LMFDB - Embedded newform 575.4.a.r.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, 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("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.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:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 4 x^{16} - 96 x^{15} + 368 x^{14} + 3705 x^{13} - 13440 x^{12} - 73933 x^{11} + 248806 x^{10} + \cdots - 2150912 $ x^17 - 4x^16 - 96x^15 + 368x^14 + 3705x^13 - 13440x^12 - 73933x^11 + 248806x^10 + 820568x^9 - 2474496x^8 - 5096659x^7 + 12861046x^6 + 16846350x^5 - 30611400x^4 - 25094976x^3 + 22642432x^2 + 5724640x - 2150912
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.752118$ of defining polynomial
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.752118 q^{2} +6.31519 q^{3} -7.43432 q^{4} +4.74977 q^{6} +25.4579 q^{7} -11.6084 q^{8} +12.8816 q^{9} +29.8906 q^{11} -46.9491 q^{12} +14.6921 q^{13} +19.1473 q^{14} +50.7436 q^{16} -44.0791 q^{17} +9.68849 q^{18} +9.77747 q^{19} +160.771 q^{21} +22.4812 q^{22} +23.0000 q^{23} -73.3095 q^{24} +11.0502 q^{26} -89.1603 q^{27} -189.262 q^{28} +124.275 q^{29} +184.024 q^{31} +131.033 q^{32} +188.765 q^{33} -33.1527 q^{34} -95.7659 q^{36} +65.7137 q^{37} +7.35382 q^{38} +92.7836 q^{39} -130.103 q^{41} +120.919 q^{42} +58.2708 q^{43} -222.216 q^{44} +17.2987 q^{46} -163.160 q^{47} +320.456 q^{48} +305.103 q^{49} -278.368 q^{51} -109.226 q^{52} +622.062 q^{53} -67.0591 q^{54} -295.526 q^{56} +61.7466 q^{57} +93.4692 q^{58} +725.566 q^{59} +606.658 q^{61} +138.408 q^{62} +327.938 q^{63} -307.397 q^{64} +141.973 q^{66} +944.820 q^{67} +327.698 q^{68} +145.249 q^{69} -925.819 q^{71} -149.535 q^{72} -81.1101 q^{73} +49.4245 q^{74} -72.6889 q^{76} +760.950 q^{77} +69.7842 q^{78} +1335.37 q^{79} -910.868 q^{81} -97.8529 q^{82} -78.3594 q^{83} -1195.22 q^{84} +43.8265 q^{86} +784.818 q^{87} -346.983 q^{88} -1408.27 q^{89} +374.030 q^{91} -170.989 q^{92} +1162.15 q^{93} -122.716 q^{94} +827.496 q^{96} -1406.08 q^{97} +229.474 q^{98} +385.038 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9} - 4 q^{11} + 342 q^{12} + 208 q^{13} - 118 q^{14} + 220 q^{16} + 268 q^{17} + 180 q^{18} - 72 q^{19} - 16 q^{21} + 318 q^{22}+ \cdots + 1908 q^{99}+O(q^{100}) $ 17 * q + 4 * q^2 + 12 * q^3 + 72 * q^4 + 12 * q^6 + 72 * q^7 + 48 * q^8 + 155 * q^9 - 4 * q^11 + 342 * q^12 + 208 * q^13 - 118 * q^14 + 220 * q^16 + 268 * q^17 + 180 * q^18 - 72 * q^19 - 16 * q^21 + 318 * q^22 + 391 * q^23 - 54 * q^24 - 72 * q^26 + 408 * q^27 + 928 * q^28 - 28 * q^29 - 40 * q^31 + 268 * q^32 + 1382 * q^33 - 132 * q^34 + 932 * q^36 + 586 * q^37 + 596 * q^38 + 600 * q^39 + 134 * q^41 - 104 * q^42 + 1264 * q^43 + 806 * q^44 + 92 * q^46 + 184 * q^47 + 2928 * q^48 + 737 * q^49 - 552 * q^51 + 1558 * q^52 + 1102 * q^53 - 782 * q^54 - 1150 * q^56 + 1188 * q^57 + 3882 * q^58 + 354 * q^59 + 550 * q^61 - 1114 * q^62 + 3102 * q^63 - 50 * q^64 - 708 * q^66 + 898 * q^67 + 3774 * q^68 + 276 * q^69 + 680 * q^71 + 102 * q^72 + 4980 * q^73 - 794 * q^74 + 54 * q^76 + 584 * q^77 + 4108 * q^78 - 1984 * q^79 + 2457 * q^81 + 1798 * q^82 + 2634 * q^83 + 974 * q^84 - 3074 * q^86 - 472 * q^87 + 7584 * q^88 - 598 * q^89 - 272 * q^91 + 1656 * q^92 + 2384 * q^93 + 1170 * q^94 + 1480 * q^96 + 6266 * q^97 + 3486 * q^98 + 1908 * 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 $

$2$	0.752118	0.265914	0.132957	−	0.991122i	$-0.457553\pi$
$2$	0.752118	0.265914	0.132957	+	0.991122i	$0.457553\pi$
$3$	6.31519	1.21536	0.607679	−	0.794183i	$-0.292101\pi$
$3$	6.31519	1.21536	0.607679	+	0.794183i	$0.292101\pi$
$4$	−7.43432	−0.929290
$5$	0	0
$5$	0	0
$6$	4.74977	0.323181
$7$	25.4579	1.37460	0.687298	−	0.726376i	$-0.258797\pi$
$7$	25.4579	1.37460	0.687298	+	0.726376i	$0.258797\pi$
$8$	−11.6084	−0.513025
$9$	12.8816	0.477096
$10$	0	0
$11$	29.8906	0.819304	0.409652	−	0.912242i	$-0.365650\pi$
$11$	29.8906	0.819304	0.409652	+	0.912242i	$0.365650\pi$
$12$	−46.9491	−1.12942
$13$	14.6921	0.313451	0.156726	−	0.987642i	$-0.449906\pi$
$13$	14.6921	0.313451	0.156726	+	0.987642i	$0.449906\pi$
$14$	19.1473	0.365524
$15$	0	0
$16$	50.7436	0.792869
$17$	−44.0791	−0.628868	−0.314434	−	0.949279i	$-0.601815\pi$
$17$	−44.0791	−0.628868	−0.314434	+	0.949279i	$0.601815\pi$
$18$	9.68849	0.126867
$19$	9.77747	0.118058	0.0590291	−	0.998256i	$-0.481200\pi$
$19$	9.77747	0.118058	0.0590291	+	0.998256i	$0.481200\pi$
$20$	0	0
$21$	160.771	1.67063
$22$	22.4812	0.217864
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−73.3095	−0.623510
$25$	0	0
$26$	11.0502	0.0833510
$27$	−89.1603	−0.635515
$28$	−189.262	−1.27740
$29$	124.275	0.795766	0.397883	−	0.917436i	$-0.369745\pi$
$29$	124.275	0.795766	0.397883	+	0.917436i	$0.369745\pi$
$30$	0	0
$31$	184.024	1.06618	0.533092	−	0.846057i	$-0.321030\pi$
$31$	184.024	1.06618	0.533092	+	0.846057i	$0.321030\pi$
$32$	131.033	0.723860
$33$	188.765	0.995748
$34$	−33.1527	−0.167225
$35$	0	0
$36$	−95.7659	−0.443361
$37$	65.7137	0.291980	0.145990	−	0.989286i	$-0.453363\pi$
$37$	65.7137	0.291980	0.145990	+	0.989286i	$0.453363\pi$
$38$	7.35382	0.0313933
$39$	92.7836	0.380955
$40$	0	0
$41$	−130.103	−0.495578	−0.247789	−	0.968814i	$-0.579704\pi$
$41$	−130.103	−0.495578	−0.247789	+	0.968814i	$0.579704\pi$
$42$	120.919	0.444243
$43$	58.2708	0.206656	0.103328	−	0.994647i	$-0.467051\pi$
$43$	58.2708	0.206656	0.103328	+	0.994647i	$0.467051\pi$
$44$	−222.216	−0.761371
$45$	0	0
$46$	17.2987	0.0554469
$47$	−163.160	−0.506369	−0.253184	−	0.967418i	$-0.581478\pi$
