LMFDB - Embedded newform 1560.4.a.n.1.1

Newspace parameters

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
Defining polynomial:	$ x^{4} - x^{3} - 41x^{2} - 30x - 4 $ x^4 - x^3 - 41x^2 - 30x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.174957$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} -35.1436 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} -35.1436 q^{7} +9.00000 q^{9} +13.7440 q^{11} +13.0000 q^{13} +15.0000 q^{15} +87.1577 q^{17} +11.6869 q^{19} -105.431 q^{21} -108.262 q^{23} +25.0000 q^{25} +27.0000 q^{27} -150.600 q^{29} -128.191 q^{31} +41.2319 q^{33} -175.718 q^{35} -11.7756 q^{37} +39.0000 q^{39} +366.620 q^{41} +301.237 q^{43} +45.0000 q^{45} -156.862 q^{47} +892.074 q^{49} +261.473 q^{51} -229.058 q^{53} +68.7198 q^{55} +35.0607 q^{57} -377.723 q^{59} -406.355 q^{61} -316.293 q^{63} +65.0000 q^{65} -352.506 q^{67} -324.786 q^{69} -842.944 q^{71} +50.4272 q^{73} +75.0000 q^{75} -483.013 q^{77} -695.344 q^{79} +81.0000 q^{81} -675.845 q^{83} +435.789 q^{85} -451.801 q^{87} -943.389 q^{89} -456.867 q^{91} -384.572 q^{93} +58.4345 q^{95} +73.8249 q^{97} +123.696 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * 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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−35.1436	−1.89758	−0.948789	−	0.315912i	$-0.897690\pi$
$7$	−35.1436	−1.89758	−0.948789	+	0.315912i	$0.897690\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	13.7440	0.376724	0.188362	−	0.982100i	$-0.439682\pi$
$11$	13.7440	0.376724	0.188362	+	0.982100i	$0.439682\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	87.1577	1.24346	0.621731	−	0.783231i	$-0.286430\pi$
$17$	87.1577	1.24346	0.621731	+	0.783231i	$0.286430\pi$
$18$	0	0
$19$	11.6869	0.141114	0.0705568	−	0.997508i	$-0.477522\pi$
$19$	11.6869	0.141114	0.0705568	+	0.997508i	$0.477522\pi$
$20$	0	0
$21$	−105.431	−1.09557
$22$	0	0
$23$	−108.262	−0.981485	−0.490742	−	0.871305i	$-0.663274\pi$
$23$	−108.262	−0.981485	−0.490742	+	0.871305i	$0.663274\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−150.600	−0.964337	−0.482169	−	0.876078i	$-0.660151\pi$
$29$	−150.600	−0.964337	−0.482169	+	0.876078i	$0.660151\pi$
$30$	0	0
$31$	−128.191	−0.742701	−0.371350	−	0.928493i	$-0.621105\pi$
$31$	−128.191	−0.742701	−0.371350	+	0.928493i	$0.621105\pi$
$32$	0	0
$33$	41.2319	0.217502
$34$	0	0
$35$	−175.718	−0.848622
$36$	0	0
$37$	−11.7756	−0.0523216	−0.0261608	−	0.999658i	$-0.508328\pi$
$37$	−11.7756	−0.0523216	−0.0261608	+	0.999658i	$0.508328\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	366.620	1.39650	0.698249	−	0.715855i	$-0.253963\pi$
$41$	366.620	1.39650	0.698249	+	0.715855i	$0.253963\pi$
$42$	0	0
$43$	301.237	1.06833	0.534165	−	0.845380i	$-0.320626\pi$
$43$	301.237	1.06833	0.534165	+	0.845380i	$0.320626\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−156.862	−0.486822	−0.243411	−	0.969923i	$-0.578266\pi$
$47$	−156.862	−0.486822	−0.243411	+	0.969923i	$0.578266\pi$
$48$	0	0
$49$	892.074	2.60080
$50$	0	0
$51$	261.473	0.717913
$52$	0	0
$53$	−229.058	−0.593651	−0.296826	−	0.954932i	$-0.595928\pi$
$53$	−229.058	−0.593651	−0.296826	+	0.954932i	$0.595928\pi$
$54$	0	0
$55$	68.7198	0.168476
$56$	0	0
$57$	35.0607	0.0814720
$58$	0	0
$59$	−377.723	−0.833482	−0.416741	−	0.909025i	$-0.636828\pi$
$59$	−377.723	−0.833482	−0.416741	+	0.909025i	$0.636828\pi$
$60$	0	0
$61$	−406.355	−0.852925	−0.426462	−	0.904505i	$-0.640240\pi$
$61$	−406.355	−0.852925	−0.426462	+	0.904505i	$0.640240\pi$
$62$	0	0
$63$	−316.293	−0.632526
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	−352.506	−0.642768	−0.321384	−	0.946949i	$-0.604148\pi$
$67$	−352.506	−0.642768	−0.321384	+	0.946949i	$0.604148\pi$
$68$	0	0
$69$	−324.786	−0.566661
$70$	0	0
$71$	−842.944	−1.40900	−0.704500	−	0.709704i	$-0.748829\pi$
$71$	−842.944	−1.40900	−0.704500	+	0.709704i	$0.748829\pi$
$72$	0	0
$73$	50.4272	0.0808502	0.0404251	−	0.999183i	$-0.487129\pi$
$73$	50.4272	0.0808502	0.0404251	+	0.999183i	$0.487129\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−483.013	−0.714862
$78$	0	0
$79$	−695.344	−0.990283	−0.495141	−	0.868812i	$-0.664884\pi$
$79$	−695.344	−0.990283	−0.495141	+	0.868812i	$0.664884\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−675.845	−0.893779	−0.446889	−	0.894589i	$-0.647468\pi$
$83$	−675.845	−0.893779	−0.446889	+	0.894589i	$0.647468\pi$
$84$	0	0
$85$	435.789	0.556093
$86$	0	0
$87$	−451.801	−0.556760
