LMFDB - Embedded newform 525.4.a.j.1.1

Newspace parameters

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$30.9760027530$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

$f(q)$	$=$	$q-3.56155 q^{2} +3.00000 q^{3} +4.68466 q^{4} -10.6847 q^{6} +7.00000 q^{7} +11.8078 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.56155 q^{2} +3.00000 q^{3} +4.68466 q^{4} -10.6847 q^{6} +7.00000 q^{7} +11.8078 q^{8} +9.00000 q^{9} -5.19224 q^{11} +14.0540 q^{12} -54.5464 q^{13} -24.9309 q^{14} -79.5312 q^{16} -16.1619 q^{17} -32.0540 q^{18} +87.4470 q^{19} +21.0000 q^{21} +18.4924 q^{22} -176.477 q^{23} +35.4233 q^{24} +194.270 q^{26} +27.0000 q^{27} +32.7926 q^{28} +142.170 q^{29} -94.3002 q^{31} +188.793 q^{32} -15.5767 q^{33} +57.5616 q^{34} +42.1619 q^{36} -17.3305 q^{37} -311.447 q^{38} -163.639 q^{39} +210.270 q^{41} -74.7926 q^{42} -521.570 q^{43} -24.3239 q^{44} +628.533 q^{46} +105.417 q^{47} -238.594 q^{48} +49.0000 q^{49} -48.4858 q^{51} -255.531 q^{52} +108.978 q^{53} -96.1619 q^{54} +82.6543 q^{56} +262.341 q^{57} -506.348 q^{58} +210.365 q^{59} -674.304 q^{61} +335.855 q^{62} +63.0000 q^{63} -36.1449 q^{64} +55.4773 q^{66} -324.929 q^{67} -75.7131 q^{68} -529.432 q^{69} +793.965 q^{71} +106.270 q^{72} -315.417 q^{73} +61.7235 q^{74} +409.659 q^{76} -36.3457 q^{77} +582.810 q^{78} -425.840 q^{79} +81.0000 q^{81} -748.887 q^{82} +283.029 q^{83} +98.3778 q^{84} +1857.60 q^{86} +426.511 q^{87} -61.3087 q^{88} -843.131 q^{89} -381.825 q^{91} -826.736 q^{92} -282.901 q^{93} -375.447 q^{94} +566.378 q^{96} -1537.33 q^{97} -174.516 q^{98} -46.7301 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9	$ 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 31 q^{11} - 9 q^{12} - 39 q^{13} - 21 q^{14} - 23 q^{16} + 79 q^{17} - 27 q^{18} - 56 q^{19} + 42 q^{21} + 4 q^{22} - 254 q^{23} + 9 q^{24} + 203 q^{26} + 54 q^{27} - 21 q^{28} - 62 q^{29} - 135 q^{31} + 291 q^{32} - 93 q^{33} + 111 q^{34} - 27 q^{36} - 113 q^{37} - 392 q^{38} - 117 q^{39} + 235 q^{41} - 63 q^{42} - 804 q^{43} + 174 q^{44} + 585 q^{46} - 152 q^{47} - 69 q^{48} + 98 q^{49} + 237 q^{51} - 375 q^{52} - 149 q^{53} - 81 q^{54} + 21 q^{56} - 168 q^{57} - 621 q^{58} - 441 q^{59} - 223 q^{61} + 313 q^{62} + 126 q^{63} - 431 q^{64} + 12 q^{66} - 1157 q^{67} - 807 q^{68} - 762 q^{69} + 619 q^{71} + 27 q^{72} - 268 q^{73} + 8 q^{74} + 1512 q^{76} - 217 q^{77} + 609 q^{78} - 427 q^{79} + 162 q^{81} - 735 q^{82} - 1211 q^{83} - 63 q^{84} + 1699 q^{86} - 186 q^{87} + 166 q^{88} + 466 q^{89} - 273 q^{91} - 231 q^{92} - 405 q^{93} - 520 q^{94} + 873 q^{96} - 172 q^{97} - 147 q^{98} - 279 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9 - 31 * q^11 - 9 * q^12 - 39 * q^13 - 21 * q^14 - 23 * q^16 + 79 * q^17 - 27 * q^18 - 56 * q^19 + 42 * q^21 + 4 * q^22 - 254 * q^23 + 9 * q^24 + 203 * q^26 + 54 * q^27 - 21 * q^28 - 62 * q^29 - 135 * q^31 + 291 * q^32 - 93 * q^33 + 111 * q^34 - 27 * q^36 - 113 * q^37 - 392 * q^38 - 117 * q^39 + 235 * q^41 - 63 * q^42 - 804 * q^43 + 174 * q^44 + 585 * q^46 - 152 * q^47 - 69 * q^48 + 98 * q^49 + 237 * q^51 - 375 * q^52 - 149 * q^53 - 81 * q^54 + 21 * q^56 - 168 * q^57 - 621 * q^58 - 441 * q^59 - 223 * q^61 + 313 * q^62 + 126 * q^63 - 431 * q^64 + 12 * q^66 - 1157 * q^67 - 807 * q^68 - 762 * q^69 + 619 * q^71 + 27 * q^72 - 268 * q^73 + 8 * q^74 + 1512 * q^76 - 217 * q^77 + 609 * q^78 - 427 * q^79 + 162 * q^81 - 735 * q^82 - 1211 * q^83 - 63 * q^84 + 1699 * q^86 - 186 * q^87 + 166 * q^88 + 466 * q^89 - 273 * q^91 - 231 * q^92 - 405 * q^93 - 520 * q^94 + 873 * q^96 - 172 * q^97 - 147 * q^98 - 279 * 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 $

$2$	−3.56155	−1.25920	−0.629600	−	0.776920i	$-0.716781\pi$
$2$	−3.56155	−1.25920	−0.629600	+	0.776920i	$0.716781\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.68466	0.585582
$5$	0	0
$5$	0	0
$6$	−10.6847	−0.726999
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	11.8078	0.521834
$9$	9.00000	0.333333
$10$	0	0
$11$	−5.19224	−0.142320	−0.0711599	−	0.997465i	$-0.522670\pi$
$11$	−5.19224	−0.142320	−0.0711599	+	0.997465i	$0.522670\pi$
$12$	14.0540	0.338086
$13$	−54.5464	−1.16373	−0.581863	−	0.813287i	$-0.697676\pi$
$13$	−54.5464	−1.16373	−0.581863	+	0.813287i	$0.697676\pi$
$14$	−24.9309	−0.475933
$15$	0	0
$16$	−79.5312	−1.24268
$17$	−16.1619	−0.230579	−0.115289	−	0.993332i	$-0.536780\pi$
$17$	−16.1619	−0.230579	−0.115289	+	0.993332i	$0.536780\pi$
$18$	−32.0540	−0.419733
$19$	87.4470	1.05588	0.527940	−	0.849282i	$-0.322965\pi$
$19$	87.4470	1.05588	0.527940	+	0.849282i	$0.322965\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	18.4924	0.179209
$23$	−176.477	−1.59992	−0.799958	−	0.600056i	$-0.795145\pi$
$23$	−176.477	−1.59992	−0.799958	+	0.600056i	$0.795145\pi$
$24$	35.4233	0.301281
$25$	0	0
$26$	194.270	1.46536
$27$	27.0000	0.192450
$28$	32.7926	0.221329
$29$	142.170	0.910358	0.455179	−	0.890400i	$-0.349575\pi$
$29$	142.170	0.910358	0.455179	+	0.890400i	$0.349575\pi$
$30$	0	0
$31$	−94.3002	−0.546349	−0.273174	−	0.961965i	$-0.588074\pi$
$31$	−94.3002	−0.546349	−0.273174	+	0.961965i	$0.588074\pi$
$32$	188.793	1.04294
$33$	−15.5767	−0.0821684
$34$	57.5616	0.290345
$35$	0	0
$36$	42.1619	0.195194
$37$	−17.3305	−0.0770031	−0.0385016	−	0.999259i	$-0.512258\pi$
$37$	−17.3305	−0.0770031	−0.0385016	+	0.999259i	$0.512258\pi$
$38$	−311.447	−1.32956
$39$	−163.639	−0.671878
$40$	0	0
$41$	210.270	0.800942	0.400471	−	0.916309i	$-0.368846\pi$
$41$	210.270	0.800942	0.400471	+	0.916309i	$0.368846\pi$
$42$	−74.7926	−0.274780
$43$	−521.570	−1.84974	−0.924868	−	0.380287i	$-0.875825\pi$
$43$	−521.570	−1.84974	−0.924868	+	0.380287i	$0.875825\pi$