$47$	−163.160	−0.506369	−0.253184	+	0.967418i	$0.581478\pi$
$48$	320.456	0.963620
$49$	305.103	0.889514
$50$	0	0
$51$	−278.368	−0.764300
$52$	−109.226	−0.291287
$53$	622.062	1.61220	0.806102	−	0.591777i	$-0.201573\pi$
$53$	622.062	1.61220	0.806102	+	0.591777i	$0.201573\pi$
$54$	−67.0591	−0.168992
$55$	0	0
$56$	−295.526	−0.705202
$57$	61.7466	0.143483
$58$	93.4692	0.211605
$59$	725.566	1.60103	0.800514	−	0.599314i	$-0.204560\pi$
$59$	725.566	1.60103	0.800514	+	0.599314i	$0.204560\pi$
$60$	0	0
$61$	606.658	1.27335	0.636677	−	0.771130i	$-0.280308\pi$
$61$	606.658	1.27335	0.636677	+	0.771130i	$0.280308\pi$
$62$	138.408	0.283513
$63$	327.938	0.655815
$64$	−307.397	−0.600385
$65$	0	0
$66$	141.973	0.264783
$67$	944.820	1.72281	0.861404	−	0.507920i	$-0.169585\pi$
$67$	944.820	1.72281	0.861404	+	0.507920i	$0.169585\pi$
$68$	327.698	0.584400
$69$	145.249	0.253420
$70$	0	0
$71$	−925.819	−1.54753	−0.773764	−	0.633474i	$-0.781628\pi$
$71$	−925.819	−1.54753	−0.773764	+	0.633474i	$0.781628\pi$
$72$	−149.535	−0.244762
$73$	−81.1101	−0.130044	−0.0650220	−	0.997884i	$-0.520712\pi$
$73$	−81.1101	−0.130044	−0.0650220	+	0.997884i	$0.520712\pi$
$74$	49.4245	0.0776416
$75$	0	0
$76$	−72.6889	−0.109710
$77$	760.950	1.12621
$78$	69.7842	0.101301
$79$	1335.37	1.90179	0.950894	−	0.309516i	$-0.100167\pi$
$79$	1335.37	1.90179	0.950894	+	0.309516i	$0.100167\pi$
$80$	0	0
$81$	−910.868	−1.24948
$82$	−97.8529	−0.131781
$83$	−78.3594	−0.103627	−0.0518137	−	0.998657i	$-0.516500\pi$
$83$	−78.3594	−0.103627	−0.0518137	+	0.998657i	$0.516500\pi$
$84$	−1195.22	−1.55250
$85$	0	0
$86$	43.8265	0.0549527
$87$	784.818	0.967141
$88$	−346.983	−0.420324
$89$	−1408.27	−1.67726	−0.838630	−	0.544702i	$-0.816643\pi$
$89$	−1408.27	−1.67726	−0.838630	+	0.544702i	$0.816643\pi$
$90$	0	0
$91$	374.030	0.430869
$92$	−170.989	−0.193770
$93$	1162.15	1.29580
$94$	−122.716	−0.134651
$95$	0	0
$96$	827.496	0.879750
$97$	−1406.08	−1.47181	−0.735904	−	0.677086i	$-0.763242\pi$
$97$	−1406.08	−1.47181	−0.735904	+	0.677086i	$0.763242\pi$
$98$	229.474	0.236534
$99$	385.038	0.390887
$100$	0	0
$101$	22.0778	0.0217507	0.0108753	−	0.999941i	$-0.496538\pi$
$101$	22.0778	0.0217507	0.0108753	+	0.999941i	$0.496538\pi$
$102$	−209.366	−0.203238
$103$	401.810	0.384384	0.192192	−	0.981357i	$-0.438440\pi$
$103$	401.810	0.384384	0.192192	+	0.981357i	$0.438440\pi$
$104$	−170.553	−0.160808
$105$	0	0
$106$	467.864	0.428708
$107$	65.4400	0.0591245	0.0295622	−	0.999563i	$-0.490589\pi$
$107$	65.4400	0.0591245	0.0295622	+	0.999563i	$0.490589\pi$
$108$	662.846	0.590578
$109$	−2108.07	−1.85244	−0.926222	−	0.376980i	$-0.876963\pi$
$109$	−2108.07	−1.85244	−0.926222	+	0.376980i	$0.876963\pi$
$110$	0	0
$111$	414.994	0.354860
$112$	1291.82	1.08987
$113$	1085.61	0.903765	0.451882	−	0.892078i	$-0.350753\pi$
$113$	1085.61	0.903765	0.451882	+	0.892078i	$0.350753\pi$
$114$	46.4408	0.0381542
$115$	0	0
$116$	−923.897	−0.739497
$117$	189.258	0.149546
$118$	545.712	0.425736
$119$	−1122.16	−0.864439
$120$	0	0
$121$	−437.554	−0.328741
$122$	456.279	0.338603
$123$	−821.625	−0.602305
$124$	−1368.09	−0.990794
$125$	0	0
$126$	246.648	0.174390
$127$	−1243.72	−0.868996	−0.434498	−	0.900673i	$-0.643074\pi$
$127$	−1243.72	−0.868996	−0.434498	+	0.900673i	$0.643074\pi$
$128$	−1279.46	−0.883511
$129$	367.991	0.251161
$130$	0	0
$131$	−959.795	−0.640135	−0.320068	−	0.947395i	$-0.603706\pi$
$131$	−959.795	−0.640135	−0.320068	+	0.947395i	$0.603706\pi$
$132$	−1403.34	−0.925339
$133$	248.914	0.162282
$134$	710.617	0.458119
$135$	0	0
$136$	511.689	0.322625
$137$	−853.640	−0.532346	−0.266173	−	0.963925i	$-0.585759\pi$
$137$	−853.640	−0.532346	−0.266173	+	0.963925i	$0.585759\pi$
$138$	109.245	0.0673879
$139$	1074.03	0.655380	0.327690	−	0.944785i	$-0.393730\pi$
$139$	1074.03	0.655380	0.327690	+	0.944785i	$0.393730\pi$
$140$	0	0
$141$	−1030.39	−0.615420
$142$	−696.325	−0.411509
$143$	439.156	0.256812
$144$	653.659	0.378275
$145$	0	0
$146$	−61.0044	−0.0345805
$147$	1926.79	1.08108
$148$	−488.536	−0.271334
$149$	1702.18	0.935890	0.467945	−	0.883758i	$-0.344994\pi$
$149$	1702.18	0.935890	0.467945	+	0.883758i	$0.344994\pi$
$150$	0	0
$151$	2347.50	1.26514	0.632572	−	0.774502i	$-0.281999\pi$
$151$	2347.50	1.26514	0.632572	+	0.774502i	$0.281999\pi$
$152$	−113.501	−0.0605669
$153$	−567.809	−0.300030
$154$	572.325	0.299476
$155$	0	0
$156$	−689.783	−0.354018
$157$	1365.29	0.694024	0.347012	−	0.937861i	$-0.387196\pi$
$157$	1365.29	0.694024	0.347012	+	0.937861i	$0.387196\pi$
$158$	1004.36	0.505712
$159$	3928.44	1.95941
$160$	0	0
$161$	585.531	0.286623
$162$	−685.080	−0.332253
$163$	−1834.48	−0.881518	−0.440759	−	0.897625i	$-0.645291\pi$
$163$	−1834.48	−0.881518	−0.440759	+	0.897625i	$0.645291\pi$
$164$	967.227	0.460535
$165$	0	0
$166$	−58.9356	−0.0275560
$167$	−1238.32	−0.573796	−0.286898	−	0.957961i	$-0.592624\pi$
$167$	−1238.32	−0.573796	−0.286898	+	0.957961i	$0.592624\pi$
$168$	−1866.30	−0.857074
$169$	−1981.14	−0.901748
$170$	0	0
$171$	125.950	0.0563252
$172$	−433.203	−0.192043
$173$	1817.77	0.798860	0.399430	−	0.916764i	$-0.369208\pi$
$173$	1817.77	0.798860	0.399430	+	0.916764i	$0.369208\pi$
$174$	590.276	0.257176
$175$	0	0
$176$	1516.76	0.649601
$177$	4582.09	1.94582
$178$	−1059.18	−0.446007
$179$	1569.68	0.655438	0.327719	−	0.944775i	$-0.393720\pi$
$179$	1569.68	0.655438	0.327719	+	0.944775i	$0.393720\pi$
$180$	0	0
$181$	74.5516	0.0306154	0.0153077	−	0.999883i	$-0.495127\pi$
$181$	74.5516	0.0306154	0.0153077	+	0.999883i	$0.495127\pi$