$88$	0	0
$89$	−943.389	−1.12358	−0.561792	−	0.827278i	$-0.689888\pi$
$89$	−943.389	−1.12358	−0.561792	+	0.827278i	$0.689888\pi$
$90$	0	0
$91$	−456.867	−0.526293
$92$	0	0
$93$	−384.572	−0.428798
$94$	0	0
$95$	58.4345	0.0631080
$96$	0	0
$97$	73.8249	0.0772761	0.0386381	−	0.999253i	$-0.487698\pi$
$97$	73.8249	0.0772761	0.0386381	+	0.999253i	$0.487698\pi$
$98$	0	0
$99$	123.696	0.125575
$100$	0	0
$101$	−0.889523	−0.000876345 0	−0.000438172	−	1.00000i	$-0.500139\pi$
$101$	−0.889523	−0.000876345 0	−0.000438172		1.00000i	$0.500139\pi$
$102$	0	0
$103$	−458.408	−0.438527	−0.219263	−	0.975666i	$-0.570365\pi$
$103$	−458.408	−0.438527	−0.219263	+	0.975666i	$0.570365\pi$
$104$	0	0
$105$	−527.154	−0.489952
$106$	0	0
$107$	967.396	0.874035	0.437017	−	0.899453i	$-0.356035\pi$
$107$	967.396	0.874035	0.437017	+	0.899453i	$0.356035\pi$
$108$	0	0
$109$	626.068	0.550151	0.275075	−	0.961423i	$-0.411297\pi$
$109$	626.068	0.550151	0.275075	+	0.961423i	$0.411297\pi$
$110$	0	0
$111$	−35.3268	−0.0302079
$112$	0	0
$113$	−2241.20	−1.86579	−0.932896	−	0.360146i	$-0.882727\pi$
$113$	−2241.20	−1.86579	−0.932896	+	0.360146i	$0.882727\pi$
$114$	0	0
$115$	−541.309	−0.438933
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−3063.04	−2.35957
$120$	0	0
$121$	−1142.10	−0.858079
$122$	0	0
$123$	1099.86	0.806268
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	895.939	0.625998	0.312999	−	0.949754i	$-0.398666\pi$
$127$	895.939	0.625998	0.312999	+	0.949754i	$0.398666\pi$
$128$	0	0
$129$	903.711	0.616801
$130$	0	0
$131$	−636.783	−0.424702	−0.212351	−	0.977193i	$-0.568112\pi$
$131$	−636.783	−0.424702	−0.212351	+	0.977193i	$0.568112\pi$
$132$	0	0
$133$	−410.720	−0.267774
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	505.892	0.315483	0.157742	−	0.987480i	$-0.449579\pi$
$137$	505.892	0.315483	0.157742	+	0.987480i	$0.449579\pi$
$138$	0	0
$139$	−1521.56	−0.928470	−0.464235	−	0.885712i	$-0.653671\pi$
$139$	−1521.56	−0.928470	−0.464235	+	0.885712i	$0.653671\pi$
$140$	0	0
$141$	−470.585	−0.281067
$142$	0	0
$143$	178.672	0.104484
$144$	0	0
$145$	−753.002	−0.431265
$146$	0	0
$147$	2676.22	1.50157
$148$	0	0
$149$	−3105.92	−1.70770	−0.853849	−	0.520521i	$-0.825738\pi$
$149$	−3105.92	−1.70770	−0.853849	+	0.520521i	$0.825738\pi$
$150$	0	0
$151$	−1634.44	−0.880856	−0.440428	−	0.897788i	$-0.645173\pi$
$151$	−1634.44	−0.880856	−0.440428	+	0.897788i	$0.645173\pi$
$152$	0	0
$153$	784.420	0.414487
$154$	0	0
$155$	−640.953	−0.332146
$156$	0	0
$157$	3132.82	1.59253	0.796263	−	0.604951i	$-0.206807\pi$
$157$	3132.82	1.59253	0.796263	+	0.604951i	$0.206807\pi$
$158$	0	0
$159$	−687.174	−0.342745
$160$	0	0
$161$	3804.71	1.86244
$162$	0	0
$163$	1419.27	0.681997	0.340999	−	0.940064i	$-0.389235\pi$
$163$	1419.27	0.681997	0.340999	+	0.940064i	$0.389235\pi$
$164$	0	0
$165$	206.159	0.0972697
$166$	0	0
$167$	3871.88	1.79410	0.897051	−	0.441927i	$-0.145705\pi$
$167$	3871.88	1.79410	0.897051	+	0.441927i	$0.145705\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	105.182	0.0470379
$172$	0	0
$173$	−1619.92	−0.711909	−0.355955	−	0.934503i	$-0.615844\pi$
$173$	−1619.92	−0.711909	−0.355955	+	0.934503i	$0.615844\pi$
$174$	0	0
$175$	−878.591	−0.379515
$176$	0	0
$177$	−1133.17	−0.481211
$178$	0	0
$179$	−2139.73	−0.893470	−0.446735	−	0.894666i	$-0.647413\pi$
$179$	−2139.73	−0.893470	−0.446735	+	0.894666i	$0.647413\pi$
$180$	0	0
$181$	−464.563	−0.190777	−0.0953887	−	0.995440i	$-0.530409\pi$
$181$	−464.563	−0.190777	−0.0953887	+	0.995440i	$0.530409\pi$
$182$	0	0
$183$	−1219.06	−0.492436
$184$	0	0
$185$	−58.8780	−0.0233989
$186$	0	0
$187$	1197.89	0.468442
$188$	0	0
$189$	−948.878	−0.365189
$190$	0	0
$191$	2698.18	1.02216	0.511082	−	0.859532i	$-0.329245\pi$
$191$	2698.18	1.02216	0.511082	+	0.859532i	$0.329245\pi$
$192$	0	0
$193$	−3896.41	−1.45321	−0.726606	−	0.687054i	$-0.758903\pi$
$193$	−3896.41	−1.45321	−0.726606	+	0.687054i	$0.758903\pi$
$194$	0	0
$195$	195.000	0.0716115
$196$	0	0
$197$	−1213.46	−0.438862	−0.219431	−	0.975628i	$-0.570420\pi$
$197$	−1213.46	−0.438862	−0.219431	+	0.975628i	$0.570420\pi$
$198$	0	0
$199$	−4368.96	−1.55632	−0.778160	−	0.628066i	$-0.783847\pi$
$199$	−4368.96	−1.55632	−0.778160	+	0.628066i	$0.783847\pi$
$200$	0	0
$201$	−1057.52	−0.371102
$202$	0	0
$203$	5292.64	1.82990
$204$	0	0
$205$	1833.10	0.624533
$206$	0	0
$207$	−974.357	−0.327162
$208$	0	0
$209$	160.624	0.0531609
$210$	0	0
$211$	3107.39	1.01385	0.506923	−	0.861991i	$-0.330783\pi$