$44$	−24.3239	−0.0833400
$45$	0	0
$46$	628.533	2.01461
$47$	105.417	0.327162	0.163581	−	0.986530i	$-0.447696\pi$
$47$	105.417	0.327162	0.163581	+	0.986530i	$0.447696\pi$
$48$	−238.594	−0.717459
$49$	49.0000	0.142857
$50$	0	0
$51$	−48.4858	−0.133125
$52$	−255.531	−0.681458
$53$	108.978	0.282440	0.141220	−	0.989978i	$-0.454898\pi$
$53$	108.978	0.282440	0.141220	+	0.989978i	$0.454898\pi$
$54$	−96.1619	−0.242333
$55$	0	0
$56$	82.6543	0.197235
$57$	262.341	0.609612
$58$	−506.348	−1.14632
$59$	210.365	0.464189	0.232094	−	0.972693i	$-0.425442\pi$
$59$	210.365	0.464189	0.232094	+	0.972693i	$0.425442\pi$
$60$	0	0
$61$	−674.304	−1.41534	−0.707670	−	0.706543i	$-0.750254\pi$
$61$	−674.304	−1.41534	−0.707670	+	0.706543i	$0.750254\pi$
$62$	335.855	0.687962
$63$	63.0000	0.125988
$64$	−36.1449	−0.0705955
$65$	0	0
$66$	55.4773	0.103466
$67$	−324.929	−0.592484	−0.296242	−	0.955113i	$-0.595733\pi$
$67$	−324.929	−0.592484	−0.296242	+	0.955113i	$0.595733\pi$
$68$	−75.7131	−0.135023
$69$	−529.432	−0.923712
$70$	0	0
$71$	793.965	1.32713	0.663565	−	0.748118i	$-0.269042\pi$
$71$	793.965	1.32713	0.663565	+	0.748118i	$0.269042\pi$
$72$	106.270	0.173945
$73$	−315.417	−0.505709	−0.252854	−	0.967504i	$-0.581369\pi$
$73$	−315.417	−0.505709	−0.252854	+	0.967504i	$0.581369\pi$
$74$	61.7235	0.0969623
$75$	0	0
$76$	409.659	0.618304
$77$	−36.3457	−0.0537918
$78$	582.810	0.846028
$79$	−425.840	−0.606465	−0.303233	−	0.952917i	$-0.598066\pi$
$79$	−425.840	−0.606465	−0.303233	+	0.952917i	$0.598066\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−748.887	−1.00855
$83$	283.029	0.374295	0.187148	−	0.982332i	$-0.440076\pi$
$83$	283.029	0.374295	0.187148	+	0.982332i	$0.440076\pi$
$84$	98.3778	0.127785
$85$	0	0
$86$	1857.60	2.32919
$87$	426.511	0.525596
$88$	−61.3087	−0.0742674
$89$	−843.131	−1.00418	−0.502088	−	0.864817i	$-0.667435\pi$
$89$	−843.131	−1.00418	−0.502088	+	0.864817i	$0.667435\pi$
$90$	0	0
$91$	−381.825	−0.439847
$92$	−826.736	−0.936882
$93$	−282.901	−0.315435
$94$	−375.447	−0.411962
$95$	0	0
$96$	566.378	0.602143
$97$	−1537.33	−1.60920	−0.804601	−	0.593816i	$-0.797621\pi$
$97$	−1537.33	−1.60920	−0.804601	+	0.593816i	$0.797621\pi$
$98$	−174.516	−0.179886
$99$	−46.7301	−0.0474399
$100$	0	0
$101$	−1589.99	−1.56644	−0.783219	−	0.621745i	$-0.786424\pi$
$101$	−1589.99	−1.56644	−0.783219	+	0.621745i	$0.786424\pi$
$102$	172.685	0.167631
$103$	164.793	0.157646	0.0788228	−	0.996889i	$-0.474884\pi$
$103$	164.793	0.157646	0.0788228	+	0.996889i	$0.474884\pi$
$104$	−644.071	−0.607273
$105$	0	0
$106$	−388.132	−0.355648
$107$	−1184.08	−1.06981	−0.534904	−	0.844913i	$-0.679652\pi$
$107$	−1184.08	−1.06981	−0.534904	+	0.844913i	$0.679652\pi$
$108$	126.486	0.112695
$109$	333.247	0.292837	0.146419	−	0.989223i	$-0.453225\pi$
$109$	333.247	0.292837	0.146419	+	0.989223i	$0.453225\pi$
$110$	0	0
$111$	−51.9915	−0.0444578
$112$	−556.719	−0.469687
$113$	−1881.49	−1.56634	−0.783168	−	0.621810i	$-0.786398\pi$
$113$	−1881.49	−1.56634	−0.783168	+	0.621810i	$0.786398\pi$
$114$	−934.341	−0.767623
$115$	0	0
$116$	666.020	0.533090
$117$	−490.918	−0.387909
$118$	−749.224	−0.584506
$119$	−113.133	−0.0871507
$120$	0	0
$121$	−1304.04	−0.979745
$122$	2401.57	1.78220
$123$	630.810	0.462424
$124$	−441.764	−0.319932
$125$	0	0
$126$	−224.378	−0.158644
$127$	−1638.79	−1.14503	−0.572516	−	0.819893i	$-0.694033\pi$
$127$	−1638.79	−1.14503	−0.572516	+	0.819893i	$0.694033\pi$
$128$	−1381.61	−0.954048
$129$	−1564.71	−1.06795
$130$	0	0
$131$	−598.142	−0.398931	−0.199465	−	0.979905i	$-0.563921\pi$
$131$	−598.142	−0.398931	−0.199465	+	0.979905i	$0.563921\pi$
$132$	−72.9716	−0.0481164
$133$	612.129	0.399085
$134$	1157.25	0.746055
$135$	0	0
$136$	−190.836	−0.120324
$137$	1005.25	0.626894	0.313447	−	0.949606i	$-0.398516\pi$
$137$	1005.25	0.626894	0.313447	+	0.949606i	$0.398516\pi$
$138$	1885.60	1.16314
$139$	−1875.01	−1.14414	−0.572072	−	0.820204i	$-0.693860\pi$
$139$	−1875.01	−1.14414	−0.572072	+	0.820204i	$0.693860\pi$
$140$	0	0
$141$	316.250	0.188887
$142$	−2827.75	−1.67112
$143$	283.218	0.165621
$144$	−715.781	−0.414225
$145$	0	0
$146$	1123.37	0.636788
$147$	147.000	0.0824786
$148$	−81.1875	−0.0450917
$149$	−1051.80	−0.578299	−0.289150	−	0.957284i	$-0.593373\pi$
$149$	−1051.80	−0.578299	−0.289150	+	0.957284i	$0.593373\pi$
$150$	0	0
$151$	−750.383	−0.404406	−0.202203	−	0.979344i	$-0.564810\pi$
$151$	−750.383	−0.404406	−0.202203	+	0.979344i	$0.564810\pi$
$152$	1032.55	0.550994
$153$	−145.457	−0.0768597
$154$	129.447	0.0677346
$155$	0	0
$156$	−766.594	−0.393440
$157$	1453.90	0.739067	0.369533	−	0.929217i	$-0.379518\pi$
$157$	1453.90	0.739067	0.369533	+	0.929217i	$0.379518\pi$
$158$	1516.65	0.763660
$159$	326.935	0.163067
$160$	0	0
$161$	−1235.34	−0.604711
$162$	−288.486	−0.139911
$163$	−1300.49	−0.624921	−0.312461	−	0.949931i	$-0.601153\pi$
$163$	−1300.49	−0.624921	−0.312461	+	0.949931i	$0.601153\pi$
$164$	985.043	0.469018
$165$	0	0
$166$	−1008.02	−0.471312
$167$	2111.46	0.978381	0.489191	−	0.872177i	$-0.337292\pi$
$167$	2111.46	0.978381	0.489191	+	0.872177i	$0.337292\pi$
$168$	247.963	0.113874
$169$	778.310	0.354260
$170$	0	0
$171$	787.023	0.351960
$172$	−2443.38	−1.08317
$173$	−335.292	−0.147351	−0.0736756	−	0.997282i	$-0.523473\pi$
$173$	−335.292	−0.147351	−0.0736756	+	0.997282i	$0.523473\pi$
$174$	−1519.04	−0.661829
$175$	0	0
$176$	412.945	0.176857
$177$	631.094	0.267999
$178$	3002.85	1.26446
$179$	2322.23	0.969672	0.484836	−	0.874605i	$-0.338879\pi$
$179$	2322.23	0.969672	0.484836	+	0.874605i	$0.338879\pi$
$180$	0	0
$181$	−1525.59	−0.626500	−0.313250	−	0.949671i	$-0.601418\pi$