$182$	281.315	0.114574
$183$	3831.16	1.54758
$184$	−266.994	−0.106973
$185$	0	0
$186$	874.072	0.344570
$187$	−1317.55	−0.515234
$188$	1212.98	0.470563
$189$	−2269.83	−0.873577
$190$	0	0
$191$	−2792.12	−1.05775	−0.528875	−	0.848700i	$-0.677386\pi$
$191$	−2792.12	−1.05775	−0.528875	+	0.848700i	$0.677386\pi$
$192$	−1941.27	−0.729682
$193$	−4699.46	−1.75272	−0.876358	−	0.481660i	$-0.840034\pi$
$193$	−4699.46	−1.75272	−0.876358	+	0.481660i	$0.840034\pi$
$194$	−1057.54	−0.391374
$195$	0	0
$196$	−2268.24	−0.826617
$197$	−3539.44	−1.28007	−0.640037	−	0.768344i	$-0.721081\pi$
$197$	−3539.44	−1.28007	−0.640037	+	0.768344i	$0.721081\pi$
$198$	289.594	0.103942
$199$	−286.039	−0.101893	−0.0509466	−	0.998701i	$-0.516224\pi$
$199$	−286.039	−0.101893	−0.0509466	+	0.998701i	$0.516224\pi$
$200$	0	0
$201$	5966.72	2.09383
$202$	16.6051	0.00578382
$203$	3163.77	1.09386
$204$	2069.47	0.710256
$205$	0	0
$206$	302.209	0.102213
$207$	296.277	0.0994815
$208$	745.532	0.248526
$209$	292.254	0.0967256
$210$	0	0
$211$	523.713	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$211$	523.713	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$212$	−4624.61	−1.49820
$213$	−5846.72	−1.88080
$214$	49.2186	0.0157220
$215$	0	0
$216$	1035.01	0.326035
$217$	4684.86	1.46557
$218$	−1585.52	−0.492591
$219$	−512.225	−0.158050
$220$	0	0
$221$	−647.616	−0.197119
$222$	312.125	0.0943624
$223$	5894.52	1.77007	0.885036	−	0.465523i	$-0.154134\pi$
$223$	5894.52	1.77007	0.885036	+	0.465523i	$0.154134\pi$
$224$	3335.81	0.995015
$225$	0	0
$226$	816.506	0.240324
$227$	6239.19	1.82427	0.912136	−	0.409888i	$-0.134432\pi$
$227$	6239.19	1.82427	0.912136	+	0.409888i	$0.134432\pi$
$228$	−459.044	−0.133337
$229$	−3926.05	−1.13293	−0.566465	−	0.824086i	$-0.691689\pi$
$229$	−3926.05	−1.13293	−0.566465	+	0.824086i	$0.691689\pi$
$230$	0	0
$231$	4805.54	1.36875
$232$	−1442.63	−0.408248
$233$	−6356.06	−1.78712	−0.893561	−	0.448942i	$-0.851801\pi$
$233$	−6356.06	−1.78712	−0.893561	+	0.448942i	$0.851801\pi$
$234$	142.345	0.0397665
$235$	0	0
$236$	−5394.09	−1.48782
$237$	8433.14	2.31135
$238$	−843.997	−0.229866
$239$	−2869.32	−0.776574	−0.388287	−	0.921539i	$-0.626933\pi$
$239$	−2869.32	−0.776574	−0.388287	+	0.921539i	$0.626933\pi$
$240$	0	0
$241$	−6359.58	−1.69982	−0.849910	−	0.526927i	$-0.823344\pi$
$241$	−6359.58	−1.69982	−0.849910	+	0.526927i	$0.823344\pi$
$242$	−329.093	−0.0874168
$243$	−3344.97	−0.883045
$244$	−4510.09	−1.18332
$245$	0	0
$246$	−617.960	−0.160161
$247$	143.652	0.0370055
$248$	−2136.23	−0.546979
$249$	−494.855	−0.125944
$250$	0	0
$251$	828.982	0.208466	0.104233	−	0.994553i	$-0.466761\pi$
$251$	828.982	0.208466	0.104233	+	0.994553i	$0.466761\pi$
$252$	−2438.00	−0.609442
$253$	687.483	0.170837
$254$	−935.427	−0.231078
$255$	0	0
$256$	1496.87	0.365447
$257$	−4085.76	−0.991684	−0.495842	−	0.868413i	$-0.665140\pi$
$257$	−4085.76	−0.991684	−0.495842	+	0.868413i	$0.665140\pi$
$258$	276.773	0.0667873
$259$	1672.93	0.401355
$260$	0	0
$261$	1600.86	0.379657
$262$	−721.880	−0.170221
$263$	−4388.26	−1.02887	−0.514433	−	0.857531i	$-0.671998\pi$
$263$	−4388.26	−1.02887	−0.514433	+	0.857531i	$0.671998\pi$
$264$	−2191.26	−0.510844
$265$	0	0
$266$	187.213	0.0431532
$267$	−8893.48	−2.03847
$268$	−7024.09	−1.60099
$269$	−6359.22	−1.44137	−0.720685	−	0.693262i	$-0.756173\pi$
$269$	−6359.22	−1.44137	−0.720685	+	0.693262i	$0.756173\pi$
$270$	0	0
$271$	3747.58	0.840033	0.420017	−	0.907516i	$-0.362024\pi$
$271$	3747.58	0.840033	0.420017	+	0.907516i	$0.362024\pi$
$272$	−2236.73	−0.498610
$273$	2362.07	0.523660
$274$	−642.038	−0.141558
$275$	0	0
$276$	−1079.83	−0.235500
$277$	−598.267	−0.129770	−0.0648852	−	0.997893i	$-0.520668\pi$
$277$	−598.267	−0.129770	−0.0648852	+	0.997893i	$0.520668\pi$
$278$	807.796	0.174275
$279$	2370.53	0.508673
$280$	0	0
$281$	−4834.80	−1.02641	−0.513203	−	0.858267i	$-0.671541\pi$
$281$	−4834.80	−1.02641	−0.513203	+	0.858267i	$0.671541\pi$
$282$	−774.973	−0.163649
$283$	−6394.70	−1.34320	−0.671600	−	0.740914i	$-0.734393\pi$
$283$	−6394.70	−1.34320	−0.671600	+	0.740914i	$0.734393\pi$
$284$	6882.83	1.43810
$285$	0	0
$286$	330.297	0.0682898
$287$	−3312.15	−0.681219
$288$	1687.91	0.345351
$289$	−2970.03	−0.604525
$290$	0	0
$291$	−8879.63	−1.78877
$292$	602.998	0.120849
$293$	749.489	0.149439	0.0747195	−	0.997205i	$-0.476194\pi$
$293$	749.489	0.149439	0.0747195	+	0.997205i	$0.476194\pi$
$294$	1449.17	0.287474
$295$	0	0
$296$	−762.833	−0.149793
$297$	−2665.05	−0.520680
$298$	1280.24	0.248866
$299$	337.919	0.0653591
$300$	0	0
$301$	1483.45	0.284069
$302$	1765.60	0.336419
$303$	139.425	0.0264349
$304$	496.144	0.0936047
$305$	0	0
$306$	−427.060	−0.0797823
$307$	797.542	0.148268	0.0741338	−	0.997248i	$-0.476381\pi$
$307$	797.542	0.148268	0.0741338	+	0.997248i	$0.476381\pi$
$308$	−5657.15	−1.04658
$309$	2537.51	0.467164
$310$	0	0
$311$	4132.60	0.753500	0.376750	−	0.926315i	$-0.377042\pi$
$311$	4132.60	0.753500	0.376750	+	0.926315i	$0.377042\pi$
$312$	−1077.07	−0.195440
$313$	532.636	0.0961865	0.0480932	−	0.998843i	$-0.484686\pi$
$313$	532.636	0.0961865	0.0480932	+	0.998843i	$0.484686\pi$
$314$	1026.86	0.184551
$315$	0	0
$316$	−9927.59	−1.76731
$317$	2138.53	0.378901	0.189450	−	0.981890i	$-0.439329\pi$
$317$	2138.53	0.378901	0.189450	+	0.981890i	$0.439329\pi$
$318$	2954.65	0.521033
$319$	3714.64	0.651974
$320$	0	0
$321$	413.266	0.0718575
$322$	440.389	0.0762171
$323$	−430.982	−0.0742430
$324$	6771.68	1.16112
$325$	0	0
$326$	−1379.75	−0.234408
$327$	−13312.8	−2.25138