$211$	3107.39	1.01385	0.506923	+	0.861991i	$0.330783\pi$
$212$	0	0
$213$	−2528.83	−0.813486
$214$	0	0
$215$	1506.19	0.477772
$216$	0	0
$217$	4505.08	1.40933
$218$	0	0
$219$	151.282	0.0466789
$220$	0	0
$221$	1133.05	0.344874
$222$	0	0
$223$	−5594.97	−1.68012	−0.840060	−	0.542493i	$-0.817480\pi$
$223$	−5594.97	−1.68012	−0.840060	+	0.542493i	$0.817480\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2353.86	−0.688244	−0.344122	−	0.938925i	$-0.611823\pi$
$227$	−2353.86	−0.688244	−0.344122	+	0.938925i	$0.611823\pi$
$228$	0	0
$229$	−1065.47	−0.307459	−0.153729	−	0.988113i	$-0.549128\pi$
$229$	−1065.47	−0.307459	−0.153729	+	0.988113i	$0.549128\pi$
$230$	0	0
$231$	−1449.04	−0.412726
$232$	0	0
$233$	−5318.61	−1.49542	−0.747712	−	0.664023i	$-0.768848\pi$
$233$	−5318.61	−1.49542	−0.747712	+	0.664023i	$0.768848\pi$
$234$	0	0
$235$	−784.309	−0.217713
$236$	0	0
$237$	−2086.03	−0.571740
$238$	0	0
$239$	623.495	0.168747	0.0843735	−	0.996434i	$-0.473111\pi$
$239$	623.495	0.168747	0.0843735	+	0.996434i	$0.473111\pi$
$240$	0	0
$241$	−955.412	−0.255367	−0.127684	−	0.991815i	$-0.540754\pi$
$241$	−955.412	−0.255367	−0.127684	+	0.991815i	$0.540754\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	4460.37	1.16311
$246$	0	0
$247$	151.930	0.0391379
$248$	0	0
$249$	−2027.54	−0.516023
$250$	0	0
$251$	3310.75	0.832560	0.416280	−	0.909236i	$-0.363334\pi$
$251$	3310.75	0.832560	0.416280	+	0.909236i	$0.363334\pi$
$252$	0	0
$253$	−1487.95	−0.369749
$254$	0	0
$255$	1307.37	0.321061
$256$	0	0
$257$	125.710	0.0305119	0.0152560	−	0.999884i	$-0.495144\pi$
$257$	125.710	0.0305119	0.0152560	+	0.999884i	$0.495144\pi$
$258$	0	0
$259$	413.838	0.0992842
$260$	0	0
$261$	−1355.40	−0.321446
$262$	0	0
$263$	2980.02	0.698691	0.349346	−	0.936994i	$-0.386404\pi$
$263$	2980.02	0.698691	0.349346	+	0.936994i	$0.386404\pi$
$264$	0	0
$265$	−1145.29	−0.265489
$266$	0	0
$267$	−2830.17	−0.648702
$268$	0	0
$269$	1769.95	0.401174	0.200587	−	0.979676i	$-0.435715\pi$
$269$	1769.95	0.401174	0.200587	+	0.979676i	$0.435715\pi$
$270$	0	0
$271$	−2492.03	−0.558597	−0.279299	−	0.960204i	$-0.590102\pi$
$271$	−2492.03	−0.558597	−0.279299	+	0.960204i	$0.590102\pi$
$272$	0	0
$273$	−1370.60	−0.303856
$274$	0	0
$275$	343.599	0.0753448
$276$	0	0
$277$	8647.77	1.87579	0.937896	−	0.346918i	$-0.112772\pi$
$277$	8647.77	1.87579	0.937896	+	0.346918i	$0.112772\pi$
$278$	0	0
$279$	−1153.72	−0.247567
$280$	0	0
$281$	−4778.49	−1.01445	−0.507226	−	0.861813i	$-0.669329\pi$
$281$	−4778.49	−1.01445	−0.507226	+	0.861813i	$0.669329\pi$
$282$	0	0
$283$	5654.65	1.18775	0.593877	−	0.804556i	$-0.297597\pi$
$283$	5654.65	1.18775	0.593877	+	0.804556i	$0.297597\pi$
$284$	0	0
$285$	175.304	0.0364354
$286$	0	0
$287$	−12884.3	−2.64996
$288$	0	0
$289$	2683.47	0.546198
$290$	0	0
$291$	221.475	0.0446154
$292$	0	0
$293$	1062.17	0.211783	0.105892	−	0.994378i	$-0.466230\pi$
$293$	1062.17	0.211783	0.105892	+	0.994378i	$0.466230\pi$
$294$	0	0
$295$	−1888.62	−0.372744
$296$	0	0
$297$	371.087	0.0725005
$298$	0	0
$299$	−1407.40	−0.272215
$300$	0	0
$301$	−10586.6	−2.02724
$302$	0	0
$303$	−2.66857	−0.000505958 0
$304$	0	0
$305$	−2031.77	−0.381440
$306$	0	0
$307$	−8643.05	−1.60679	−0.803396	−	0.595445i	$-0.796976\pi$
$307$	−8643.05	−1.60679	−0.803396	+	0.595445i	$0.796976\pi$
$308$	0	0
$309$	−1375.22	−0.253184
$310$	0	0
$311$	3893.30	0.709868	0.354934	−	0.934891i	$-0.384503\pi$
$311$	3893.30	0.709868	0.354934	+	0.934891i	$0.384503\pi$
$312$	0	0
$313$	−3637.44	−0.656871	−0.328435	−	0.944526i	$-0.606521\pi$
$313$	−3637.44	−0.656871	−0.328435	+	0.944526i	$0.606521\pi$
$314$	0	0
$315$	−1581.46	−0.282874
$316$	0	0
$317$	6003.11	1.06362	0.531811	−	0.846863i	$-0.321512\pi$
$317$	6003.11	1.06362	0.531811	+	0.846863i	$0.321512\pi$
$318$	0	0
$319$	−2069.85	−0.363289
$320$	0	0
$321$	2902.19	0.504624
$322$	0	0
$323$	1018.60	0.175470
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	1878.20	0.317630
$328$	0	0
$329$	5512.69	0.923782
$330$	0	0
$331$	−6466.23	−1.07377	−0.536883	−	0.843657i	$-0.680398\pi$
$331$	−6466.23	−1.07377	−0.536883	+	0.843657i	$0.680398\pi$
$332$	0	0
$333$	−105.980	−0.0174405
$334$	0	0
$335$	−1762.53	−0.287455
$336$	0	0
$337$	−2923.30	−0.472529	−0.236265	−	0.971689i	$-0.575923\pi$
$337$	−2923.30	−0.472529	−0.236265	+	0.971689i	$0.575923\pi$
$338$	0	0
$339$	−6723.60	−1.07722
$340$	0	0
$341$	−1761.85	−0.279793
$342$	0	0
$343$	−19296.5	−3.03764
$344$	0	0