$181$	−1525.59	−0.626500	−0.313250	+	0.949671i	$0.601418\pi$
$182$	1359.89	0.553855
$183$	−2022.91	−0.817147
$184$	−2083.80	−0.834891
$185$	0	0
$186$	1007.57	0.397195
$187$	83.9165	0.0328160
$188$	493.841	0.191580
$189$	189.000	0.0727393
$190$	0	0
$191$	293.912	0.111344	0.0556721	−	0.998449i	$-0.482270\pi$
$191$	293.912	0.111344	0.0556721	+	0.998449i	$0.482270\pi$
$192$	−108.435	−0.0407583
$193$	3664.91	1.36687	0.683435	−	0.730012i	$-0.260485\pi$
$193$	3664.91	1.36687	0.683435	+	0.730012i	$0.260485\pi$
$194$	5475.29	2.02630
$195$	0	0
$196$	229.548	0.0836546
$197$	5101.89	1.84515	0.922576	−	0.385816i	$-0.126080\pi$
$197$	5101.89	1.84515	0.922576	+	0.385816i	$0.126080\pi$
$198$	166.432	0.0597363
$199$	5025.86	1.79032	0.895161	−	0.445743i	$-0.147061\pi$
$199$	5025.86	1.79032	0.895161	+	0.445743i	$0.147061\pi$
$200$	0	0
$201$	−974.787	−0.342071
$202$	5662.85	1.97246
$203$	995.193	0.344083
$204$	−227.139	−0.0779556
$205$	0	0
$206$	−586.918	−0.198507
$207$	−1588.30	−0.533305
$208$	4338.14	1.44614
$209$	−454.045	−0.150273
$210$	0	0
$211$	−3267.98	−1.06624	−0.533122	−	0.846039i	$-0.678981\pi$
$211$	−3267.98	−1.06624	−0.533122	+	0.846039i	$0.678981\pi$
$212$	510.526	0.165392
$213$	2381.89	0.766219
$214$	4217.17	1.34710
$215$	0	0
$216$	318.810	0.100427
$217$	−660.101	−0.206500
$218$	−1186.88	−0.368741
$219$	−946.250	−0.291971
$220$	0	0
$221$	881.575	0.268331
$222$	185.170	0.0559812
$223$	−5457.65	−1.63888	−0.819442	−	0.573162i	$-0.805717\pi$
$223$	−5457.65	−1.63888	−0.819442	+	0.573162i	$0.805717\pi$
$224$	1321.55	0.394195
$225$	0	0
$226$	6701.04	1.97233
$227$	−281.023	−0.0821682	−0.0410841	−	0.999156i	$-0.513081\pi$
$227$	−281.023	−0.0821682	−0.0410841	+	0.999156i	$0.513081\pi$
$228$	1228.98	0.356978
$229$	2776.64	0.801248	0.400624	−	0.916243i	$-0.368793\pi$
$229$	2776.64	0.801248	0.400624	+	0.916243i	$0.368793\pi$
$230$	0	0
$231$	−109.037	−0.0310567
$232$	1678.71	0.475056
$233$	−5781.09	−1.62546	−0.812729	−	0.582642i	$-0.802019\pi$
$233$	−5781.09	−1.62546	−0.812729	+	0.582642i	$0.802019\pi$
$234$	1748.43	0.488455
$235$	0	0
$236$	985.486	0.271821
$237$	−1277.52	−0.350143
$238$	402.931	0.109740
$239$	1588.17	0.429833	0.214916	−	0.976632i	$-0.431052\pi$
$239$	1588.17	0.429833	0.214916	+	0.976632i	$0.431052\pi$
$240$	0	0
$241$	−4330.01	−1.15735	−0.578673	−	0.815560i	$-0.696429\pi$
$241$	−4330.01	−1.15735	−0.578673	+	0.815560i	$0.696429\pi$
$242$	4644.41	1.23369
$243$	243.000	0.0641500
$244$	−3158.88	−0.828798
$245$	0	0
$246$	−2246.66	−0.582284
$247$	−4769.92	−1.22876
$248$	−1113.47	−0.285104
$249$	849.088	0.216099
$250$	0	0
$251$	1400.53	0.352195	0.176097	−	0.984373i	$-0.443653\pi$
$251$	1400.53	0.352195	0.176097	+	0.984373i	$0.443653\pi$
$252$	295.133	0.0737764
$253$	916.312	0.227700
$254$	5836.64	1.44182
$255$	0	0
$256$	5209.83	1.27193
$257$	−4304.86	−1.04486	−0.522431	−	0.852681i	$-0.674975\pi$
$257$	−4304.86	−1.04486	−0.522431	+	0.852681i	$0.674975\pi$
$258$	5572.80	1.34476
$259$	−121.313	−0.0291045
$260$	0	0
$261$	1279.53	0.303453
$262$	2130.31	0.502333
$263$	1724.69	0.404369	0.202184	−	0.979347i	$-0.435196\pi$
$263$	1724.69	0.404369	0.202184	+	0.979347i	$0.435196\pi$
$264$	−183.926	−0.0428783
$265$	0	0
$266$	−2180.13	−0.502527
$267$	−2529.39	−0.579761
$268$	−1522.18	−0.346948
$269$	8004.82	1.81436	0.907180	−	0.420744i	$-0.138231\pi$
$269$	8004.82	1.81436	0.907180	+	0.420744i	$0.138231\pi$
$270$	0	0
$271$	−1963.65	−0.440160	−0.220080	−	0.975482i	$-0.570632\pi$
$271$	−1963.65	−0.440160	−0.220080	+	0.975482i	$0.570632\pi$
$272$	1285.38	0.286535
$273$	−1145.47	−0.253946
$274$	−3580.26	−0.789384
$275$	0	0
$276$	−2480.21	−0.540909
$277$	3278.33	0.711104	0.355552	−	0.934656i	$-0.384293\pi$
$277$	3278.33	0.711104	0.355552	+	0.934656i	$0.384293\pi$
$278$	6677.93	1.44070
$279$	−848.702	−0.182116
$280$	0	0
$281$	2859.04	0.606961	0.303480	−	0.952838i	$-0.401851\pi$
$281$	2859.04	0.606961	0.303480	+	0.952838i	$0.401851\pi$
$282$	−1126.34	−0.237846
$283$	5433.66	1.14134	0.570668	−	0.821181i	$-0.306685\pi$
$283$	5433.66	1.14134	0.570668	+	0.821181i	$0.306685\pi$
$284$	3719.45	0.777144
$285$	0	0
$286$	−1008.70	−0.208550
$287$	1471.89	0.302728
$288$	1699.13	0.347647
$289$	−4651.79	−0.946833
$290$	0	0
$291$	−4612.00	−0.929073
$292$	−1477.62	−0.296134
$293$	−8583.43	−1.71143	−0.855715	−	0.517447i	$-0.826883\pi$
$293$	−8583.43	−1.71143	−0.855715	+	0.517447i	$0.826883\pi$
$294$	−523.548	−0.103857
$295$	0	0
$296$	−204.634	−0.0401829
$297$	−140.190	−0.0273895
$298$	3746.03	0.728194
$299$	9626.20	1.86186
$300$	0	0
$301$	−3650.99	−0.699135
$302$	2672.53	0.509228
$303$	−4769.98	−0.904384
$304$	−6954.77	−1.31212
$305$	0	0
$306$	518.054	0.0967816
$307$	5269.83	0.979691	0.489846	−	0.871809i	$-0.337053\pi$
$307$	5269.83	0.979691	0.489846	+	0.871809i	$0.337053\pi$
$308$	−170.267	−0.0314995
$309$	494.378	0.0910167
$310$	0	0
$311$	−4761.43	−0.868154	−0.434077	−	0.900876i	$-0.642925\pi$
$311$	−4761.43	−0.868154	−0.434077	+	0.900876i	$0.642925\pi$
$312$	−1932.21	−0.350609
$313$	7602.95	1.37298	0.686492	−	0.727137i	$-0.259150\pi$
$313$	7602.95	1.37298	0.686492	+	0.727137i	$0.259150\pi$
$314$	−5178.13	−0.930632
$315$	0	0
$316$	−1994.91	−0.355135
$317$	−8064.55	−1.42886	−0.714432	−	0.699704i	$-0.753315\pi$
$317$	−8064.55	−1.42886	−0.714432	+	0.699704i	$0.753315\pi$
$318$	−1164.39	−0.205333
$319$	−738.182	−0.129562
$320$	0	0
$321$	−3552.24	−0.617654
$322$	4399.73	0.761452
$323$	−1413.31	−0.243464
$324$	379.457	0.0650647
$325$	0	0
$326$	4631.76	0.786901
$327$	999.741	0.169070
$328$	2482.82	0.417959
$329$	737.917	0.123655
$330$	0	0
$331$	6960.79	1.15589	0.577945	−	0.816076i	$-0.303855\pi$