$328$	1510.29	0.254244
$329$	−4153.71	−0.696053
$330$	0	0
$331$	−10652.1	−1.76885	−0.884427	−	0.466678i	$-0.845451\pi$
$331$	−10652.1	−1.76885	−0.884427	+	0.466678i	$0.845451\pi$
$332$	582.549	0.0962998
$333$	846.497	0.139303
$334$	−931.362	−0.152580
$335$	0	0
$336$	8158.12	1.32459
$337$	3552.87	0.574294	0.287147	−	0.957887i	$-0.407293\pi$
$337$	3552.87	0.574294	0.287147	+	0.957887i	$0.407293\pi$
$338$	−1490.05	−0.239788
$339$	6855.82	1.09840
$340$	0	0
$341$	5500.58	0.873529
$342$	94.7290	0.0149776
$343$	−964.765	−0.151873
$344$	−676.433	−0.106020
$345$	0	0
$346$	1367.18	0.212428
$347$	12605.6	1.95015	0.975076	−	0.221871i	$-0.0712163\pi$
$347$	12605.6	1.95015	0.975076	+	0.221871i	$0.0712163\pi$
$348$	−5834.58	−0.898754
$349$	2550.49	0.391188	0.195594	−	0.980685i	$-0.437337\pi$
$349$	2550.49	0.391188	0.195594	+	0.980685i	$0.437337\pi$
$350$	0	0
$351$	−1309.96	−0.199203
$352$	3916.64	0.593062
$353$	9359.21	1.41116	0.705582	−	0.708629i	$-0.250686\pi$
$353$	9359.21	1.41116	0.705582	+	0.708629i	$0.250686\pi$
$354$	3446.27	0.517422
$355$	0	0
$356$	10469.5	1.55866
$357$	−7086.65	−1.05060
$358$	1180.59	0.174290
$359$	6568.88	0.965717	0.482859	−	0.875698i	$-0.339599\pi$
$359$	6568.88	0.965717	0.482859	+	0.875698i	$0.339599\pi$
$360$	0	0
$361$	−6763.40	−0.986062
$362$	56.0717	0.00814105
$363$	−2763.24	−0.399538
$364$	−2780.66	−0.400402
$365$	0	0
$366$	2881.49	0.411524
$367$	2062.02	0.293287	0.146644	−	0.989189i	$-0.453153\pi$
$367$	2062.02	0.293287	0.146644	+	0.989189i	$0.453153\pi$
$368$	1167.10	0.165325
$369$	−1675.94	−0.236438
$370$	0	0
$371$	15836.4	2.21613
$372$	−8639.77	−1.20417
$373$	3945.27	0.547663	0.273831	−	0.961778i	$-0.411709\pi$
$373$	3945.27	0.547663	0.273831	+	0.961778i	$0.411709\pi$
$374$	−990.953	−0.137008
$375$	0	0
$376$	1894.03	0.259780
$377$	1825.86	0.249434
$378$	−1707.18	−0.232296
$379$	2470.90	0.334885	0.167443	−	0.985882i	$-0.446449\pi$
$379$	2470.90	0.334885	0.167443	+	0.985882i	$0.446449\pi$
$380$	0	0
$381$	−7854.34	−1.05614
$382$	−2100.00	−0.281271
$383$	−9582.90	−1.27849	−0.639247	−	0.769001i	$-0.720754\pi$
$383$	−9582.90	−1.27849	−0.639247	+	0.769001i	$0.720754\pi$
$384$	−8080.03	−1.07378
$385$	0	0
$386$	−3534.55	−0.466072
$387$	750.621	0.0985948
$388$	10453.2	1.36774
$389$	−7282.26	−0.949166	−0.474583	−	0.880211i	$-0.657401\pi$
$389$	−7282.26	−0.949166	−0.474583	+	0.880211i	$0.657401\pi$
$390$	0	0
$391$	−1013.82	−0.131128
$392$	−3541.77	−0.456343
$393$	−6061.29	−0.777994
$394$	−2662.08	−0.340390
$395$	0	0
$396$	−2862.50	−0.363247
$397$	8813.21	1.11416	0.557081	−	0.830458i	$-0.311921\pi$
$397$	8813.21	1.11416	0.557081	+	0.830458i	$0.311921\pi$
$398$	−215.135	−0.0270948
$399$	1571.94	0.197231
$400$	0	0
$401$	7998.99	0.996136	0.498068	−	0.867138i	$-0.334043\pi$
$401$	7998.99	0.996136	0.498068	+	0.867138i	$0.334043\pi$
$402$	4487.68	0.556779
$403$	2703.71	0.334197
$404$	−164.133	−0.0202127
$405$	0	0
$406$	2379.53	0.290872
$407$	1964.22	0.239220
$408$	3231.41	0.392105
$409$	−2359.31	−0.285233	−0.142616	−	0.989778i	$-0.545551\pi$
$409$	−2359.31	−0.285233	−0.142616	+	0.989778i	$0.545551\pi$
$410$	0	0
$411$	−5390.89	−0.646991
$412$	−2987.18	−0.357204
$413$	18471.4	2.20077
$414$	222.835	0.0264535
$415$	0	0
$416$	1925.15	0.226895
$417$	6782.69	0.796522
$418$	219.810	0.0257207
$419$	−12005.6	−1.39979	−0.699897	−	0.714244i	$-0.746771\pi$
$419$	−12005.6	−1.39979	−0.699897	+	0.714244i	$0.746771\pi$
$420$	0	0
$421$	−8767.51	−1.01497	−0.507485	−	0.861661i	$-0.669425\pi$
$421$	−8767.51	−1.01497	−0.507485	+	0.861661i	$0.669425\pi$
$422$	393.894	0.0454371
$423$	−2101.76	−0.241587
$424$	−7221.17	−0.827101
$425$	0	0
$426$	−4397.42	−0.500131
$427$	15444.2	1.75035
$428$	−486.502	−0.0549438
$429$	2773.35	0.312118
$430$	0	0
$431$	12409.0	1.38683	0.693413	−	0.720540i	$-0.256106\pi$
$431$	12409.0	1.38683	0.693413	+	0.720540i	$0.256106\pi$
$432$	−4524.32	−0.503881
$433$	4722.10	0.524087	0.262044	−	0.965056i	$-0.415604\pi$
$433$	4722.10	0.524087	0.262044	+	0.965056i	$0.415604\pi$
$434$	3523.57	0.389716
$435$	0	0
$436$	15672.0	1.72146
$437$	224.882	0.0246168
$438$	−385.254	−0.0420278
$439$	−12403.2	−1.34845	−0.674226	−	0.738525i	$-0.735523\pi$
$439$	−12403.2	−1.34845	−0.674226	+	0.738525i	$0.735523\pi$
$440$	0	0
$441$	3930.22	0.424384
$442$	−487.084	−0.0524168
$443$	1894.04	0.203135	0.101567	−	0.994829i	$-0.467614\pi$
$443$	1894.04	0.203135	0.101567	+	0.994829i	$0.467614\pi$
$444$	−3085.20	−0.329768
$445$	0	0
$446$	4433.37	0.470687
$447$	10749.6	1.13744
$448$	−7825.67	−0.825286
$449$	−14069.5	−1.47880	−0.739401	−	0.673265i	$-0.764891\pi$
$449$	−14069.5	−1.47880	−0.739401	+	0.673265i	$0.764891\pi$
$450$	0	0
$451$	−3888.85	−0.406029
$452$	−8070.75	−0.839859
$453$	14824.9	1.53760
$454$	4692.61	0.485099
$455$	0	0
$456$	−716.781	−0.0736104
$457$	12073.9	1.23587	0.617936	−	0.786228i	$-0.287969\pi$
$457$	12073.9	1.23587	0.617936	+	0.786228i	$0.287969\pi$
$458$	−2952.86	−0.301262
$459$	3930.11	0.399655
$460$	0	0
$461$	−3191.86	−0.322472	−0.161236	−	0.986916i	$-0.551548\pi$
$461$	−3191.86	−0.322472	−0.161236	+	0.986916i	$0.551548\pi$
$462$	3614.34	0.363970
$463$	5072.32	0.509137	0.254569	−	0.967055i	$-0.418067\pi$
$463$	5072.32	0.509137	0.254569	+	0.967055i	$0.418067\pi$
$464$	6306.14	0.630938
$465$	0	0
$466$	−4780.51	−0.475221
$467$	4040.20	0.400339	0.200169	−	0.979761i	$-0.435851\pi$
$467$	4040.20	0.400339	0.200169	+	0.979761i	$0.435851\pi$
$468$	−1407.01	−0.138972
$469$	24053.1	2.36817
$470$	0	0
$471$	8622.05	0.843488