$345$	−1623.93	−0.253418
$346$	0	0
$347$	6051.65	0.936223	0.468112	−	0.883669i	$-0.344934\pi$
$347$	6051.65	0.936223	0.468112	+	0.883669i	$0.344934\pi$
$348$	0	0
$349$	−4539.56	−0.696267	−0.348134	−	0.937445i	$-0.613184\pi$
$349$	−4539.56	−0.696267	−0.348134	+	0.937445i	$0.613184\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	5484.84	0.826993	0.413497	−	0.910506i	$-0.364307\pi$
$353$	5484.84	0.826993	0.413497	+	0.910506i	$0.364307\pi$
$354$	0	0
$355$	−4214.72	−0.630124
$356$	0	0
$357$	−9189.12	−1.36230
$358$	0	0
$359$	−8106.78	−1.19181	−0.595905	−	0.803055i	$-0.703206\pi$
$359$	−8106.78	−1.19181	−0.595905	+	0.803055i	$0.703206\pi$
$360$	0	0
$361$	−6722.42	−0.980087
$362$	0	0
$363$	−3426.31	−0.495412
$364$	0	0
$365$	252.136	0.0361573
$366$	0	0
$367$	−6563.94	−0.933610	−0.466805	−	0.884360i	$-0.654595\pi$
$367$	−6563.94	−0.933610	−0.466805	+	0.884360i	$0.654595\pi$
$368$	0	0
$369$	3299.58	0.465499
$370$	0	0
$371$	8049.92	1.12650
$372$	0	0
$373$	14069.6	1.95307	0.976535	−	0.215358i	$-0.0690917\pi$
$373$	14069.6	1.95307	0.976535	+	0.215358i	$0.0690917\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−1957.80	−0.267459
$378$	0	0
$379$	−5008.62	−0.678828	−0.339414	−	0.940637i	$-0.610229\pi$
$379$	−5008.62	−0.678828	−0.339414	+	0.940637i	$0.610229\pi$
$380$	0	0
$381$	2687.82	0.361420
$382$	0	0
$383$	−6718.96	−0.896405	−0.448202	−	0.893932i	$-0.647936\pi$
$383$	−6718.96	−0.896405	−0.448202	+	0.893932i	$0.647936\pi$
$384$	0	0
$385$	−2415.06	−0.319696
$386$	0	0
$387$	2711.13	0.356110
$388$	0	0
$389$	12362.2	1.61128	0.805640	−	0.592406i	$-0.201822\pi$
$389$	12362.2	1.61128	0.805640	+	0.592406i	$0.201822\pi$
$390$	0	0
$391$	−9435.86	−1.22044
$392$	0	0
$393$	−1910.35	−0.245202
$394$	0	0
$395$	−3476.72	−0.442868
$396$	0	0
$397$	2905.75	0.367344	0.183672	−	0.982988i	$-0.441202\pi$
$397$	2905.75	0.367344	0.183672	+	0.982988i	$0.441202\pi$
$398$	0	0
$399$	−1232.16	−0.154599
$400$	0	0
$401$	10275.9	1.27969	0.639844	−	0.768505i	$-0.278999\pi$
$401$	10275.9	1.27969	0.639844	+	0.768505i	$0.278999\pi$
$402$	0	0
$403$	−1666.48	−0.205988
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−161.844	−0.0197108
$408$	0	0
$409$	2615.14	0.316162	0.158081	−	0.987426i	$-0.449469\pi$
$409$	2615.14	0.316162	0.158081	+	0.987426i	$0.449469\pi$
$410$	0	0
$411$	1517.67	0.182144
$412$	0	0
$413$	13274.6	1.58160
$414$	0	0
$415$	−3379.23	−0.399710
$416$	0	0
$417$	−4564.69	−0.536052
$418$	0	0
$419$	784.644	0.0914854	0.0457427	−	0.998953i	$-0.485435\pi$
$419$	784.644	0.0914854	0.0457427	+	0.998953i	$0.485435\pi$
$420$	0	0
$421$	6500.38	0.752515	0.376258	−	0.926515i	$-0.377211\pi$
$421$	6500.38	0.752515	0.376258	+	0.926515i	$0.377211\pi$
$422$	0	0
$423$	−1411.76	−0.162274
$424$	0	0
$425$	2178.94	0.248692
$426$	0	0
$427$	14280.8	1.61849
$428$	0	0
$429$	536.015	0.0603241
$430$	0	0
$431$	12519.0	1.39912	0.699558	−	0.714575i	$-0.253380\pi$
$431$	12519.0	1.39912	0.699558	+	0.714575i	$0.253380\pi$
$432$	0	0
$433$	13600.5	1.50947	0.754734	−	0.656030i	$-0.227766\pi$
$433$	13600.5	1.50947	0.754734	+	0.656030i	$0.227766\pi$
$434$	0	0
$435$	−2259.01	−0.248991
$436$	0	0
$437$	−1265.25	−0.138501
$438$	0	0
$439$	4044.70	0.439733	0.219867	−	0.975530i	$-0.429438\pi$
$439$	4044.70	0.439733	0.219867	+	0.975530i	$0.429438\pi$
$440$	0	0
$441$	8028.67	0.866933
$442$	0	0
$443$	−12860.3	−1.37926	−0.689629	−	0.724162i	$-0.742227\pi$
$443$	−12860.3	−1.37926	−0.689629	+	0.724162i	$0.742227\pi$
$444$	0	0
$445$	−4716.95	−0.502482
$446$	0	0
$447$	−9317.76	−0.985940
$448$	0	0
$449$	16819.8	1.76788	0.883939	−	0.467602i	$-0.154882\pi$
$449$	16819.8	1.76788	0.883939	+	0.467602i	$0.154882\pi$
$450$	0	0
$451$	5038.81	0.526094
$452$	0	0
$453$	−4903.33	−0.508562
$454$	0	0
$455$	−2284.34	−0.235365
$456$	0	0
$457$	−11489.0	−1.17600	−0.588000	−	0.808861i	$-0.700084\pi$
$457$	−11489.0	−1.17600	−0.588000	+	0.808861i	$0.700084\pi$
$458$	0	0
$459$	2353.26	0.239304
$460$	0	0
$461$	9109.43	0.920322	0.460161	−	0.887836i	$-0.347792\pi$
$461$	9109.43	0.920322	0.460161	+	0.887836i	$0.347792\pi$
$462$	0	0
$463$	12617.5	1.26649	0.633244	−	0.773952i	$-0.281723\pi$
$463$	12617.5	1.26649	0.633244	+	0.773952i	$0.281723\pi$
$464$	0	0
$465$	−1922.86	−0.191764
$466$	0	0
$467$	−13067.9	−1.29488	−0.647440	−	0.762116i	$-0.724160\pi$
$467$	−13067.9	−1.29488	−0.647440	+	0.762116i	$0.724160\pi$
$468$	0	0
$469$	12388.3	1.21970
$470$	0	0
$471$	9398.47	0.919445
$472$	0	0