$331$	6960.79	1.15589	0.577945	+	0.816076i	$0.303855\pi$
$332$	1325.90	0.219181
$333$	−155.974	−0.0256677
$334$	−7520.08	−1.23198
$335$	0	0
$336$	−1670.16	−0.271174
$337$	−4731.61	−0.764828	−0.382414	−	0.923991i	$-0.624907\pi$
$337$	−4731.61	−0.764828	−0.382414	+	0.923991i	$0.624907\pi$
$338$	−2771.99	−0.446084
$339$	−5644.48	−0.904325
$340$	0	0
$341$	489.629	0.0777563
$342$	−2803.02	−0.443187
$343$	343.000	0.0539949
$344$	−6158.58	−0.965256
$345$	0	0
$346$	1194.16	0.185544
$347$	−9796.67	−1.51560	−0.757800	−	0.652487i	$-0.773726\pi$
$347$	−9796.67	−1.51560	−0.757800	+	0.652487i	$0.773726\pi$
$348$	1998.06	0.307779
$349$	12702.4	1.94827	0.974134	−	0.225971i	$-0.0725554\pi$
$349$	12702.4	1.94827	0.974134	+	0.225971i	$0.0725554\pi$
$350$	0	0
$351$	−1472.75	−0.223959
$352$	−980.256	−0.148431
$353$	9970.21	1.50329	0.751644	−	0.659569i	$-0.229261\pi$
$353$	9970.21	1.50329	0.751644	+	0.659569i	$0.229261\pi$
$354$	−2247.67	−0.337465
$355$	0	0
$356$	−3949.78	−0.588028
$357$	−339.400	−0.0503165
$358$	−8270.73	−1.22101
$359$	4388.21	0.645128	0.322564	−	0.946548i	$-0.395455\pi$
$359$	4388.21	0.645128	0.322564	+	0.946548i	$0.395455\pi$
$360$	0	0
$361$	787.970	0.114881
$362$	5433.49	0.788889
$363$	−3912.12	−0.565656
$364$	−1788.72	−0.257567
$365$	0	0
$366$	7204.71	1.02895
$367$	9441.30	1.34287	0.671433	−	0.741065i	$-0.265679\pi$
$367$	9441.30	1.34287	0.671433	+	0.741065i	$0.265679\pi$
$368$	14035.5	1.98818
$369$	1892.43	0.266981
$370$	0	0
$371$	762.847	0.106752
$372$	−1325.29	−0.184713
$373$	3219.40	0.446901	0.223451	−	0.974715i	$-0.428268\pi$
$373$	3219.40	0.446901	0.223451	+	0.974715i	$0.428268\pi$
$374$	−298.873	−0.0413218
$375$	0	0
$376$	1244.73	0.170724
$377$	−7754.89	−1.05941
$378$	−673.133	−0.0915933
$379$	−14011.4	−1.89899	−0.949495	−	0.313783i	$-0.898403\pi$
$379$	−14011.4	−1.89899	−0.949495	+	0.313783i	$0.898403\pi$
$380$	0	0
$381$	−4916.37	−0.661085
$382$	−1046.78	−0.140204
$383$	−5322.87	−0.710147	−0.355073	−	0.934838i	$-0.615544\pi$
$383$	−5322.87	−0.710147	−0.355073	+	0.934838i	$0.615544\pi$
$384$	−4144.83	−0.550820
$385$	0	0
$386$	−13052.8	−1.72116
$387$	−4694.13	−0.616579
$388$	−7201.88	−0.942320
$389$	−3844.51	−0.501091	−0.250545	−	0.968105i	$-0.580610\pi$
$389$	−3844.51	−0.501091	−0.250545	+	0.968105i	$0.580610\pi$
$390$	0	0
$391$	2852.21	0.368907
$392$	578.580	0.0745478
$393$	−1794.43	−0.230323
$394$	−18170.7	−2.32341
$395$	0	0
$396$	−218.915	−0.0277800
$397$	8046.40	1.01722	0.508611	−	0.860996i	$-0.330159\pi$
$397$	8046.40	1.01722	0.508611	+	0.860996i	$0.330159\pi$
$398$	−17899.9	−2.25437
$399$	1836.39	0.230412
$400$	0	0
$401$	7741.38	0.964055	0.482027	−	0.876156i	$-0.339901\pi$
$401$	7741.38	0.964055	0.482027	+	0.876156i	$0.339901\pi$
$402$	3471.76	0.430735
$403$	5143.74	0.635801
$404$	−7448.58	−0.917279
$405$	0	0
$406$	−3544.43	−0.433269
$407$	89.9840	0.0109591
$408$	−572.509	−0.0694691
$409$	−8966.94	−1.08407	−0.542037	−	0.840354i	$-0.682347\pi$
$409$	−8966.94	−1.08407	−0.542037	+	0.840354i	$0.682347\pi$
$410$	0	0
$411$	3015.76	0.361937
$412$	771.997	0.0923145
$413$	1472.55	0.175447
$414$	5656.80	0.671537
$415$	0	0
$416$	−10298.0	−1.21370
$417$	−5625.02	−0.660571
$418$	1617.11	0.189223
$419$	−12413.6	−1.44736	−0.723681	−	0.690135i	$-0.757551\pi$
$419$	−12413.6	−1.44736	−0.723681	+	0.690135i	$0.757551\pi$
$420$	0	0
$421$	−1672.14	−0.193575	−0.0967875	−	0.995305i	$-0.530857\pi$
$421$	−1672.14	−0.193575	−0.0967875	+	0.995305i	$0.530857\pi$
$422$	11639.1	1.34261
$423$	948.750	0.109054
$424$	1286.79	0.147387
$425$	0	0
$426$	−8483.24	−0.964823
$427$	−4720.13	−0.534948
$428$	−5547.02	−0.626461
$429$	849.653	0.0956216
$430$	0	0
$431$	16021.8	1.79059	0.895296	−	0.445472i	$-0.146964\pi$
$431$	16021.8	1.79059	0.895296	+	0.445472i	$0.146964\pi$
$432$	−2147.34	−0.239153
$433$	−10882.7	−1.20782	−0.603912	−	0.797051i	$-0.706392\pi$
$433$	−10882.7	−1.20782	−0.603912	+	0.797051i	$0.706392\pi$
$434$	2350.99	0.260025
$435$	0	0
$436$	1561.15	0.171480
$437$	−15432.4	−1.68932
$438$	3370.12	0.367650
$439$	7738.40	0.841307	0.420653	−	0.907221i	$-0.361801\pi$
$439$	7738.40	0.841307	0.420653	+	0.907221i	$0.361801\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−3139.78	−0.337882
$443$	−8766.56	−0.940207	−0.470103	−	0.882611i	$-0.655783\pi$
$443$	−8766.56	−0.940207	−0.470103	+	0.882611i	$0.655783\pi$
$444$	−243.562	−0.0260337
$445$	0	0
$446$	19437.7	2.06368
$447$	−3155.39	−0.333881
$448$	−253.014	−0.0266826
$449$	3099.58	0.325786	0.162893	−	0.986644i	$-0.447917\pi$
$449$	3099.58	0.325786	0.162893	+	0.986644i	$0.447917\pi$
$450$	0	0
$451$	−1091.77	−0.113990
$452$	−8814.15	−0.917219
$453$	−2251.15	−0.233484
$454$	1000.88	0.103466
$455$	0	0
$456$	3097.66	0.318117
$457$	6122.94	0.626737	0.313369	−	0.949632i	$-0.398542\pi$
$457$	6122.94	0.626737	0.313369	+	0.949632i	$0.398542\pi$
$458$	−9889.16	−1.00893
$459$	−436.372	−0.0443749
$460$	0	0
$461$	−10412.2	−1.05194	−0.525970	−	0.850503i	$-0.676297\pi$
$461$	−10412.2	−1.05194	−0.525970	+	0.850503i	$0.676297\pi$
$462$	388.341	0.0391066
$463$	11278.5	1.13209	0.566043	−	0.824376i	$-0.308474\pi$
$463$	11278.5	1.13209	0.566043	+	0.824376i	$0.308474\pi$
$464$	−11307.0	−1.13128
$465$	0	0
$466$	20589.6	2.04677
$467$	−14923.2	−1.47872	−0.739359	−	0.673311i	$-0.764872\pi$
$467$	−14923.2	−1.47872	−0.739359	+	0.673311i	$0.764872\pi$
$468$	−2299.78	−0.227153
$469$	−2274.50	−0.223938
$470$	0	0
$471$	4361.69	0.426701
$472$	2483.93	0.242230
$473$	2708.11	0.263254
$474$	4549.95	0.440899
$475$	0	0
$476$	−529.992	−0.0510339
$477$	980.804	0.0941466
$478$	−5656.34	−0.541245
$479$	−4674.21	−0.445867	−0.222933	−	0.974834i	$-0.571563\pi$