$472$	−8422.69	−0.821368
$473$	1741.75	0.169314
$474$	6342.72	0.614622
$475$	0	0
$476$	8342.50	0.803314
$477$	8013.16	0.769176
$478$	−2158.07	−0.206502
$479$	−10825.0	−1.03258	−0.516292	−	0.856412i	$-0.672688\pi$
$479$	−10825.0	−1.03258	−0.516292	+	0.856412i	$0.672688\pi$
$480$	0	0
$481$	965.474	0.0915214
$482$	−4783.16	−0.452006
$483$	3697.74	0.348350
$484$	3252.92	0.305496
$485$	0	0
$486$	−2515.81	−0.234814
$487$	−1166.89	−0.108577	−0.0542885	−	0.998525i	$-0.517289\pi$
$487$	−1166.89	−0.108577	−0.0542885	+	0.998525i	$0.517289\pi$
$488$	−7042.35	−0.653263
$489$	−11585.1	−1.07136
$490$	0	0
$491$	−10558.3	−0.970446	−0.485223	−	0.874390i	$-0.661262\pi$
$491$	−10558.3	−0.970446	−0.485223	+	0.874390i	$0.661262\pi$
$492$	6108.22	0.559715
$493$	−5477.91	−0.500432
$494$	108.043	0.00984028
$495$	0	0
$496$	9338.05	0.845344
$497$	−23569.4	−2.12723
$498$	−372.189	−0.0334904
$499$	17354.5	1.55691	0.778453	−	0.627703i	$-0.216005\pi$
$499$	17354.5	1.55691	0.778453	+	0.627703i	$0.216005\pi$
$500$	0	0
$501$	−7820.21	−0.697368
$502$	623.493	0.0554340
$503$	3700.54	0.328030	0.164015	−	0.986458i	$-0.447555\pi$
$503$	3700.54	0.328030	0.164015	+	0.986458i	$0.447555\pi$
$504$	−3806.85	−0.336450
$505$	0	0
$506$	517.069	0.0454279
$507$	−12511.3	−1.09595
$508$	9246.23	0.807549
$509$	6465.51	0.563023	0.281512	−	0.959558i	$-0.409164\pi$
$509$	6465.51	0.563023	0.281512	+	0.959558i	$0.409164\pi$
$510$	0	0
$511$	−2064.89	−0.178758
$512$	11361.5	0.980688
$513$	−871.763	−0.0750278
$514$	−3072.98	−0.263703
$515$	0	0
$516$	−2735.76	−0.233402
$517$	−4876.95	−0.414870
$518$	1258.24	0.106726
$519$	11479.6	0.970901
$520$	0	0
$521$	−11016.6	−0.926384	−0.463192	−	0.886258i	$-0.653296\pi$
$521$	−11016.6	−0.926384	−0.463192	+	0.886258i	$0.653296\pi$
$522$	1204.03	0.100956
$523$	14924.7	1.24782	0.623911	−	0.781495i	$-0.285543\pi$
$523$	14924.7	1.24782	0.623911	+	0.781495i	$0.285543\pi$
$524$	7135.42	0.594871
$525$	0	0
$526$	−3300.49	−0.273590
$527$	−8111.62	−0.670489
$528$	9578.60	0.789498
$529$	529.000	0.0434783
$530$	0	0
$531$	9346.46	0.763845
$532$	−1850.50	−0.150807
$533$	−1911.49	−0.155339
$534$	−6688.95	−0.542058
$535$	0	0
$536$	−10967.9	−0.883844
$537$	9912.82	0.796592
$538$	−4782.89	−0.383281
$539$	9119.71	0.728783
$540$	0	0
$541$	−10203.5	−0.810876	−0.405438	−	0.914123i	$-0.632881\pi$
$541$	−10203.5	−0.810876	−0.405438	+	0.914123i	$0.632881\pi$
$542$	2818.62	0.223377
$543$	470.808	0.0372086
$544$	−5775.80	−0.455212
$545$	0	0
$546$	1776.56	0.139249
$547$	10666.2	0.833733	0.416866	−	0.908968i	$-0.363128\pi$
$547$	10666.2	0.833733	0.416866	+	0.908968i	$0.363128\pi$
$548$	6346.23	0.494703
$549$	7814.73	0.607513
$550$	0	0
$551$	1215.09	0.0939467
$552$	−1686.12	−0.130011
$553$	33995.8	2.61419
$554$	−449.968	−0.0345078
$555$	0	0
$556$	−7984.66	−0.609038
$557$	−1967.58	−0.149675	−0.0748376	−	0.997196i	$-0.523844\pi$
$557$	−1967.58	−0.149675	−0.0748376	+	0.997196i	$0.523844\pi$
$558$	1782.92	0.135263
$559$	856.122	0.0647766
$560$	0	0
$561$	−8320.57	−0.626194
$562$	−3636.35	−0.272936
$563$	−15947.2	−1.19377	−0.596887	−	0.802325i	$-0.703596\pi$
$563$	−15947.2	−1.19377	−0.596887	+	0.802325i	$0.703596\pi$
$564$	7660.22	0.571903
$565$	0	0
$566$	−4809.57	−0.357176
$567$	−23188.8	−1.71752
$568$	10747.3	0.793921
$569$	−25178.4	−1.85506	−0.927532	−	0.373744i	$-0.878074\pi$
$569$	−25178.4	−1.85506	−0.927532	+	0.373744i	$0.878074\pi$
$570$	0	0
$571$	10048.6	0.736464	0.368232	−	0.929734i	$-0.379963\pi$
$571$	10048.6	0.736464	0.368232	+	0.929734i	$0.379963\pi$
$572$	−3264.83	−0.238653
$573$	−17632.7	−1.28555
$574$	−2491.13	−0.181146
$575$	0	0
$576$	−3959.76	−0.286441
$577$	1912.51	0.137988	0.0689938	−	0.997617i	$-0.478021\pi$
$577$	1912.51	0.137988	0.0689938	+	0.997617i	$0.478021\pi$
$578$	−2233.82	−0.160752
$579$	−29677.9	−2.13018
$580$	0	0
$581$	−1994.86	−0.142446
$582$	−6678.54	−0.475660
$583$	18593.8	1.32088
$584$	941.561	0.0667159
$585$	0	0
$586$	563.704	0.0397379
$587$	13861.5	0.974662	0.487331	−	0.873217i	$-0.337971\pi$
$587$	13861.5	0.974662	0.487331	+	0.873217i	$0.337971\pi$
$588$	−14324.3	−1.00464
$589$	1799.29	0.125872
$590$	0	0
$591$	−22352.2	−1.55575
$592$	3334.55	0.231502
$593$	5991.56	0.414914	0.207457	−	0.978244i	$-0.433481\pi$
$593$	5991.56	0.414914	0.207457	+	0.978244i	$0.433481\pi$
$594$	−2004.44	−0.138456
$595$	0	0
$596$	−12654.5	−0.869713
$597$	−1806.39	−0.123837
$598$	254.155	0.0173799
$599$	15017.3	1.02435	0.512177	−	0.858880i	$-0.328839\pi$
$599$	15017.3	1.02435	0.512177	+	0.858880i	$0.328839\pi$
$600$	0	0
$601$	11885.0	0.806655	0.403328	−	0.915056i	$-0.367853\pi$
$601$	11885.0	0.806655	0.403328	+	0.915056i	$0.367853\pi$
$602$	1115.73	0.0755378
$603$	12170.8	0.821946
$604$	−17452.0	−1.17568
$605$	0	0
$606$	104.864	0.00702941
$607$	24838.3	1.66088	0.830440	−	0.557108i	$-0.188089\pi$
$607$	24838.3	1.66088	0.830440	+	0.557108i	$0.188089\pi$
$608$	1281.17	0.0854577
$609$	19979.8	1.32943
$610$	0	0
$611$	−2397.17	−0.158722
$612$	4221.27	0.278815
$613$	−19056.2	−1.25559	−0.627793	−	0.778380i	$-0.716042\pi$
$613$	−19056.2	−1.25559	−0.627793	+	0.778380i	$0.716042\pi$
$614$	599.846	0.0394264
$615$	0	0
$616$	−8833.44	−0.577775
$617$	−538.180	−0.0351156	−0.0175578	−	0.999846i	$-0.505589\pi$
$617$	−538.180	−0.0351156	−0.0175578	+	0.999846i	$0.505589\pi$
$618$	1908.50	0.124225
$619$	8769.31	0.569416	0.284708	−	0.958614i	$-0.408103\pi$
$619$	8769.31	0.569416	0.284708	+	0.958614i	$0.408103\pi$
$620$	0	0
$621$	−2050.69	−0.132514
$622$	3108.21	0.200366