$473$	4140.19	0.402466
$474$	0	0
$475$	292.173	0.0282227
$476$	0	0
$477$	−2061.52	−0.197884
$478$	0	0
$479$	−10980.0	−1.04737	−0.523684	−	0.851913i	$-0.675443\pi$
$479$	−10980.0	−1.04737	−0.523684	+	0.851913i	$0.675443\pi$
$480$	0	0
$481$	−153.083	−0.0145114
$482$	0	0
$483$	11414.1	1.07528
$484$	0	0
$485$	369.125	0.0345589
$486$	0	0
$487$	−12455.9	−1.15899	−0.579497	−	0.814975i	$-0.696751\pi$
$487$	−12455.9	−1.15899	−0.579497	+	0.814975i	$0.696751\pi$
$488$	0	0
$489$	4257.80	0.393751
$490$	0	0
$491$	2632.88	0.241996	0.120998	−	0.992653i	$-0.461391\pi$
$491$	2632.88	0.241996	0.120998	+	0.992653i	$0.461391\pi$
$492$	0	0
$493$	−13126.0	−1.19912
$494$	0	0
$495$	618.478	0.0561587
$496$	0	0
$497$	29624.1	2.67369
$498$	0	0
$499$	4716.61	0.423136	0.211568	−	0.977363i	$-0.432143\pi$
$499$	4716.61	0.423136	0.211568	+	0.977363i	$0.432143\pi$
$500$	0	0
$501$	11615.6	1.03583
$502$	0	0
$503$	3448.42	0.305681	0.152841	−	0.988251i	$-0.451158\pi$
$503$	3448.42	0.305681	0.152841	+	0.988251i	$0.451158\pi$
$504$	0	0
$505$	−4.44761	−0.000391913 0
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−16933.1	−1.47455	−0.737274	−	0.675593i	$-0.763888\pi$
$509$	−16933.1	−1.47455	−0.737274	+	0.675593i	$0.763888\pi$
$510$	0	0
$511$	−1772.20	−0.153419
$512$	0	0
$513$	315.546	0.0271573
$514$	0	0
$515$	−2292.04	−0.196115
$516$	0	0
$517$	−2155.90	−0.183397
$518$	0	0
$519$	−4859.76	−0.411021
$520$	0	0
$521$	−5108.30	−0.429556	−0.214778	−	0.976663i	$-0.568903\pi$
$521$	−5108.30	−0.429556	−0.214778	+	0.976663i	$0.568903\pi$
$522$	0	0
$523$	−6314.75	−0.527963	−0.263981	−	0.964528i	$-0.585036\pi$
$523$	−6314.75	−0.527963	−0.263981	+	0.964528i	$0.585036\pi$
$524$	0	0
$525$	−2635.77	−0.219113
$526$	0	0
$527$	−11172.8	−0.923520
$528$	0	0
$529$	−446.375	−0.0366874
$530$	0	0
$531$	−3399.51	−0.277827
$532$	0	0
$533$	4766.06	0.387319
$534$	0	0
$535$	4836.98	0.390880
$536$	0	0
$537$	−6419.20	−0.515845
$538$	0	0
$539$	12260.6	0.979783
$540$	0	0
$541$	6811.49	0.541310	0.270655	−	0.962676i	$-0.412760\pi$
$541$	6811.49	0.541310	0.270655	+	0.962676i	$0.412760\pi$
$542$	0	0
$543$	−1393.69	−0.110145
$544$	0	0
$545$	3130.34	0.246035
$546$	0	0
$547$	5424.93	0.424046	0.212023	−	0.977265i	$-0.431995\pi$
$547$	5424.93	0.424046	0.212023	+	0.977265i	$0.431995\pi$
$548$	0	0
$549$	−3657.19	−0.284308
$550$	0	0
$551$	−1760.05	−0.136081
$552$	0	0
$553$	24436.9	1.87914
$554$	0	0
$555$	−176.634	−0.0135094
$556$	0	0
$557$	−14188.1	−1.07930	−0.539649	−	0.841890i	$-0.681443\pi$
$557$	−14188.1	−1.07930	−0.539649	+	0.841890i	$0.681443\pi$
$558$	0	0
$559$	3916.08	0.296302
$560$	0	0
$561$	3593.68	0.270455
$562$	0	0
$563$	3216.80	0.240803	0.120401	−	0.992725i	$-0.461582\pi$
$563$	3216.80	0.240803	0.120401	+	0.992725i	$0.461582\pi$
$564$	0	0
$565$	−11206.0	−0.834407
$566$	0	0
$567$	−2846.63	−0.210842
$568$	0	0
$569$	−5551.73	−0.409034	−0.204517	−	0.978863i	$-0.565562\pi$
$569$	−5551.73	−0.409034	−0.204517	+	0.978863i	$0.565562\pi$
$570$	0	0
$571$	381.117	0.0279322	0.0139661	−	0.999902i	$-0.495554\pi$
$571$	381.117	0.0279322	0.0139661	+	0.999902i	$0.495554\pi$
$572$	0	0
$573$	8094.54	0.590147
$574$	0	0
$575$	−2706.55	−0.196297
$576$	0	0
$577$	3178.27	0.229312	0.114656	−	0.993405i	$-0.463423\pi$
$577$	3178.27	0.229312	0.114656	+	0.993405i	$0.463423\pi$
$578$	0	0
$579$	−11689.2	−0.839012
$580$	0	0
$581$	23751.6	1.69601
$582$	0	0
$583$	−3148.16	−0.223643
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	438.511	0.0308336	0.0154168	−	0.999881i	$-0.495092\pi$
$587$	438.511	0.0308336	0.0154168	+	0.999881i	$0.495092\pi$
$588$	0	0
$589$	−1498.15	−0.104805
$590$	0	0
$591$	−3640.39	−0.253377
$592$	0	0
$593$	−7029.95	−0.486822	−0.243411	−	0.969923i	$-0.578266\pi$
$593$	−7029.95	−0.486822	−0.243411	+	0.969923i	$0.578266\pi$
$594$	0	0
$595$	−15315.2	−1.05523
$596$	0	0
$597$	−13106.9	−0.898542
$598$	0	0
$599$	−14506.4	−0.989507	−0.494754	−	0.869033i	$-0.664742\pi$
$599$	−14506.4	−0.989507	−0.494754	+	0.869033i	$0.664742\pi$
$600$	0	0
$601$	−17430.6	−1.18304	−0.591520	−	0.806290i	$-0.701472\pi$
$601$	−17430.6	−1.18304	−0.591520	+	0.806290i	$0.701472\pi$
$602$	0	0
$603$	−3172.56	−0.214256
$604$	0	0
$605$	−5710.52	−0.383745
$606$	0	0
$607$	13330.5	0.891383	0.445691	−	0.895187i	$-0.352958\pi$
$607$	13330.5	0.891383	0.445691	+	0.895187i	$0.352958\pi$
$608$	0	0
$609$	15877.9	1.05650
$610$	0	0
$611$	−2039.20	−0.135020
$612$	0	0
$613$	−14016.8	−0.923542	−0.461771	−	0.886999i	$-0.652786\pi$