$479$	−4674.21	−0.445867	−0.222933	+	0.974834i	$0.571563\pi$
$480$	0	0
$481$	945.316	0.0896106
$482$	15421.6	1.45733
$483$	−3706.02	−0.349130
$484$	−6108.99	−0.573721
$485$	0	0
$486$	−865.457	−0.0807777
$487$	−17081.7	−1.58941	−0.794706	−	0.606994i	$-0.792375\pi$
$487$	−17081.7	−1.58941	−0.794706	+	0.606994i	$0.792375\pi$
$488$	−7962.02	−0.738573
$489$	−3901.47	−0.360799
$490$	0	0
$491$	18203.9	1.67318	0.836588	−	0.547832i	$-0.184547\pi$
$491$	18203.9	1.67318	0.836588	+	0.547832i	$0.184547\pi$
$492$	2955.13	0.270787
$493$	−2297.75	−0.209909
$494$	16988.3	1.54725
$495$	0	0
$496$	7499.81	0.678934
$497$	5557.75	0.501608
$498$	−3024.07	−0.272112
$499$	7109.47	0.637803	0.318901	−	0.947788i	$-0.396686\pi$
$499$	7109.47	0.637803	0.318901	+	0.947788i	$0.396686\pi$
$500$	0	0
$501$	6334.38	0.564869
$502$	−4988.07	−0.443483
$503$	15402.0	1.36529	0.682647	−	0.730748i	$-0.260829\pi$
$503$	15402.0	1.36529	0.682647	+	0.730748i	$0.260829\pi$
$504$	743.889	0.0657450
$505$	0	0
$506$	−3263.49	−0.286719
$507$	2334.93	0.204532
$508$	−7677.17	−0.670511
$509$	6404.72	0.557730	0.278865	−	0.960330i	$-0.410042\pi$
$509$	6404.72	0.557730	0.278865	+	0.960330i	$0.410042\pi$
$510$	0	0
$511$	−2207.92	−0.191140
$512$	−7502.22	−0.647567
$513$	2361.07	0.203204
$514$	15332.0	1.31569
$515$	0	0
$516$	−7330.13	−0.625370
$517$	−547.348	−0.0465616
$518$	432.064	0.0366483
$519$	−1005.88	−0.0850732
$520$	0	0
$521$	−8916.72	−0.749806	−0.374903	−	0.927064i	$-0.622324\pi$
$521$	−8916.72	−0.749806	−0.374903	+	0.927064i	$0.622324\pi$
$522$	−4557.13	−0.382107
$523$	6929.40	0.579353	0.289677	−	0.957125i	$-0.406452\pi$
$523$	6929.40	0.579353	0.289677	+	0.957125i	$0.406452\pi$
$524$	−2802.09	−0.233607
$525$	0	0
$526$	−6142.58	−0.509181
$527$	1524.07	0.125977
$528$	1238.83	0.102109
$529$	18977.2	1.55973
$530$	0	0
$531$	1893.28	0.154730
$532$	2867.61	0.233697
$533$	−11469.5	−0.932078
$534$	9008.56	0.730035
$535$	0	0
$536$	−3836.69	−0.309178
$537$	6966.68	0.559840
$538$	−28509.6	−2.28464
$539$	−254.420	−0.0203314
$540$	0	0
$541$	−6929.23	−0.550667	−0.275334	−	0.961349i	$-0.588788\pi$
$541$	−6929.23	−0.550667	−0.275334	+	0.961349i	$0.588788\pi$
$542$	6993.65	0.554249
$543$	−4576.78	−0.361710
$544$	−3051.25	−0.240480
$545$	0	0
$546$	4079.67	0.319769
$547$	8509.95	0.665190	0.332595	−	0.943070i	$-0.392076\pi$
$547$	8509.95	0.665190	0.332595	+	0.943070i	$0.392076\pi$
$548$	4709.26	0.367098
$549$	−6068.74	−0.471780
$550$	0	0
$551$	12432.4	0.961228
$552$	−6251.41	−0.482024
$553$	−2980.88	−0.229222
$554$	−11676.0	−0.895422
$555$	0	0
$556$	−8783.76	−0.669990
$557$	4043.94	0.307625	0.153813	−	0.988100i	$-0.450845\pi$
$557$	4043.94	0.307625	0.153813	+	0.988100i	$0.450845\pi$
$558$	3022.70	0.229321
$559$	28449.8	2.15259
$560$	0	0
$561$	251.750	0.0189463
$562$	−10182.6	−0.764285
$563$	−15878.9	−1.18866	−0.594331	−	0.804221i	$-0.702583\pi$
$563$	−15878.9	−1.18866	−0.594331	+	0.804221i	$0.702583\pi$
$564$	1481.52	0.110609
$565$	0	0
$566$	−19352.3	−1.43717
$567$	567.000	0.0419961
$568$	9374.95	0.692543
$569$	11611.6	0.855510	0.427755	−	0.903895i	$-0.359305\pi$
$569$	11611.6	0.855510	0.427755	+	0.903895i	$0.359305\pi$
$570$	0	0
$571$	17395.7	1.27493	0.637466	−	0.770478i	$-0.279982\pi$
$571$	17395.7	1.27493	0.637466	+	0.770478i	$0.279982\pi$
$572$	1326.78	0.0969850
$573$	881.736	0.0642846
$574$	−5242.21	−0.381195
$575$	0	0
$576$	−325.304	−0.0235318
$577$	11474.0	0.827848	0.413924	−	0.910311i	$-0.364158\pi$
$577$	11474.0	0.827848	0.413924	+	0.910311i	$0.364158\pi$
$578$	16567.6	1.19225
$579$	10994.7	0.789162
$580$	0	0
$581$	1981.20	0.141470
$582$	16425.9	1.16989
$583$	−565.841	−0.0401968
$584$	−3724.37	−0.263896
$585$	0	0
$586$	30570.3	2.15503
$587$	−11870.4	−0.834659	−0.417330	−	0.908755i	$-0.637034\pi$
$587$	−11870.4	−0.834659	−0.417330	+	0.908755i	$0.637034\pi$
$588$	688.645	0.0482980
$589$	−8246.26	−0.576878
$590$	0	0
$591$	15305.7	1.06530
$592$	1378.32	0.0956899
$593$	5760.65	0.398923	0.199462	−	0.979906i	$-0.436081\pi$
$593$	5760.65	0.398923	0.199462	+	0.979906i	$0.436081\pi$
$594$	499.295	0.0344888
$595$	0	0
$596$	−4927.31	−0.338642
$597$	15077.6	1.03364
$598$	−34284.2	−2.34446
$599$	−21696.5	−1.47996	−0.739978	−	0.672631i	$-0.765164\pi$
$599$	−21696.5	−1.47996	−0.739978	+	0.672631i	$0.765164\pi$
$600$	0	0
$601$	−12403.0	−0.841812	−0.420906	−	0.907104i	$-0.638288\pi$
$601$	−12403.0	−0.841812	−0.420906	+	0.907104i	$0.638288\pi$
$602$	13003.2	0.880350
$603$	−2924.36	−0.197495
$604$	−3515.29	−0.236813
$605$	0	0
$606$	16988.5	1.13880
$607$	−17066.5	−1.14120	−0.570600	−	0.821228i	$-0.693289\pi$
$607$	−17066.5	−1.14120	−0.570600	+	0.821228i	$0.693289\pi$
$608$	16509.3	1.10122
$609$	2985.58	0.198656
$610$	0	0
$611$	−5750.10	−0.380727
$612$	−681.418	−0.0450077
$613$	−2707.50	−0.178393	−0.0891965	−	0.996014i	$-0.528430\pi$
$613$	−2707.50	−0.178393	−0.0891965	+	0.996014i	$0.528430\pi$
$614$	−18768.8	−1.23363
$615$	0	0
$616$	−429.161	−0.0280704
$617$	23226.3	1.51549	0.757743	−	0.652553i	$-0.226302\pi$
$617$	23226.3	1.51549	0.757743	+	0.652553i	$0.226302\pi$
$618$	−1760.75	−0.114608
$619$	−2298.43	−0.149243	−0.0746216	−	0.997212i	$-0.523775\pi$
$619$	−2298.43	−0.149243	−0.0746216	+	0.997212i	$0.523775\pi$
$620$	0	0
$621$	−4764.89	−0.307904
$622$	16958.1	1.09318
$623$	−5901.91	−0.379543
$624$	13014.4	0.834926
$625$	0	0
$626$	−27078.3	−1.72886
$627$	−1362.14	−0.0867599
$628$	6811.01	0.432785
$629$	280.094	0.0177553
$630$	0	0
$631$	−663.913	−0.0418858	−0.0209429	−	0.999781i	$-0.506667\pi$
$631$	−663.913	−0.0418858	−0.0209429	+	0.999781i	$0.506667\pi$
$632$	−5028.22	−0.316474
$633$	−9803.95	−0.615596
$634$	28722.3	1.79923
$635$	0	0
$636$	1531.58	0.0954890