$623$	−35851.5	−2.30555
$624$	4708.17	0.302048
$625$	0	0
$626$	400.605	0.0255773
$627$	1845.64	0.117556
$628$	−10150.0	−0.644950
$629$	−2896.60	−0.183617
$630$	0	0
$631$	−11235.2	−0.708821	−0.354411	−	0.935090i	$-0.615318\pi$
$631$	−11235.2	−0.708821	−0.354411	+	0.935090i	$0.615318\pi$
$632$	−15501.6	−0.975665
$633$	3307.35	0.207670
$634$	1608.42	0.100755
$635$	0	0
$636$	−29205.3	−1.82086
$637$	4482.62	0.278819
$638$	2793.85	0.173369
$639$	−11926.0	−0.738320
$640$	0	0
$641$	12206.6	0.752153	0.376077	−	0.926589i	$-0.377273\pi$
$641$	12206.6	0.752153	0.376077	+	0.926589i	$0.377273\pi$
$642$	310.825	0.0191079
$643$	26075.8	1.59927	0.799633	−	0.600489i	$-0.205027\pi$
$643$	26075.8	1.59927	0.799633	+	0.600489i	$0.205027\pi$
$644$	−4353.02	−0.266356
$645$	0	0
$646$	−324.150	−0.0197423
$647$	−5223.06	−0.317372	−0.158686	−	0.987329i	$-0.550726\pi$
$647$	−5223.06	−0.317372	−0.158686	+	0.987329i	$0.550726\pi$
$648$	10573.7	0.641012
$649$	21687.6	1.31173
$650$	0	0
$651$	29585.8	1.78120
$652$	13638.1	0.819186
$653$	−17785.4	−1.06584	−0.532922	−	0.846164i	$-0.678906\pi$
$653$	−17785.4	−1.06584	−0.532922	+	0.846164i	$0.678906\pi$
$654$	−10012.8	−0.598674
$655$	0	0
$656$	−6601.90	−0.392928
$657$	−1044.83	−0.0620436
$658$	−3124.08	−0.185090
$659$	−11310.0	−0.668554	−0.334277	−	0.942475i	$-0.608492\pi$
$659$	−11310.0	−0.668554	−0.334277	+	0.942475i	$0.608492\pi$
$660$	0	0
$661$	15765.0	0.927666	0.463833	−	0.885923i	$-0.346474\pi$
$661$	15765.0	0.927666	0.463833	+	0.885923i	$0.346474\pi$
$662$	−8011.62	−0.470363
$663$	−4089.82	−0.239571
$664$	909.631	0.0531634
$665$	0	0
$666$	636.666	0.0370425
$667$	2858.32	0.165929
$668$	9206.05	0.533223
$669$	37225.0	2.15127
$670$	0	0
$671$	18133.4	1.04326
$672$	21066.3	1.20930
$673$	22578.2	1.29320	0.646602	−	0.762827i	$-0.276189\pi$
$673$	22578.2	1.29320	0.646602	+	0.762827i	$0.276189\pi$
$674$	2672.18	0.152713
$675$	0	0
$676$	14728.4	0.837986
$677$	−15197.6	−0.862764	−0.431382	−	0.902169i	$-0.641974\pi$
$677$	−15197.6	−0.862764	−0.431382	+	0.902169i	$0.641974\pi$
$678$	5156.39	0.292079
$679$	−35795.7	−2.02314
$680$	0	0
$681$	39401.7	2.21714
$682$	4137.09	0.232284
$683$	7020.00	0.393284	0.196642	−	0.980475i	$-0.436996\pi$
$683$	7020.00	0.393284	0.196642	+	0.980475i	$0.436996\pi$
$684$	−936.349	−0.0523424
$685$	0	0
$686$	−725.618	−0.0403852
$687$	−24793.8	−1.37692
$688$	2956.87	0.163851
$689$	9139.42	0.505347
$690$	0	0
$691$	26464.0	1.45693	0.728464	−	0.685084i	$-0.240235\pi$
$691$	26464.0	1.45693	0.728464	+	0.685084i	$0.240235\pi$
$692$	−13513.9	−0.742372
$693$	9802.26	0.537312
$694$	9480.89	0.518573
$695$	0	0
$696$	−9110.50	−0.496168
$697$	5734.82	0.311653
$698$	1918.27	0.104022
$699$	−40139.7	−2.17199
$700$	0	0
$701$	−528.232	−0.0284608	−0.0142304	−	0.999899i	$-0.504530\pi$
$701$	−528.232	−0.0284608	−0.0142304	+	0.999899i	$0.504530\pi$
$702$	−985.242	−0.0529709
$703$	642.514	0.0344706
$704$	−9188.27	−0.491897
$705$	0	0
$706$	7039.24	0.375248
$707$	562.053	0.0298984
$708$	−34064.7	−1.80823
$709$	11155.8	0.590921	0.295461	−	0.955355i	$-0.404527\pi$
$709$	11155.8	0.590921	0.295461	+	0.955355i	$0.404527\pi$
$710$	0	0
$711$	17201.8	0.907336
$712$	16347.8	0.860476
$713$	4232.55	0.222315
$714$	−5330.00	−0.279370
$715$	0	0
$716$	−11669.5	−0.609092
$717$	−18120.3	−0.943816
$718$	4940.58	0.256798
$719$	13883.2	0.720103	0.360052	−	0.932932i	$-0.382759\pi$
$719$	13883.2	0.720103	0.360052	+	0.932932i	$0.382759\pi$
$720$	0	0
$721$	10229.2	0.528372
$722$	−5086.88	−0.262208
$723$	−40162.0	−2.06589
$724$	−554.241	−0.0284505
$725$	0	0
$726$	−2078.28	−0.106243
$727$	30552.9	1.55866	0.779329	−	0.626615i	$-0.215560\pi$
$727$	30552.9	1.55866	0.779329	+	0.626615i	$0.215560\pi$
$728$	−4341.91	−0.221046
$729$	3469.30	0.176259
$730$	0	0
$731$	−2568.52	−0.129959
$732$	−28482.1	−1.43815
$733$	−31880.0	−1.60643	−0.803216	−	0.595688i	$-0.796879\pi$
$733$	−31880.0	−1.60643	−0.803216	+	0.595688i	$0.796879\pi$
$734$	1550.88	0.0779893
$735$	0	0
$736$	3013.75	0.150935
$737$	28241.2	1.41150
$738$	−1260.50	−0.0628723
$739$	39570.5	1.96972	0.984862	−	0.173342i	$-0.0554567\pi$
$739$	39570.5	1.96972	0.984862	+	0.173342i	$0.0554567\pi$
$740$	0	0
$741$	907.189	0.0449749
$742$	11910.8	0.589300
$743$	−25014.3	−1.23511	−0.617555	−	0.786528i	$-0.711877\pi$
$743$	−25014.3	−1.23511	−0.617555	+	0.786528i	$0.711877\pi$
$744$	−13490.7	−0.664776
$745$	0	0
$746$	2967.31	0.145631
$747$	−1009.40	−0.0494402
$748$	9795.08	0.478802
$749$	1665.96	0.0812723
$750$	0	0
$751$	34198.5	1.66168	0.830840	−	0.556511i	$-0.187860\pi$
$751$	34198.5	1.66168	0.830840	+	0.556511i	$0.187860\pi$
$752$	−8279.33	−0.401484
$753$	5235.18	0.253361
$754$	1373.26	0.0663279
$755$	0	0
$756$	16874.7	0.811806
$757$	−3276.06	−0.157293	−0.0786463	−	0.996903i	$-0.525060\pi$
$757$	−3276.06	−0.157293	−0.0786463	+	0.996903i	$0.525060\pi$
$758$	1858.41	0.0890507
$759$	4341.58	0.207628
$760$	0	0
$761$	−13305.7	−0.633810	−0.316905	−	0.948457i	$-0.602644\pi$
$761$	−13305.7	−0.633810	−0.316905	+	0.948457i	$0.602644\pi$
$762$	−5907.40	−0.280843
$763$	−53666.9	−2.54636
$764$	20757.5	0.982957
$765$	0	0
$766$	−7207.48	−0.339970
$767$	10660.1	0.501844
$768$	9453.01	0.444149
$769$	−10307.7	−0.483360	−0.241680	−	0.970356i	$-0.577698\pi$
$769$	−10307.7	−0.483360	−0.241680	+	0.970356i	$0.577698\pi$
$770$	0	0
$771$	−25802.3	−1.20525
$772$	34937.2	1.62878
$773$	−10286.0	−0.478604	−0.239302	−	0.970945i	$-0.576919\pi$
$773$	−10286.0	−0.478604	−0.239302	+	0.970945i	$0.576919\pi$
$774$	564.556	0.0262178