$613$	−14016.8	−0.923542	−0.461771	+	0.886999i	$0.652786\pi$
$614$	0	0
$615$	5499.30	0.360574
$616$	0	0
$617$	14539.6	0.948688	0.474344	−	0.880339i	$-0.342685\pi$
$617$	14539.6	0.948688	0.474344	+	0.880339i	$0.342685\pi$
$618$	0	0
$619$	−21204.4	−1.37686	−0.688429	−	0.725304i	$-0.741699\pi$
$619$	−21204.4	−1.37686	−0.688429	+	0.725304i	$0.741699\pi$
$620$	0	0
$621$	−2923.07	−0.188887
$622$	0	0
$623$	33154.1	2.13209
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	481.873	0.0306924
$628$	0	0
$629$	−1026.34	−0.0650599
$630$	0	0
$631$	27682.4	1.74646	0.873231	−	0.487307i	$-0.162021\pi$
$631$	27682.4	1.74646	0.873231	+	0.487307i	$0.162021\pi$
$632$	0	0
$633$	9322.17	0.585345
$634$	0	0
$635$	4479.69	0.279955
$636$	0	0
$637$	11597.0	0.721332
$638$	0	0
$639$	−7586.49	−0.469667
$640$	0	0
$641$	−25577.7	−1.57606	−0.788032	−	0.615634i	$-0.788900\pi$
$641$	−25577.7	−1.57606	−0.788032	+	0.615634i	$0.788900\pi$
$642$	0	0
$643$	13204.5	0.809852	0.404926	−	0.914349i	$-0.367297\pi$
$643$	13204.5	0.809852	0.404926	+	0.914349i	$0.367297\pi$
$644$	0	0
$645$	4518.56	0.275842
$646$	0	0
$647$	969.924	0.0589361	0.0294680	−	0.999566i	$-0.490619\pi$
$647$	969.924	0.0589361	0.0294680	+	0.999566i	$0.490619\pi$
$648$	0	0
$649$	−5191.42	−0.313992
$650$	0	0
$651$	13515.3	0.813678
$652$	0	0
$653$	−9040.07	−0.541754	−0.270877	−	0.962614i	$-0.587314\pi$
$653$	−9040.07	−0.541754	−0.270877	+	0.962614i	$0.587314\pi$
$654$	0	0
$655$	−3183.92	−0.189933
$656$	0	0
$657$	453.845	0.0269501
$658$	0	0
$659$	27518.0	1.62663	0.813316	−	0.581822i	$-0.197660\pi$
$659$	27518.0	1.62663	0.813316	+	0.581822i	$0.197660\pi$
$660$	0	0
$661$	22610.3	1.33047	0.665235	−	0.746634i	$-0.268331\pi$
$661$	22610.3	1.33047	0.665235	+	0.746634i	$0.268331\pi$
$662$	0	0
$663$	3399.15	0.199113
$664$	0	0
$665$	−2053.60	−0.119752
$666$	0	0
$667$	16304.3	0.946483
$668$	0	0
$669$	−16784.9	−0.970018
$670$	0	0
$671$	−5584.93	−0.321317
$672$	0	0
$673$	−731.086	−0.0418741	−0.0209371	−	0.999781i	$-0.506665\pi$
$673$	−731.086	−0.0418741	−0.0209371	+	0.999781i	$0.506665\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	9436.61	0.535714	0.267857	−	0.963459i	$-0.413685\pi$
$677$	9436.61	0.535714	0.267857	+	0.963459i	$0.413685\pi$
$678$	0	0
$679$	−2594.48	−0.146637
$680$	0	0
$681$	−7061.59	−0.397358
$682$	0	0
$683$	17455.9	0.977936	0.488968	−	0.872302i	$-0.337373\pi$
$683$	17455.9	0.977936	0.488968	+	0.872302i	$0.337373\pi$
$684$	0	0
$685$	2529.46	0.141088
$686$	0	0
$687$	−3196.40	−0.177512
$688$	0	0
$689$	−2977.75	−0.164649
$690$	0	0
$691$	−28075.1	−1.54563	−0.772814	−	0.634633i	$-0.781151\pi$
$691$	−28075.1	−1.54563	−0.772814	+	0.634633i	$0.781151\pi$
$692$	0	0
$693$	−4347.11	−0.238287
$694$	0	0
$695$	−7607.81	−0.415224
$696$	0	0
$697$	31953.8	1.73649
$698$	0	0
$699$	−15955.8	−0.863384
$700$	0	0
$701$	−20427.0	−1.10059	−0.550297	−	0.834969i	$-0.685486\pi$
$701$	−20427.0	−1.10059	−0.550297	+	0.834969i	$0.685486\pi$
$702$	0	0
$703$	−137.620	−0.00738329
$704$	0	0
$705$	−2352.93	−0.125697
$706$	0	0
$707$	31.2610	0.00166293
$708$	0	0
$709$	10208.4	0.540737	0.270369	−	0.962757i	$-0.412854\pi$
$709$	10208.4	0.540737	0.270369	+	0.962757i	$0.412854\pi$
$710$	0	0
$711$	−6258.10	−0.330094
$712$	0	0
$713$	13878.2	0.728950
$714$	0	0
$715$	893.358	0.0467268
$716$	0	0
$717$	1870.49	0.0974261
$718$	0	0
$719$	32325.4	1.67668	0.838341	−	0.545146i	$-0.183526\pi$
$719$	32325.4	1.67668	0.838341	+	0.545146i	$0.183526\pi$
$720$	0	0
$721$	16110.1	0.832139
$722$	0	0
$723$	−2866.24	−0.147436
$724$	0	0
$725$	−3765.01	−0.192867
$726$	0	0
$727$	−278.152	−0.0141900	−0.00709498	−	0.999975i	$-0.502258\pi$
$727$	−278.152	−0.0141900	−0.00709498	+	0.999975i	$0.502258\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	26255.1	1.32843
$732$	0	0
$733$	23210.0	1.16955	0.584775	−	0.811195i	$-0.301183\pi$
$733$	23210.0	1.16955	0.584775	+	0.811195i	$0.301183\pi$
$734$	0	0
$735$	13381.1	0.671524
$736$	0	0
$737$	−4844.83	−0.242146
$738$	0	0
$739$	−28322.4	−1.40982	−0.704910	−	0.709296i	$-0.749013\pi$
$739$	−28322.4	−1.40982	−0.704910	+	0.709296i	$0.749013\pi$
$740$	0	0
$741$	455.789	0.0225963
$742$	0	0
$743$	−13502.7	−0.666710	−0.333355	−	0.942801i	$-0.608181\pi$
$743$	−13502.7	−0.666710	−0.333355	+	0.942801i	$0.608181\pi$
$744$	0	0
$745$	−15529.6	−0.763706
$746$	0	0
$747$	−6082.61	−0.297926
$748$	0	0
$749$	−33997.8	−1.65855
$750$	0	0
$751$	3344.10	0.162487	0.0812436	−	0.996694i	$-0.474111\pi$