$637$	−2672.77	−0.166247
$638$	2629.08	0.163144
$639$	7145.68	0.442377
$640$	0	0
$641$	15215.6	0.937566	0.468783	−	0.883313i	$-0.344693\pi$
$641$	15215.6	0.937566	0.468783	+	0.883313i	$0.344693\pi$
$642$	12651.5	0.777749
$643$	−12904.0	−0.791420	−0.395710	−	0.918375i	$-0.629502\pi$
$643$	−12904.0	−0.791420	−0.395710	+	0.918375i	$0.629502\pi$
$644$	−5787.15	−0.354108
$645$	0	0
$646$	5033.58	0.306569
$647$	9425.54	0.572730	0.286365	−	0.958121i	$-0.407553\pi$
$647$	9425.54	0.572730	0.286365	+	0.958121i	$0.407553\pi$
$648$	956.429	0.0579816
$649$	−1092.26	−0.0660632
$650$	0	0
$651$	−1980.30	−0.119223
$652$	−6092.35	−0.365943
$653$	29894.7	1.79153	0.895765	−	0.444528i	$-0.146628\pi$
$653$	29894.7	1.79153	0.895765	+	0.444528i	$0.146628\pi$
$654$	−3560.63	−0.212892
$655$	0	0
$656$	−16723.0	−0.995312
$657$	−2838.75	−0.168570
$658$	−2628.13	−0.155707
$659$	−11593.6	−0.685313	−0.342656	−	0.939461i	$-0.611327\pi$
$659$	−11593.6	−0.685313	−0.342656	+	0.939461i	$0.611327\pi$
$660$	0	0
$661$	−17149.2	−1.00911	−0.504557	−	0.863378i	$-0.668344\pi$
$661$	−17149.2	−1.00911	−0.504557	+	0.863378i	$0.668344\pi$
$662$	−24791.2	−1.45550
$663$	2644.72	0.154921
$664$	3341.94	0.195320
$665$	0	0
$666$	555.511	0.0323208
$667$	−25089.9	−1.45650
$668$	9891.47	0.572923
$669$	−16372.9	−0.946210
$670$	0	0
$671$	3501.15	0.201431
$672$	3964.64	0.227589
$673$	16475.0	0.943633	0.471817	−	0.881697i	$-0.343598\pi$
$673$	16475.0	0.943633	0.471817	+	0.881697i	$0.343598\pi$
$674$	16851.9	0.963071
$675$	0	0
$676$	3646.11	0.207448
$677$	4559.89	0.258864	0.129432	−	0.991588i	$-0.458685\pi$
$677$	4559.89	0.258864	0.129432	+	0.991588i	$0.458685\pi$
$678$	20103.1	1.13872
$679$	−10761.3	−0.608221
$680$	0	0
$681$	−843.070	−0.0474398
$682$	−1743.84	−0.0979106
$683$	27895.9	1.56282	0.781411	−	0.624017i	$-0.214500\pi$
$683$	27895.9	1.56282	0.781411	+	0.624017i	$0.214500\pi$
$684$	3686.93	0.206101
$685$	0	0
$686$	−1221.61	−0.0679904
$687$	8329.93	0.462601
$688$	41481.1	2.29862
$689$	−5944.37	−0.328683
$690$	0	0
$691$	−28178.5	−1.55132	−0.775659	−	0.631152i	$-0.782582\pi$
$691$	−28178.5	−1.55132	−0.775659	+	0.631152i	$0.782582\pi$
$692$	−1570.73	−0.0862862
$693$	−327.111	−0.0179306
$694$	34891.4	1.90844
$695$	0	0
$696$	5036.14	0.274274
$697$	−3398.37	−0.184680
$698$	−45240.4	−2.45326
$699$	−17343.3	−0.938458
$700$	0	0
$701$	3912.96	0.210828	0.105414	−	0.994428i	$-0.466383\pi$
$701$	3912.96	0.210828	0.105414	+	0.994428i	$0.466383\pi$
$702$	5245.29	0.282009
$703$	−1515.50	−0.0813060
$704$	187.673	0.0100471
$705$	0	0
$706$	−35509.4	−1.89294
$707$	−11130.0	−0.592058
$708$	2956.46	0.156936
$709$	−7782.72	−0.412251	−0.206126	−	0.978526i	$-0.566086\pi$
$709$	−7782.72	−0.412251	−0.206126	+	0.978526i	$0.566086\pi$
$710$	0	0
$711$	−3832.56	−0.202155
$712$	−9955.49	−0.524014
$713$	16641.8	0.874112
$714$	1208.79	0.0633584
$715$	0	0
$716$	10878.8	0.567823
$717$	4764.50	0.248164
$718$	−15628.9	−0.812345
$719$	−27868.8	−1.44552	−0.722762	−	0.691097i	$-0.757128\pi$
$719$	−27868.8	−1.44552	−0.722762	+	0.691097i	$0.757128\pi$
$720$	0	0
$721$	1153.55	0.0595844
$722$	−2806.40	−0.144658
$723$	−12990.0	−0.668194
$724$	−7146.89	−0.366868
$725$	0	0
$726$	13933.2	0.712274
$727$	34202.4	1.74484	0.872419	−	0.488759i	$-0.162550\pi$
$727$	34202.4	1.74484	0.872419	+	0.488759i	$0.162550\pi$
$728$	−4508.50	−0.229527
$729$	729.000	0.0370370
$730$	0	0
$731$	8429.58	0.426510
$732$	−9476.65	−0.478507
$733$	12544.2	0.632101	0.316051	−	0.948742i	$-0.397643\pi$
$733$	12544.2	0.632101	0.316051	+	0.948742i	$0.397643\pi$
$734$	−33625.7	−1.69094
$735$	0	0
$736$	−33317.6	−1.66862
$737$	1687.11	0.0843221
$738$	−6739.99	−0.336182
$739$	4563.19	0.227144	0.113572	−	0.993530i	$-0.463771\pi$
$739$	4563.19	0.227144	0.113572	+	0.993530i	$0.463771\pi$
$740$	0	0
$741$	−14309.7	−0.709422
$742$	−2716.92	−0.134422
$743$	−10369.7	−0.512017	−0.256009	−	0.966674i	$-0.582408\pi$
$743$	−10369.7	−0.512017	−0.256009	+	0.966674i	$0.582408\pi$
$744$	−3340.42	−0.164605
$745$	0	0
$746$	−11466.1	−0.562738
$747$	2547.26	0.124765
$748$	393.120	0.0192164
$749$	−8288.57	−0.404349
$750$	0	0
$751$	−36808.0	−1.78847	−0.894237	−	0.447595i	$-0.852281\pi$
$751$	−36808.0	−1.78847	−0.894237	+	0.447595i	$0.852281\pi$
$752$	−8383.92	−0.406556
$753$	4201.60	0.203340
$754$	27619.4	1.33401
$755$	0	0
$756$	885.400	0.0425948
$757$	12516.6	0.600955	0.300477	−	0.953789i	$-0.402854\pi$
$757$	12516.6	0.600955	0.300477	+	0.953789i	$0.402854\pi$
$758$	49902.3	2.39121
$759$	2748.93	0.131462
$760$	0	0
$761$	11745.1	0.559473	0.279736	−	0.960077i	$-0.409753\pi$
$761$	11745.1	0.559473	0.279736	+	0.960077i	$0.409753\pi$
$762$	17509.9	0.832438
$763$	2332.73	0.110682
$764$	1376.88	0.0652011
$765$	0	0
$766$	18957.7	0.894216
$767$	−11474.6	−0.540189
$768$	15629.5	0.734350
$769$	36497.1	1.71147	0.855735	−	0.517414i	$-0.173105\pi$
$769$	36497.1	1.71147	0.855735	+	0.517414i	$0.173105\pi$
$770$	0	0
$771$	−12914.6	−0.603252
$772$	17168.8	0.800414
$773$	29858.1	1.38929	0.694644	−	0.719353i	$-0.255562\pi$
$773$	29858.1	1.38929	0.694644	+	0.719353i	$0.255562\pi$
$774$	16718.4	0.776396
$775$	0	0
$776$	−18152.5	−0.839737
$777$	−363.940	−0.0168035
$778$	13692.4	0.630973
$779$	18387.5	0.845699
$780$	0	0
$781$	−4122.45	−0.188877
$782$	−10158.3	−0.464527
$783$	3838.60	0.175199
$784$	−3897.03	−0.177525
$785$	0	0
$786$	6390.94	0.290022
$787$	−3168.00	−0.143491	−0.0717453	−	0.997423i	$-0.522857\pi$
$787$	−3168.00	−0.143491	−0.0717453	+	0.997423i	$0.522857\pi$
$788$	23900.6	1.08049
$789$	5174.07	0.233463
$790$	0	0
$791$	−13170.5	−0.592019
$792$	−551.778	−0.0247558
$793$	36780.8	1.64707
$794$	−28657.7	−1.28089
$795$	0	0