$775$	0	0
$776$	16322.3	0.755074
$777$	10564.9	0.487790
$778$	−5477.12	−0.252396
$779$	−1272.08	−0.0585070
$780$	0	0
$781$	−27673.2	−1.26790
$782$	−762.512	−0.0348688
$783$	−11080.4	−0.505722
$784$	15482.1	0.705269
$785$	0	0
$786$	−4558.81	−0.206879
$787$	−12901.1	−0.584338	−0.292169	−	0.956367i	$-0.594377\pi$
$787$	−12901.1	−0.584338	−0.292169	+	0.956367i	$0.594377\pi$
$788$	26313.3	1.18956
$789$	−27712.7	−1.25044
$790$	0	0
$791$	27637.3	1.24231
$792$	−4469.69	−0.200535
$793$	8913.10	0.399134
$794$	6628.58	0.296271
$795$	0	0
$796$	2126.50	0.0946883
$797$	−6308.44	−0.280372	−0.140186	−	0.990125i	$-0.544770\pi$
$797$	−6308.44	−0.280372	−0.140186	+	0.990125i	$0.544770\pi$
$798$	1182.28	0.0524466
$799$	7191.95	0.318439
$800$	0	0
$801$	−18140.7	−0.800214
$802$	6016.19	0.264887
$803$	−2424.43	−0.106546
$804$	−44358.5	−1.94577
$805$	0	0
$806$	2033.51	0.0888676
$807$	−40159.7	−1.75178
$808$	−256.288	−0.0111587
$809$	30826.3	1.33967	0.669836	−	0.742509i	$-0.266364\pi$
$809$	30826.3	1.33967	0.669836	+	0.742509i	$0.266364\pi$
$810$	0	0
$811$	−37409.2	−1.61975	−0.809873	−	0.586605i	$-0.800464\pi$
$811$	−37409.2	−1.61975	−0.809873	+	0.586605i	$0.800464\pi$
$812$	−23520.5	−1.01651
$813$	23666.6	1.02094
$814$	1477.33	0.0636121
$815$	0	0
$816$	−14125.4	−0.605990
$817$	569.741	0.0243974
$818$	−1774.48	−0.0758474
$819$	4818.11	0.205566
$820$	0	0
$821$	−6138.76	−0.260955	−0.130478	−	0.991451i	$-0.541651\pi$
$821$	−6138.76	−0.260955	−0.130478	+	0.991451i	$0.541651\pi$
$822$	−4054.59	−0.172044
$823$	12789.2	0.541682	0.270841	−	0.962624i	$-0.412698\pi$
$823$	12789.2	0.541682	0.270841	+	0.962624i	$0.412698\pi$
$824$	−4664.39	−0.197198
$825$	0	0
$826$	13892.7	0.585215
$827$	−13392.6	−0.563125	−0.281563	−	0.959543i	$-0.590853\pi$
$827$	−13392.6	−0.563125	−0.281563	+	0.959543i	$0.590853\pi$
$828$	−2202.62	−0.0924471
$829$	12506.0	0.523945	0.261973	−	0.965075i	$-0.415627\pi$
$829$	12506.0	0.523945	0.261973	+	0.965075i	$0.415627\pi$
$830$	0	0
$831$	−3778.17	−0.157718
$832$	−4516.32	−0.188191
$833$	−13448.7	−0.559387
$834$	5101.38	0.211806
$835$	0	0
$836$	−2172.71	−0.0898861
$837$	−16407.7	−0.677576
$838$	−9029.65	−0.372225
$839$	−30719.0	−1.26405	−0.632024	−	0.774949i	$-0.717776\pi$
$839$	−30719.0	−1.26405	−0.632024	+	0.774949i	$0.717776\pi$
$840$	0	0
$841$	−8944.82	−0.366756
$842$	−6594.20	−0.269895
$843$	−30532.7	−1.24745
$844$	−3893.45	−0.158789
$845$	0	0
$846$	−1580.77	−0.0642413
$847$	−11139.2	−0.451886
$848$	31565.7	1.27827
$849$	−40383.8	−1.63247
$850$	0	0
$851$	1511.41	0.0608820
$852$	43466.4	1.74781
$853$	28733.4	1.15336	0.576678	−	0.816972i	$-0.304349\pi$
$853$	28733.4	1.15336	0.576678	+	0.816972i	$0.304349\pi$
$854$	11615.9	0.465442
$855$	0	0
$856$	−759.656	−0.0303324
$857$	−18108.0	−0.721772	−0.360886	−	0.932610i	$-0.617526\pi$
$857$	−18108.0	−0.721772	−0.360886	+	0.932610i	$0.617526\pi$
$858$	2085.89	0.0829966
$859$	−209.950	−0.00833924	−0.00416962	−	0.999991i	$-0.501327\pi$
$859$	−209.950	−0.00833924	−0.00416962	+	0.999991i	$0.501327\pi$
$860$	0	0
$861$	−20916.8	−0.827925
$862$	9333.07	0.368777
$863$	−21321.9	−0.841026	−0.420513	−	0.907286i	$-0.638150\pi$
$863$	−21321.9	−0.841026	−0.420513	+	0.907286i	$0.638150\pi$
$864$	−11682.9	−0.460024
$865$	0	0
$866$	3551.58	0.139362
$867$	−18756.3	−0.734715
$868$	−34828.8	−1.36194
$869$	39915.1	1.55814
$870$	0	0
$871$	13881.4	0.540016
$872$	24471.4	0.950350
$873$	−18112.5	−0.702194
$874$	169.138	0.00654596
$875$	0	0
$876$	3808.05	0.146874
$877$	−8116.82	−0.312526	−0.156263	−	0.987715i	$-0.549945\pi$
$877$	−8116.82	−0.312526	−0.156263	+	0.987715i	$0.549945\pi$
$878$	−9328.64	−0.358572
$879$	4733.16	0.181622
$880$	0	0
$881$	−33635.7	−1.28628	−0.643141	−	0.765748i	$-0.722369\pi$
$881$	−33635.7	−1.28628	−0.643141	+	0.765748i	$0.722369\pi$
$882$	2955.99	0.112850
$883$	10516.5	0.400804	0.200402	−	0.979714i	$-0.435775\pi$
$883$	10516.5	0.400804	0.200402	+	0.979714i	$0.435775\pi$
$884$	4814.58	0.183181
$885$	0	0
$886$	1424.55	0.0540164
$887$	−31975.5	−1.21041	−0.605204	−	0.796070i	$-0.706909\pi$
$887$	−31975.5	−1.21041	−0.605204	+	0.796070i	$0.706909\pi$
$888$	−4817.43	−0.182052
$889$	−31662.5	−1.19452
$890$	0	0
$891$	−27226.3	−1.02370
$892$	−43821.7	−1.64491
$893$	−1595.29	−0.0597810
$894$	8084.94	0.302462
$895$	0	0
$896$	−32572.3	−1.21447
$897$	2134.02	0.0794347
$898$	−10582.0	−0.393234
$899$	22869.5	0.848433
$900$	0	0
$901$	−27419.9	−1.01386
$902$	−2924.88	−0.107969
$903$	9368.27	0.345245
$904$	−12602.2	−0.463654
$905$	0	0
$906$	11150.1	0.408870
$907$	−47837.5	−1.75129	−0.875644	−	0.482958i	$-0.839562\pi$
$907$	−47837.5	−1.75129	−0.875644	+	0.482958i	$0.839562\pi$
$908$	−46384.1	−1.69528
$909$	284.397	0.0103772
$910$	0	0
$911$	−19733.6	−0.717677	−0.358839	−	0.933400i	$-0.616827\pi$
$911$	−19733.6	−0.717677	−0.358839	+	0.933400i	$0.616827\pi$
$912$	3133.25	0.113763
$913$	−2342.21	−0.0849023
$914$	9081.01	0.328636
$915$	0	0
$916$	29187.5	1.05282
$917$	−24434.3	−0.879927
$918$	2955.91	0.106274
$919$	−25597.5	−0.918807	−0.459404	−	0.888228i	$-0.651937\pi$
$919$	−25597.5	−0.918807	−0.459404	+	0.888228i	$0.651937\pi$
$920$	0	0
$921$	5036.63	0.180198
$922$	−2400.66	−0.0857499
$923$	−13602.2	−0.485074
$924$	−35725.9	−1.27197
$925$	0	0
$926$	3814.98	0.135387
$927$	5175.96	0.183388
$928$	16284.0	0.576023
$929$	−3535.76	−0.124870	−0.0624352	−	0.998049i	$-0.519887\pi$
$929$	−3535.76	−0.124870	−0.0624352	+	0.998049i	$0.519887\pi$
$930$	0	0
$931$	2983.14	0.105015
$932$	47253.0	1.66075