$751$	3344.10	0.162487	0.0812436	+	0.996694i	$0.474111\pi$
$752$	0	0
$753$	9932.25	0.480679
$754$	0	0
$755$	−8172.22	−0.393931
$756$	0	0
$757$	16645.4	0.799190	0.399595	−	0.916692i	$-0.369151\pi$
$757$	16645.4	0.799190	0.399595	+	0.916692i	$0.369151\pi$
$758$	0	0
$759$	−4463.84	−0.213475
$760$	0	0
$761$	21705.9	1.03395	0.516977	−	0.855999i	$-0.327057\pi$
$761$	21705.9	1.03395	0.516977	+	0.855999i	$0.327057\pi$
$762$	0	0
$763$	−22002.3	−1.04395
$764$	0	0
$765$	3922.10	0.185364
$766$	0	0
$767$	−4910.41	−0.231166
$768$	0	0
$769$	−27050.4	−1.26848	−0.634240	−	0.773136i	$-0.718687\pi$
$769$	−27050.4	−1.26848	−0.634240	+	0.773136i	$0.718687\pi$
$770$	0	0
$771$	377.129	0.0176161
$772$	0	0
$773$	22869.0	1.06409	0.532045	−	0.846716i	$-0.321424\pi$
$773$	22869.0	1.06409	0.532045	+	0.846716i	$0.321424\pi$
$774$	0	0
$775$	−3204.77	−0.148540
$776$	0	0
$777$	1241.51	0.0573218
$778$	0	0
$779$	4284.65	0.197065
$780$	0	0
$781$	−11585.4	−0.530804
$782$	0	0
$783$	−4066.21	−0.185587
$784$	0	0
$785$	15664.1	0.712199
$786$	0	0
$787$	−35597.3	−1.61234	−0.806168	−	0.591687i	$-0.798462\pi$
$787$	−35597.3	−1.61234	−0.806168	+	0.591687i	$0.798462\pi$
$788$	0	0
$789$	8940.05	0.403390
$790$	0	0
$791$	78763.9	3.54048
$792$	0	0
$793$	−5282.61	−0.236559
$794$	0	0
$795$	−3435.87	−0.153280
$796$	0	0
$797$	34475.5	1.53223	0.766114	−	0.642705i	$-0.222188\pi$
$797$	34475.5	1.53223	0.766114	+	0.642705i	$0.222188\pi$
$798$	0	0
$799$	−13671.7	−0.605345
$800$	0	0
$801$	−8490.50	−0.374528
$802$	0	0
$803$	693.070	0.0304582
$804$	0	0
$805$	19023.6	0.832910
$806$	0	0
$807$	5309.85	0.231618
$808$	0	0
$809$	15321.5	0.665855	0.332928	−	0.942952i	$-0.391964\pi$
$809$	15321.5	0.665855	0.332928	+	0.942952i	$0.391964\pi$
$810$	0	0
$811$	−38189.4	−1.65353	−0.826764	−	0.562548i	$-0.809821\pi$
$811$	−38189.4	−1.65353	−0.826764	+	0.562548i	$0.809821\pi$
$812$	0	0
$813$	−7476.08	−0.322506
$814$	0	0
$815$	7096.33	0.304998
$816$	0	0
$817$	3520.53	0.150756
$818$	0	0
$819$	−4111.80	−0.175431
$820$	0	0
$821$	−2531.59	−0.107616	−0.0538082	−	0.998551i	$-0.517136\pi$
$821$	−2531.59	−0.107616	−0.0538082	+	0.998551i	$0.517136\pi$
$822$	0	0
$823$	41111.4	1.74126	0.870628	−	0.491942i	$-0.163713\pi$
$823$	41111.4	1.74126	0.870628	+	0.491942i	$0.163713\pi$
$824$	0	0
$825$	1030.80	0.0435003
$826$	0	0
$827$	12919.5	0.543233	0.271617	−	0.962406i	$-0.412442\pi$
$827$	12919.5	0.543233	0.271617	+	0.962406i	$0.412442\pi$
$828$	0	0
$829$	17503.1	0.733303	0.366651	−	0.930358i	$-0.380504\pi$
$829$	17503.1	0.733303	0.366651	+	0.930358i	$0.380504\pi$
$830$	0	0
$831$	25943.3	1.08299
$832$	0	0
$833$	77751.2	3.23400
$834$	0	0
$835$	19359.4	0.802347
$836$	0	0
$837$	−3461.15	−0.142933
$838$	0	0
$839$	−7315.80	−0.301036	−0.150518	−	0.988607i	$-0.548094\pi$
$839$	−7315.80	−0.301036	−0.150518	+	0.988607i	$0.548094\pi$
$840$	0	0
$841$	−1708.53	−0.0700535
$842$	0	0
$843$	−14335.5	−0.585694
$844$	0	0
$845$	845.000	0.0344010
$846$	0	0
$847$	40137.7	1.62827
$848$	0	0
$849$	16964.0	0.685750
$850$	0	0
$851$	1274.85	0.0513528
$852$	0	0
$853$	45161.1	1.81276	0.906382	−	0.422458i	$-0.138833\pi$
$853$	45161.1	1.81276	0.906382	+	0.422458i	$0.138833\pi$
$854$	0	0
$855$	525.911	0.0210360
$856$	0	0
$857$	−11777.2	−0.469431	−0.234716	−	0.972064i	$-0.575416\pi$
$857$	−11777.2	−0.469431	−0.234716	+	0.972064i	$0.575416\pi$
$858$	0	0
$859$	25218.0	1.00166	0.500830	−	0.865545i	$-0.333028\pi$
$859$	25218.0	1.00166	0.500830	+	0.865545i	$0.333028\pi$
$860$	0	0
$861$	−38653.0	−1.52996
$862$	0	0
$863$	26065.2	1.02812	0.514062	−	0.857753i	$-0.328140\pi$
$863$	26065.2	1.02812	0.514062	+	0.857753i	$0.328140\pi$
$864$	0	0
$865$	−8099.61	−0.318376
$866$	0	0
$867$	8050.42	0.315348
$868$	0	0
$869$	−9556.79	−0.373063
$870$	0	0
$871$	−4582.58	−0.178272
$872$	0	0
$873$	664.424	0.0257587
$874$	0	0
$875$	−4392.95	−0.169724
$876$	0	0
$877$	−20173.2	−0.776740	−0.388370	−	0.921504i	$-0.626962\pi$
$877$	−20173.2	−0.776740	−0.388370	+	0.921504i	$0.626962\pi$
$878$	0	0
$879$	3186.50	0.122273
$880$	0	0
$881$	42933.8	1.64186	0.820929	−	0.571030i	$-0.193456\pi$
$881$	42933.8	1.64186	0.820929	+	0.571030i	$0.193456\pi$
$882$	0	0
$883$	22251.5	0.848043	0.424022	−	0.905652i	$-0.360618\pi$
$883$	22251.5	0.848043	0.424022	+	0.905652i	$0.360618\pi$
$884$	0	0
$885$	−5665.85	−0.215204
$886$	0	0
$887$	36828.7	1.39412	0.697061	−	0.717012i	$-0.254491\pi$
$887$	36828.7	1.39412	0.697061	+	0.717012i	$0.254491\pi$