$796$	23544.5	1.04838
$797$	−29317.0	−1.30296	−0.651481	−	0.758665i	$-0.725852\pi$
$797$	−29317.0	−1.30296	−0.651481	+	0.758665i	$0.725852\pi$
$798$	−6540.39	−0.290134
$799$	−1703.74	−0.0754366
$800$	0	0
$801$	−7588.18	−0.334725
$802$	−27571.3	−1.21394
$803$	1637.72	0.0719724
$804$	−4566.54	−0.200310
$805$	0	0
$806$	−18319.7	−0.800600
$807$	24014.5	1.04752
$808$	−18774.3	−0.817422
$809$	16657.3	0.723904	0.361952	−	0.932197i	$-0.382110\pi$
$809$	16657.3	0.723904	0.361952	+	0.932197i	$0.382110\pi$
$810$	0	0
$811$	5144.55	0.222749	0.111375	−	0.993779i	$-0.464475\pi$
$811$	5144.55	0.222749	0.111375	+	0.993779i	$0.464475\pi$
$812$	4662.14	0.201489
$813$	−5890.95	−0.254126
$814$	−320.483	−0.0137997
$815$	0	0
$816$	3856.13	0.165431
$817$	−45609.7	−1.95310
$818$	31936.2	1.36507
$819$	−3436.42	−0.146616
$820$	0	0
$821$	−5217.18	−0.221779	−0.110890	−	0.993833i	$-0.535370\pi$
$821$	−5217.18	−0.221779	−0.110890	+	0.993833i	$0.535370\pi$
$822$	−10740.8	−0.455751
$823$	−42326.5	−1.79272	−0.896360	−	0.443327i	$-0.853798\pi$
$823$	−42326.5	−1.79272	−0.896360	+	0.443327i	$0.853798\pi$
$824$	1945.83	0.0822649
$825$	0	0
$826$	−5244.57	−0.220922
$827$	31675.8	1.33189	0.665946	−	0.746000i	$-0.268028\pi$
$827$	31675.8	1.33189	0.665946	+	0.746000i	$0.268028\pi$
$828$	−7440.62	−0.312294
$829$	−3471.22	−0.145429	−0.0727144	−	0.997353i	$-0.523166\pi$
$829$	−3471.22	−0.145429	−0.0727144	+	0.997353i	$0.523166\pi$
$830$	0	0
$831$	9835.00	0.410556
$832$	1971.57	0.0821539
$833$	−791.934	−0.0329399
$834$	20033.8	0.831791
$835$	0	0
$836$	−2127.05	−0.0879970
$837$	−2546.11	−0.105145
$838$	44211.7	1.82252
$839$	−20964.9	−0.862682	−0.431341	−	0.902189i	$-0.641959\pi$
$839$	−20964.9	−0.862682	−0.431341	+	0.902189i	$0.641959\pi$
$840$	0	0
$841$	−4176.57	−0.171248
$842$	5955.41	0.243749
$843$	8577.12	0.350429
$844$	−15309.4	−0.624373
$845$	0	0
$846$	−3379.02	−0.137321
$847$	−9128.28	−0.370309
$848$	−8667.17	−0.350981
$849$	16301.0	0.658950
$850$	0	0
$851$	3058.44	0.123199
$852$	11158.4	0.448685
$853$	−1084.77	−0.0435426	−0.0217713	−	0.999763i	$-0.506931\pi$
$853$	−1084.77	−0.0435426	−0.0217713	+	0.999763i	$0.506931\pi$
$854$	16811.0	0.673607
$855$	0	0
$856$	−13981.4	−0.558263
$857$	12661.6	0.504679	0.252340	−	0.967639i	$-0.418800\pi$
$857$	12661.6	0.504679	0.252340	+	0.967639i	$0.418800\pi$
$858$	−3026.09	−0.120407
$859$	−39678.7	−1.57604	−0.788021	−	0.615648i	$-0.788894\pi$
$859$	−39678.7	−1.57604	−0.788021	+	0.615648i	$0.788894\pi$
$860$	0	0
$861$	4415.67	0.174780
$862$	−57062.6	−2.25471
$863$	41614.0	1.64143	0.820717	−	0.571334i	$-0.193574\pi$
$863$	41614.0	1.64143	0.820717	+	0.571334i	$0.193574\pi$
$864$	5097.40	0.200714
$865$	0	0
$866$	38759.2	1.52089
$867$	−13955.4	−0.546654
$868$	−3092.35	−0.120923
$869$	2211.06	0.0863120
$870$	0	0
$871$	17723.7	0.689489
$872$	3934.90	0.152813
$873$	−13836.0	−0.536400
$874$	54963.3	2.12719
$875$	0	0
$876$	−4432.86	−0.170973
$877$	37061.0	1.42698	0.713490	−	0.700665i	$-0.247113\pi$
$877$	37061.0	1.42698	0.713490	+	0.700665i	$0.247113\pi$
$878$	−27560.7	−1.05937
$879$	−25750.3	−0.988095
$880$	0	0
$881$	−25468.7	−0.973962	−0.486981	−	0.873412i	$-0.661902\pi$
$881$	−25468.7	−0.973962	−0.486981	+	0.873412i	$0.661902\pi$
$882$	−1570.64	−0.0599619
$883$	−34428.3	−1.31212	−0.656062	−	0.754707i	$-0.727779\pi$
$883$	−34428.3	−1.31212	−0.656062	+	0.754707i	$0.727779\pi$
$884$	4129.88	0.157130
$885$	0	0
$886$	31222.6	1.18391
$887$	−41295.4	−1.56321	−0.781603	−	0.623777i	$-0.785597\pi$
$887$	−41295.4	−1.56321	−0.781603	+	0.623777i	$0.785597\pi$
$888$	−613.903	−0.0231996
$889$	−11471.5	−0.432782
$890$	0	0
$891$	−420.571	−0.0158133
$892$	−25567.2	−0.959702
$893$	9218.37	0.345443
$894$	11238.1	0.420423
$895$	0	0
$896$	−9671.26	−0.360596
$897$	28878.6	1.07495
$898$	−11039.3	−0.410230
$899$	−13406.7	−0.497373
$900$	0	0
$901$	−1761.30	−0.0651247
$902$	3888.40	0.143536
$903$	−10953.0	−0.403646
$904$	−22216.2	−0.817368
$905$	0	0
$906$	8017.59	0.294003
$907$	53733.8	1.96715	0.983573	−	0.180509i	$-0.0577746\pi$
$907$	53733.8	1.96715	0.983573	+	0.180509i	$0.0577746\pi$
$908$	−1316.50	−0.0481162
$909$	−14309.9	−0.522146
$910$	0	0
$911$	24296.8	0.883634	0.441817	−	0.897105i	$-0.354334\pi$
$911$	24296.8	0.883634	0.441817	+	0.897105i	$0.354334\pi$
$912$	−20864.3	−0.757550
$913$	−1469.55	−0.0532696
$914$	−21807.2	−0.789187
$915$	0	0
$916$	13007.6	0.469197
$917$	−4186.99	−0.150782
$918$	1554.16	0.0558769
$919$	4280.46	0.153645	0.0768223	−	0.997045i	$-0.475523\pi$
$919$	4280.46	0.153645	0.0768223	+	0.997045i	$0.475523\pi$
$920$	0	0
$921$	15809.5	0.565625
$922$	37083.6	1.32460
$923$	−43307.9	−1.54442
$924$	−510.801	−0.0181863
$925$	0	0
$926$	−40168.9	−1.42552
$927$	1483.13	0.0525485
$928$	26840.7	0.949450
$929$	−31884.5	−1.12604	−0.563022	−	0.826442i	$-0.690361\pi$
$929$	−31884.5	−1.12604	−0.563022	+	0.826442i	$0.690361\pi$
$930$	0	0
$931$	4284.90	0.150840
$932$	−27082.4	−0.951839
$933$	−14284.3	−0.501229
$934$	53149.6	1.86200
$935$	0	0
$936$	−5796.64	−0.202424
$937$	44523.1	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$937$	44523.1	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$938$	8100.76	0.281982
$939$	22808.8	0.792693
$940$	0	0
$941$	46374.9	1.60657	0.803283	−	0.595598i	$-0.203085\pi$
$941$	46374.9	1.60657	0.803283	+	0.595598i	$0.203085\pi$
$942$	−15534.4	−0.537301
$943$	−37107.9	−1.28144
$944$	−16730.6	−0.576836
$945$	0	0
$946$	−9645.09	−0.331489
$947$	20348.2	0.698234	0.349117	−	0.937079i	$-0.386482\pi$
$947$	20348.2	0.698234	0.349117	+	0.937079i	$0.386482\pi$
$948$	−5984.74	−0.205037
$949$	17204.8	0.588507
$950$	0	0
$951$	−24193.6	−0.824956