$933$	26098.2	0.915772
$934$	3038.71	0.106456
$935$	0	0
$936$	−2196.99	−0.0767211
$937$	7480.62	0.260812	0.130406	−	0.991461i	$-0.458372\pi$
$937$	7480.62	0.260812	0.130406	+	0.991461i	$0.458372\pi$
$938$	18090.8	0.629729
$939$	3363.70	0.116901
$940$	0	0
$941$	−30827.2	−1.06795	−0.533973	−	0.845501i	$-0.679302\pi$
$941$	−30827.2	−1.06795	−0.533973	+	0.845501i	$0.679302\pi$
$942$	6484.80	0.224295
$943$	−2992.37	−0.103335
$944$	36817.9	1.26941
$945$	0	0
$946$	1310.00	0.0450230
$947$	−12784.3	−0.438685	−0.219342	−	0.975648i	$-0.570391\pi$
$947$	−12784.3	−0.438685	−0.219342	+	0.975648i	$0.570391\pi$
$948$	−62694.6	−2.14792
$949$	−1191.68	−0.0407625
$950$	0	0
$951$	13505.2	0.460500
$952$	13026.5	0.443479
$953$	−7938.01	−0.269819	−0.134909	−	0.990858i	$-0.543074\pi$
$953$	−7938.01	−0.269819	−0.134909	+	0.990858i	$0.543074\pi$
$954$	6026.84	0.204535
$955$	0	0
$956$	21331.5	0.721662
$957$	23458.6	0.792383
$958$	−8141.70	−0.274579
$959$	−21731.8	−0.731760
$960$	0	0
$961$	4073.87	0.136748
$962$	726.151	0.0243368
$963$	842.972	0.0282081
$964$	47279.2	1.57963
$965$	0	0
$966$	2781.14	0.0926311
$967$	47112.7	1.56674	0.783372	−	0.621553i	$-0.213498\pi$
$967$	47112.7	1.56674	0.783372	+	0.621553i	$0.213498\pi$
$968$	5079.32	0.168652
$969$	−2721.73	−0.0902319
$970$	0	0
$971$	−8974.52	−0.296608	−0.148304	−	0.988942i	$-0.547381\pi$
$971$	−8974.52	−0.296608	−0.148304	+	0.988942i	$0.547381\pi$
$972$	24867.6	0.820605
$973$	27342.5	0.900883
$974$	−877.642	−0.0288721
$975$	0	0
$976$	30784.0	1.00960
$977$	−11925.1	−0.390499	−0.195249	−	0.980754i	$-0.562552\pi$
$977$	−11925.1	−0.390499	−0.195249	+	0.980754i	$0.562552\pi$
$978$	−8713.35	−0.284890
$979$	−42093.9	−1.37419
$980$	0	0
$981$	−27155.3	−0.883794
$982$	−7941.08	−0.258055
$983$	21935.4	0.711730	0.355865	−	0.934537i	$-0.384186\pi$
$983$	21935.4	0.711730	0.355865	+	0.934537i	$0.384186\pi$
$984$	9537.78	0.308997
$985$	0	0
$986$	−4120.04	−0.133072
$987$	−26231.5	−0.845954
$988$	−1067.95	−0.0343888
$989$	1340.23	0.0430908
$990$	0	0
$991$	14039.4	0.450027	0.225013	−	0.974356i	$-0.427757\pi$
$991$	14039.4	0.450027	0.225013	+	0.974356i	$0.427757\pi$
$992$	24113.2	0.771768
$993$	−67269.9	−2.14979
$994$	−17727.0	−0.565659
$995$	0	0
$996$	3678.91	0.117039
$997$	−3609.91	−0.114671	−0.0573354	−	0.998355i	$-0.518260\pi$
$997$	−3609.91	−0.114671	−0.0573354	+	0.998355i	$0.518260\pi$
$998$	13052.7	0.414003
$999$	−5859.05	−0.185558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.a.r.1.10		17
5.2	odd	4		115.4.b.a.24.20	yes	34
5.3	odd	4		115.4.b.a.24.15	✓	34
5.4	even	2		575.4.a.q.1.8		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.b.a.24.15	✓	34	5.3	odd	4
115.4.b.a.24.20	yes	34	5.2	odd	4
575.4.a.q.1.8		17	5.4	even	2
575.4.a.r.1.10		17	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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q+0.752118 q^{2} +6.31519 q^{3} -7.43432 q^{4} +4.74977 q^{6} +25.4579 q^{7} -11.6084 q^{8} +12.8816 q^{9} +29.8906 q^{11} -46.9491 q^{12} +14.6921 q^{13} +19.1473 q^{14} +50.7436 q^{16} -44.0791 q^{17} +9.68849 q^{18} +9.77747 q^{19} +160.771 q^{21} +22.4812 q^{22} +23.0000 q^{23} -73.3095 q^{24} +11.0502 q^{26} -89.1603 q^{27} -189.262 q^{28} +124.275 q^{29} +184.024 q^{31} +131.033 q^{32} +188.765 q^{33} -33.1527 q^{34} -95.7659 q^{36} +65.7137 q^{37} +7.35382 q^{38} +92.7836 q^{39} -130.103 q^{41} +120.919 q^{42} +58.2708 q^{43} -222.216 q^{44} +17.2987 q^{46} -163.160 q^{47} +320.456 q^{48} +305.103 q^{49} -278.368 q^{51} -109.226 q^{52} +622.062 q^{53} -67.0591 q^{54} -295.526 q^{56} +61.7466 q^{57} +93.4692 q^{58} +725.566 q^{59} +606.658 q^{61} +138.408 q^{62} +327.938 q^{63} -307.397 q^{64} +141.973 q^{66} +944.820 q^{67} +327.698 q^{68} +145.249 q^{69} -925.819 q^{71} -149.535 q^{72} -81.1101 q^{73} +49.4245 q^{74} -72.6889 q^{76} +760.950 q^{77} +69.7842 q^{78} +1335.37 q^{79} -910.868 q^{81} -97.8529 q^{82} -78.3594 q^{83} -1195.22 q^{84} +43.8265 q^{86} +784.818 q^{87} -346.983 q^{88} -1408.27 q^{89} +374.030 q^{91} -170.989 q^{92} +1162.15 q^{93} -122.716 q^{94} +827.496 q^{96} -1406.08 q^{97} +229.474 q^{98} +385.038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9} - 4 q^{11} + 342 q^{12} + 208 q^{13} - 118 q^{14} + 220 q^{16} + 268 q^{17} + 180 q^{18} - 72 q^{19} - 16 q^{21} + 318 q^{22}+ \cdots + 1908 q^{99}+O(q^{100}) \) 17 * q + 4 * q^2 + 12 * q^3 + 72 * q^4 + 12 * q^6 + 72 * q^7 + 48 * q^8 + 155 * q^9 - 4 * q^11 + 342 * q^12 + 208 * q^13 - 118 * q^14 + 220 * q^16 + 268 * q^17 + 180 * q^18 - 72 * q^19 - 16 * q^21 + 318 * q^22 + 391 * q^23 - 54 * q^24 - 72 * q^26 + 408 * q^27 + 928 * q^28 - 28 * q^29 - 40 * q^31 + 268 * q^32 + 1382 * q^33 - 132 * q^34 + 932 * q^36 + 586 * q^37 + 596 * q^38 + 600 * q^39 + 134 * q^41 - 104 * q^42 + 1264 * q^43 + 806 * q^44 + 92 * q^46 + 184 * q^47 + 2928 * q^48 + 737 * q^49 - 552 * q^51 + 1558 * q^52 + 1102 * q^53 - 782 * q^54 - 1150 * q^56 + 1188 * q^57 + 3882 * q^58 + 354 * q^59 + 550 * q^61 - 1114 * q^62 + 3102 * q^63 - 50 * q^64 - 708 * q^66 + 898 * q^67 + 3774 * q^68 + 276 * q^69 + 680 * q^71 + 102 * q^72 + 4980 * q^73 - 794 * q^74 + 54 * q^76 + 584 * q^77 + 4108 * q^78 - 1984 * q^79 + 2457 * q^81 + 1798 * q^82 + 2634 * q^83 + 974 * q^84 - 3074 * q^86 - 472 * q^87 + 7584 * q^88 - 598 * q^89 - 272 * q^91 + 1656 * q^92 + 2384 * q^93 + 1170 * q^94 + 1480 * q^96 + 6266 * q^97 + 3486 * q^98 + 1908 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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