$888$	0	0
$889$	−31486.5	−1.18788
$890$	0	0
$891$	1113.26	0.0418582
$892$	0	0
$893$	−1833.23	−0.0686973
$894$	0	0
$895$	−10698.7	−0.399572
$896$	0	0
$897$	−4222.21	−0.157163
$898$	0	0
$899$	19305.6	0.716214
$900$	0	0
$901$	−19964.2	−0.738183
$902$	0	0
$903$	−31759.7	−1.17043
$904$	0	0
$905$	−2322.82	−0.0853183
$906$	0	0
$907$	42790.0	1.56650	0.783252	−	0.621704i	$-0.213559\pi$
$907$	42790.0	1.56650	0.783252	+	0.621704i	$0.213559\pi$
$908$	0	0
$909$	−8.00570	−0.000292115 0
$910$	0	0
$911$	3041.01	0.110596	0.0552981	−	0.998470i	$-0.482389\pi$
$911$	3041.01	0.110596	0.0552981	+	0.998470i	$0.482389\pi$
$912$	0	0
$913$	−9288.79	−0.336708
$914$	0	0
$915$	−6095.32	−0.220224
$916$	0	0
$917$	22378.9	0.805905
$918$	0	0
$919$	−24088.9	−0.864656	−0.432328	−	0.901716i	$-0.642308\pi$
$919$	−24088.9	−0.864656	−0.432328	+	0.901716i	$0.642308\pi$
$920$	0	0
$921$	−25929.2	−0.927682
$922$	0	0
$923$	−10958.3	−0.390786
$924$	0	0
$925$	−294.390	−0.0104643
$926$	0	0
$927$	−4125.67	−0.146176
$928$	0	0
$929$	2324.68	0.0820992	0.0410496	−	0.999157i	$-0.486930\pi$
$929$	2324.68	0.0820992	0.0410496	+	0.999157i	$0.486930\pi$
$930$	0	0
$931$	10425.6	0.367008
$932$	0	0
$933$	11679.9	0.409842
$934$	0	0
$935$	5989.47	0.209494
$936$	0	0
$937$	12489.1	0.435433	0.217717	−	0.976012i	$-0.430139\pi$
$937$	12489.1	0.435433	0.217717	+	0.976012i	$0.430139\pi$
$938$	0	0
$939$	−10912.3	−0.379244
$940$	0	0
$941$	−25862.3	−0.895949	−0.447974	−	0.894046i	$-0.647854\pi$
$941$	−25862.3	−0.895949	−0.447974	+	0.894046i	$0.647854\pi$
$942$	0	0
$943$	−39690.9	−1.37064
$944$	0	0
$945$	−4744.39	−0.163317
$946$	0	0
$947$	14137.1	0.485104	0.242552	−	0.970138i	$-0.422016\pi$
$947$	14137.1	0.485104	0.242552	+	0.970138i	$0.422016\pi$
$948$	0	0
$949$	655.554	0.0224238
$950$	0	0
$951$	18009.3	0.614083
$952$	0	0
$953$	7474.99	0.254081	0.127040	−	0.991898i	$-0.459452\pi$
$953$	7474.99	0.254081	0.127040	+	0.991898i	$0.459452\pi$
$954$	0	0
$955$	13490.9	0.457126
$956$	0	0
$957$	−6209.54	−0.209745
$958$	0	0
$959$	−17778.9	−0.598654
$960$	0	0
$961$	−13358.2	−0.448396
$962$	0	0
$963$	8706.56	0.291345
$964$	0	0
$965$	−19482.1	−0.649896
$966$	0	0
$967$	12889.8	0.428653	0.214327	−	0.976762i	$-0.431244\pi$
$967$	12889.8	0.428653	0.214327	+	0.976762i	$0.431244\pi$
$968$	0	0
$969$	3055.81	0.101307
$970$	0	0
$971$	−11727.4	−0.387591	−0.193795	−	0.981042i	$-0.562080\pi$
$971$	−11727.4	−0.387591	−0.193795	+	0.981042i	$0.562080\pi$
$972$	0	0
$973$	53473.2	1.76184
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	22896.3	0.749763	0.374882	−	0.927073i	$-0.377683\pi$
$977$	22896.3	0.749763	0.374882	+	0.927073i	$0.377683\pi$
$978$	0	0
$979$	−12965.9	−0.423281
$980$	0	0
$981$	5634.61	0.183384
$982$	0	0
$983$	−36022.9	−1.16882	−0.584412	−	0.811457i	$-0.698675\pi$
$983$	−36022.9	−1.16882	−0.584412	+	0.811457i	$0.698675\pi$
$984$	0	0
$985$	−6067.32	−0.196265
$986$	0	0
$987$	16538.1	0.533346
$988$	0	0
$989$	−32612.5	−1.04855
$990$	0	0
$991$	−868.766	−0.0278479	−0.0139240	−	0.999903i	$-0.504432\pi$
$991$	−868.766	−0.0278479	−0.0139240	+	0.999903i	$0.504432\pi$
$992$	0	0
$993$	−19398.7	−0.619939
$994$	0	0
$995$	−21844.8	−0.696007
$996$	0	0
$997$	−42992.0	−1.36567	−0.682834	−	0.730574i	$-0.739253\pi$
$997$	−42992.0	−1.36567	−0.682834	+	0.730574i	$0.739253\pi$
$998$	0	0
$999$	−317.941	−0.0100693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.n.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.n.1.1	✓	4	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} -35.1436 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} -35.1436 q^{7} +9.00000 q^{9} +13.7440 q^{11} +13.0000 q^{13} +15.0000 q^{15} +87.1577 q^{17} +11.6869 q^{19} -105.431 q^{21} -108.262 q^{23} +25.0000 q^{25} +27.0000 q^{27} -150.600 q^{29} -128.191 q^{31} +41.2319 q^{33} -175.718 q^{35} -11.7756 q^{37} +39.0000 q^{39} +366.620 q^{41} +301.237 q^{43} +45.0000 q^{45} -156.862 q^{47} +892.074 q^{49} +261.473 q^{51} -229.058 q^{53} +68.7198 q^{55} +35.0607 q^{57} -377.723 q^{59} -406.355 q^{61} -316.293 q^{63} +65.0000 q^{65} -352.506 q^{67} -324.786 q^{69} -842.944 q^{71} +50.4272 q^{73} +75.0000 q^{75} -483.013 q^{77} -695.344 q^{79} +81.0000 q^{81} -675.845 q^{83} +435.789 q^{85} -451.801 q^{87} -943.389 q^{89} -456.867 q^{91} -384.572 q^{93} +58.4345 q^{95} +73.8249 q^{97} +123.696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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