$952$	−1335.85	−0.0454782
$953$	45012.9	1.53002	0.765010	−	0.644018i	$-0.222734\pi$
$953$	45012.9	1.53002	0.765010	+	0.644018i	$0.222734\pi$
$954$	−3493.18	−0.118549
$955$	0	0
$956$	7440.02	0.251702
$957$	−2214.55	−0.0748027
$958$	16647.4	0.561435
$959$	7036.76	0.236944
$960$	0	0
$961$	−20898.5	−0.701503
$962$	−3366.79	−0.112838
$963$	−10656.7	−0.356603
$964$	−20284.6	−0.677721
$965$	0	0
$966$	13199.2	0.439624
$967$	40305.8	1.34038	0.670190	−	0.742190i	$-0.266213\pi$
$967$	40305.8	1.34038	0.670190	+	0.742190i	$0.266213\pi$
$968$	−15397.8	−0.511265
$969$	−4239.93	−0.140564
$970$	0	0
$971$	33991.8	1.12343	0.561713	−	0.827332i	$-0.310142\pi$
$971$	33991.8	1.12343	0.561713	+	0.827332i	$0.310142\pi$
$972$	1138.37	0.0375651
$973$	−13125.0	−0.432445
$974$	60837.2	2.00139
$975$	0	0
$976$	53628.2	1.75881
$977$	18219.0	0.596600	0.298300	−	0.954472i	$-0.403580\pi$
$977$	18219.0	0.596600	0.298300	+	0.954472i	$0.403580\pi$
$978$	13895.3	0.454317
$979$	4377.73	0.142914
$980$	0	0
$981$	2999.22	0.0976125
$982$	−64834.1	−2.10686
$983$	−7676.89	−0.249089	−0.124545	−	0.992214i	$-0.539747\pi$
$983$	−7676.89	−0.249089	−0.124545	+	0.992214i	$0.539747\pi$
$984$	7448.45	0.241309
$985$	0	0
$986$	8183.55	0.264318
$987$	2213.75	0.0713925
$988$	−22345.4	−0.719537
$989$	92045.3	2.95942
$990$	0	0
$991$	50585.6	1.62150	0.810748	−	0.585395i	$-0.199061\pi$
$991$	50585.6	1.62150	0.810748	+	0.585395i	$0.199061\pi$
$992$	−17803.2	−0.569810
$993$	20882.4	0.667354
$994$	−19794.2	−0.631625
$995$	0	0
$996$	3977.69	0.126544
$997$	53060.5	1.68550	0.842750	−	0.538305i	$-0.180935\pi$
$997$	53060.5	1.68550	0.842750	+	0.538305i	$0.180935\pi$
$998$	−25320.8	−0.803121
$999$	−467.923	−0.0148193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.j.1.1	✓	2
3.2	odd	2		1575.4.a.x.1.2		2
5.2	odd	4		525.4.d.m.274.1		4
5.3	odd	4		525.4.d.m.274.4		4
5.4	even	2		525.4.a.m.1.2	yes	2
15.14	odd	2		1575.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.j.1.1	✓	2	1.1	even	1	trivial
525.4.a.m.1.2	yes	2	5.4	even	2
525.4.d.m.274.1		4	5.2	odd	4
525.4.d.m.274.4		4	5.3	odd	4
1575.4.a.o.1.1		2	15.14	odd	2
1575.4.a.x.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q-3.56155 q^{2} +3.00000 q^{3} +4.68466 q^{4} -10.6847 q^{6} +7.00000 q^{7} +11.8078 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.56155 q^{2} +3.00000 q^{3} +4.68466 q^{4} -10.6847 q^{6} +7.00000 q^{7} +11.8078 q^{8} +9.00000 q^{9} -5.19224 q^{11} +14.0540 q^{12} -54.5464 q^{13} -24.9309 q^{14} -79.5312 q^{16} -16.1619 q^{17} -32.0540 q^{18} +87.4470 q^{19} +21.0000 q^{21} +18.4924 q^{22} -176.477 q^{23} +35.4233 q^{24} +194.270 q^{26} +27.0000 q^{27} +32.7926 q^{28} +142.170 q^{29} -94.3002 q^{31} +188.793 q^{32} -15.5767 q^{33} +57.5616 q^{34} +42.1619 q^{36} -17.3305 q^{37} -311.447 q^{38} -163.639 q^{39} +210.270 q^{41} -74.7926 q^{42} -521.570 q^{43} -24.3239 q^{44} +628.533 q^{46} +105.417 q^{47} -238.594 q^{48} +49.0000 q^{49} -48.4858 q^{51} -255.531 q^{52} +108.978 q^{53} -96.1619 q^{54} +82.6543 q^{56} +262.341 q^{57} -506.348 q^{58} +210.365 q^{59} -674.304 q^{61} +335.855 q^{62} +63.0000 q^{63} -36.1449 q^{64} +55.4773 q^{66} -324.929 q^{67} -75.7131 q^{68} -529.432 q^{69} +793.965 q^{71} +106.270 q^{72} -315.417 q^{73} +61.7235 q^{74} +409.659 q^{76} -36.3457 q^{77} +582.810 q^{78} -425.840 q^{79} +81.0000 q^{81} -748.887 q^{82} +283.029 q^{83} +98.3778 q^{84} +1857.60 q^{86} +426.511 q^{87} -61.3087 q^{88} -843.131 q^{89} -381.825 q^{91} -826.736 q^{92} -282.901 q^{93} -375.447 q^{94} +566.378 q^{96} -1537.33 q^{97} -174.516 q^{98} -46.7301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9	\( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 31 q^{11} - 9 q^{12} - 39 q^{13} - 21 q^{14} - 23 q^{16} + 79 q^{17} - 27 q^{18} - 56 q^{19} + 42 q^{21} + 4 q^{22} - 254 q^{23} + 9 q^{24} + 203 q^{26} + 54 q^{27} - 21 q^{28} - 62 q^{29} - 135 q^{31} + 291 q^{32} - 93 q^{33} + 111 q^{34} - 27 q^{36} - 113 q^{37} - 392 q^{38} - 117 q^{39} + 235 q^{41} - 63 q^{42} - 804 q^{43} + 174 q^{44} + 585 q^{46} - 152 q^{47} - 69 q^{48} + 98 q^{49} + 237 q^{51} - 375 q^{52} - 149 q^{53} - 81 q^{54} + 21 q^{56} - 168 q^{57} - 621 q^{58} - 441 q^{59} - 223 q^{61} + 313 q^{62} + 126 q^{63} - 431 q^{64} + 12 q^{66} - 1157 q^{67} - 807 q^{68} - 762 q^{69} + 619 q^{71} + 27 q^{72} - 268 q^{73} + 8 q^{74} + 1512 q^{76} - 217 q^{77} + 609 q^{78} - 427 q^{79} + 162 q^{81} - 735 q^{82} - 1211 q^{83} - 63 q^{84} + 1699 q^{86} - 186 q^{87} + 166 q^{88} + 466 q^{89} - 273 q^{91} - 231 q^{92} - 405 q^{93} - 520 q^{94} + 873 q^{96} - 172 q^{97} - 147 q^{98} - 279 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9 - 31 * q^11 - 9 * q^12 - 39 * q^13 - 21 * q^14 - 23 * q^16 + 79 * q^17 - 27 * q^18 - 56 * q^19 + 42 * q^21 + 4 * q^22 - 254 * q^23 + 9 * q^24 + 203 * q^26 + 54 * q^27 - 21 * q^28 - 62 * q^29 - 135 * q^31 + 291 * q^32 - 93 * q^33 + 111 * q^34 - 27 * q^36 - 113 * q^37 - 392 * q^38 - 117 * q^39 + 235 * q^41 - 63 * q^42 - 804 * q^43 + 174 * q^44 + 585 * q^46 - 152 * q^47 - 69 * q^48 + 98 * q^49 + 237 * q^51 - 375 * q^52 - 149 * q^53 - 81 * q^54 + 21 * q^56 - 168 * q^57 - 621 * q^58 - 441 * q^59 - 223 * q^61 + 313 * q^62 + 126 * q^63 - 431 * q^64 + 12 * q^66 - 1157 * q^67 - 807 * q^68 - 762 * q^69 + 619 * q^71 + 27 * q^72 - 268 * q^73 + 8 * q^74 + 1512 * q^76 - 217 * q^77 + 609 * q^78 - 427 * q^79 + 162 * q^81 - 735 * q^82 - 1211 * q^83 - 63 * q^84 + 1699 * q^86 - 186 * q^87 + 166 * q^88 + 466 * q^89 - 273 * q^91 - 231 * q^92 - 405 * q^93 - 520 * q^94 + 873 * q^96 - 172 * q^97 - 147 * q^98 - 279 * q^99

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