$981$	−418.462	−0.0136192
$982$	43252.2	1.40553
$983$	12744.7	0.413523	0.206762	−	0.978391i	$-0.433708\pi$
$983$	12744.7	0.413523	0.206762	+	0.978391i	$0.433708\pi$
$984$	10488.2	0.339787
$985$	0	0
$986$	−6274.09	−0.202645
$987$	0	0
$988$	−30417.5	−0.979462
$989$	−53909.0	−1.73327
$990$	0	0
$991$	−7072.58	−0.226708	−0.113354	−	0.993555i	$-0.536159\pi$
$991$	−7072.58	−0.226708	−0.113354	+	0.993555i	$0.536159\pi$
$992$	−3375.04	−0.108022
$993$	48560.6	1.55189
$994$	0	0
$995$	0	0
$996$	16476.3	0.524168
$997$	−13372.8	−0.424795	−0.212398	−	0.977183i	$-0.568127\pi$
$997$	−13372.8	−0.424795	−0.212398	+	0.977183i	$0.568127\pi$
$998$	340.815	0.0108099
$999$	165.045	0.00522701

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.bz.1.1		2
5.4	even	2		490.4.a.r.1.2		2
7.2	even	3		350.4.e.h.151.2		4
7.4	even	3		350.4.e.h.51.2		4
7.6	odd	2		2450.4.a.bv.1.2		2
35.2	odd	12		350.4.j.g.249.3		8
35.4	even	6		70.4.e.d.51.1	yes	4
35.9	even	6		70.4.e.d.11.1	✓	4
35.18	odd	12		350.4.j.g.149.3		8
35.19	odd	6		490.4.e.u.361.2		4
35.23	odd	12		350.4.j.g.249.2		8
35.24	odd	6		490.4.e.u.471.2		4
35.32	odd	12		350.4.j.g.149.2		8
35.34	odd	2		490.4.a.t.1.1		2
105.44	odd	6		630.4.k.l.361.1		4
105.74	odd	6		630.4.k.l.541.1		4
140.39	odd	6		560.4.q.j.401.2		4
140.79	odd	6		560.4.q.j.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.e.d.11.1	✓	4	35.9	even	6
70.4.e.d.51.1	yes	4	35.4	even	6
350.4.e.h.51.2		4	7.4	even	3
350.4.e.h.151.2		4	7.2	even	3
350.4.j.g.149.2		8	35.32	odd	12
350.4.j.g.149.3		8	35.18	odd	12
350.4.j.g.249.2		8	35.23	odd	12
350.4.j.g.249.3		8	35.2	odd	12
490.4.a.r.1.2		2	5.4	even	2
490.4.a.t.1.1		2	35.34	odd	2
490.4.e.u.361.2		4	35.19	odd	6
490.4.e.u.471.2		4	35.24	odd	6
560.4.q.j.81.2		4	140.79	odd	6
560.4.q.j.401.2		4	140.39	odd	6
630.4.k.l.361.1		4	105.44	odd	6
630.4.k.l.541.1		4	105.74	odd	6
2450.4.a.bv.1.2		2	7.6	odd	2
2450.4.a.bz.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -5.78233 q^{3} +4.00000 q^{4} -11.5647 q^{6} +8.00000 q^{8} +6.43534 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -5.78233 q^{3} +4.00000 q^{4} -11.5647 q^{6} +8.00000 q^{8} +6.43534 q^{9} -20.4353 q^{11} -23.1293 q^{12} +53.1293 q^{13} +16.0000 q^{16} -27.7414 q^{17} +12.8707 q^{18} -143.129 q^{19} -40.8707 q^{22} +200.735 q^{23} -46.2586 q^{24} +106.259 q^{26} +118.912 q^{27} +113.082 q^{29} -105.470 q^{31} +32.0000 q^{32} +118.164 q^{33} -55.4827 q^{34} +25.7414 q^{36} +1.38796 q^{37} -286.259 q^{38} -307.211 q^{39} -226.729 q^{41} -268.558 q^{43} -81.7414 q^{44} +401.470 q^{46} -27.7414 q^{47} -92.5173 q^{48} +160.410 q^{51} +212.517 q^{52} -74.2586 q^{53} +237.823 q^{54} +827.621 q^{57} +226.164 q^{58} +665.362 q^{59} +509.845 q^{61} -210.940 q^{62} +64.0000 q^{64} +236.328 q^{66} +981.416 q^{67} -110.965 q^{68} -1160.72 q^{69} -144.707 q^{71} +51.4827 q^{72} -735.552 q^{73} +2.77592 q^{74} -572.517 q^{76} -614.423 q^{78} +146.069 q^{79} -861.341 q^{81} -453.457 q^{82} -712.356 q^{83} -537.116 q^{86} -653.877 q^{87} -163.483 q^{88} -606.457 q^{89} +802.940 q^{92} +609.862 q^{93} -55.4827 q^{94} -185.035 q^{96} -771.552 q^{97} -131.508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{8} + 40 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^6 + 16 * q^8 + 40 * q^9	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{8} + 40 q^{9} - 68 q^{11} + 8 q^{12} + 52 q^{13} + 32 q^{16} - 164 q^{17} + 80 q^{18} - 232 q^{19} - 136 q^{22} + 198 q^{23} + 16 q^{24} + 104 q^{26} + 170 q^{27} - 18 q^{29} + 196 q^{31} + 64 q^{32} - 252 q^{33} - 328 q^{34} + 160 q^{36} - 160 q^{37} - 464 q^{38} - 316 q^{39} + 62 q^{41} - 198 q^{43} - 272 q^{44} + 396 q^{46} - 164 q^{47} + 32 q^{48} - 900 q^{51} + 208 q^{52} - 40 q^{53} + 340 q^{54} + 136 q^{57} - 36 q^{58} - 80 q^{59} - 174 q^{61} + 392 q^{62} + 128 q^{64} - 504 q^{66} + 1054 q^{67} - 656 q^{68} - 1182 q^{69} - 832 q^{71} + 320 q^{72} - 820 q^{73} - 320 q^{74} - 928 q^{76} - 632 q^{78} - 576 q^{79} - 1370 q^{81} + 124 q^{82} + 298 q^{83} - 396 q^{86} - 1674 q^{87} - 544 q^{88} - 182 q^{89} + 792 q^{92} + 2956 q^{93} - 328 q^{94} + 64 q^{96} - 892 q^{97} - 1728 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^6 + 16 * q^8 + 40 * q^9 - 68 * q^11 + 8 * q^12 + 52 * q^13 + 32 * q^16 - 164 * q^17 + 80 * q^18 - 232 * q^19 - 136 * q^22 + 198 * q^23 + 16 * q^24 + 104 * q^26 + 170 * q^27 - 18 * q^29 + 196 * q^31 + 64 * q^32 - 252 * q^33 - 328 * q^34 + 160 * q^36 - 160 * q^37 - 464 * q^38 - 316 * q^39 + 62 * q^41 - 198 * q^43 - 272 * q^44 + 396 * q^46 - 164 * q^47 + 32 * q^48 - 900 * q^51 + 208 * q^52 - 40 * q^53 + 340 * q^54 + 136 * q^57 - 36 * q^58 - 80 * q^59 - 174 * q^61 + 392 * q^62 + 128 * q^64 - 504 * q^66 + 1054 * q^67 - 656 * q^68 - 1182 * q^69 - 832 * q^71 + 320 * q^72 - 820 * q^73 - 320 * q^74 - 928 * q^76 - 632 * q^78 - 576 * q^79 - 1370 * q^81 + 124 * q^82 + 298 * q^83 - 396 * q^86 - 1674 * q^87 - 544 * q^88 - 182 * q^89 + 792 * q^92 + 2956 * q^93 - 328 * q^94 + 64 * q^96 - 892 * q^97 - 1728 * q^99

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