LMFDB - Embedded newform 1075.6.a.a.1.4

Newspace parameters

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$172.412606299$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Defining polynomial:	$x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.58275$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

$f(q)$	$=$	$q-0.582753 q^{2} -3.05838 q^{3} -31.6604 q^{4} +1.78228 q^{6} +103.690 q^{7} +37.0983 q^{8} -233.646 q^{9} +O(q^{10})$	\(q-0.582753 q^{2} -3.05838 q^{3} -31.6604 q^{4} +1.78228 q^{6} +103.690 q^{7} +37.0983 q^{8} -233.646 q^{9} -158.323 q^{11} +96.8296 q^{12} +578.882 q^{13} -60.4256 q^{14} +991.514 q^{16} -253.871 q^{17} +136.158 q^{18} -3092.51 q^{19} -317.124 q^{21} +92.2633 q^{22} -4163.45 q^{23} -113.461 q^{24} -337.345 q^{26} +1457.77 q^{27} -3282.87 q^{28} +6771.63 q^{29} +6264.06 q^{31} -1764.95 q^{32} +484.213 q^{33} +147.944 q^{34} +7397.34 q^{36} +3294.82 q^{37} +1802.17 q^{38} -1770.44 q^{39} -6150.84 q^{41} +184.805 q^{42} +1849.00 q^{43} +5012.58 q^{44} +2426.26 q^{46} -8157.17 q^{47} -3032.43 q^{48} -6055.38 q^{49} +776.436 q^{51} -18327.6 q^{52} +30457.8 q^{53} -849.517 q^{54} +3846.72 q^{56} +9458.07 q^{57} -3946.19 q^{58} -45236.1 q^{59} -7251.18 q^{61} -3650.40 q^{62} -24226.8 q^{63} -30699.9 q^{64} -282.176 q^{66} -19685.7 q^{67} +8037.67 q^{68} +12733.4 q^{69} -48132.2 q^{71} -8667.87 q^{72} -42502.1 q^{73} -1920.07 q^{74} +97910.0 q^{76} -16416.5 q^{77} +1031.73 q^{78} -65151.9 q^{79} +52317.6 q^{81} +3584.42 q^{82} -70403.6 q^{83} +10040.3 q^{84} -1077.51 q^{86} -20710.2 q^{87} -5873.52 q^{88} +29645.1 q^{89} +60024.2 q^{91} +131816. q^{92} -19157.9 q^{93} +4753.61 q^{94} +5397.90 q^{96} +89440.4 q^{97} +3528.79 q^{98} +36991.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} + O(q^{10}) $	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

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 $

$2$	−0.582753	−0.103017	−0.0515085	−	0.998673i	$-0.516403\pi$
$2$	−0.582753	−0.103017	−0.0515085	+	0.998673i	$0.516403\pi$
$3$	−3.05838	−0.196195	−0.0980976	−	0.995177i	$-0.531276\pi$
$3$	−3.05838	−0.196195	−0.0980976	+	0.995177i	$0.531276\pi$
$4$	−31.6604	−0.989387
$5$	0	0
$5$	0	0
$6$	1.78228	0.0202115
$7$	103.690	0.799819	0.399910	−	0.916555i	$-0.369042\pi$
$7$	103.690	0.799819	0.399910	+	0.916555i	$0.369042\pi$
$8$	37.0983	0.204941
$9$	−233.646	−0.961507
$10$	0	0
$11$	−158.323	−0.394515	−0.197257	−	0.980352i	$-0.563203\pi$
$11$	−158.323	−0.394515	−0.197257	+	0.980352i	$0.563203\pi$
$12$	96.8296	0.194113
$13$	578.882	0.950017	0.475009	−	0.879981i	$-0.342445\pi$
$13$	578.882	0.950017	0.475009	+	0.879981i	$0.342445\pi$
$14$	−60.4256	−0.0823950
$15$	0	0
$16$	991.514	0.968275
$17$	−253.871	−0.213055	−0.106527	−	0.994310i	$-0.533973\pi$
$17$	−253.871	−0.213055	−0.106527	+	0.994310i	$0.533973\pi$
$18$	136.158	0.0990517
$19$	−3092.51	−1.96529	−0.982645	−	0.185494i	$-0.940612\pi$
$19$	−3092.51	−1.96529	−0.982645	+	0.185494i	$0.940612\pi$
$20$	0	0
$21$	−317.124	−0.156921
$22$	92.2633	0.0406417
$23$	−4163.45	−1.64109	−0.820547	−	0.571579i	$-0.806331\pi$
$23$	−4163.45	−1.64109	−0.820547	+	0.571579i	$0.806331\pi$
$24$	−113.461	−0.0402084
$25$	0	0
$26$	−337.345	−0.0978680
$27$	1457.77	0.384839
$28$	−3282.87	−0.791331
$29$	6771.63	1.49520	0.747598	−	0.664151i	$-0.231207\pi$
$29$	6771.63	1.49520	0.747598	+	0.664151i	$0.231207\pi$
$30$	0	0
$31$	6264.06	1.17072	0.585358	−	0.810775i	$-0.300954\pi$
$31$	6264.06	1.17072	0.585358	+	0.810775i	$0.300954\pi$
$32$	−1764.95	−0.304690
$33$	484.213	0.0774019
$34$	147.944	0.0219483
$35$	0	0
$36$	7397.34	0.951303
$37$	3294.82	0.395665	0.197833	−	0.980236i	$-0.436610\pi$
$37$	3294.82	0.395665	0.197833	+	0.980236i	$0.436610\pi$
$38$	1802.17	0.202459
$39$	−1770.44	−0.186389
$40$	0	0
$41$	−6150.84	−0.571445	−0.285723	−	0.958312i	$-0.592234\pi$
$41$	−6150.84	−0.571445	−0.285723	+	0.958312i	$0.592234\pi$
$42$	184.805	0.0161655
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	5012.58	0.390328
$45$	0	0
$46$	2426.26	0.169061
$47$	−8157.17	−0.538635	−0.269318	−	0.963051i	$-0.586798\pi$
$47$	−8157.17	−0.538635	−0.269318	+	0.963051i	$0.586798\pi$
$48$	−3032.43	−0.189971
$49$	−6055.38	−0.360289
$50$	0	0
$51$	776.436	0.0418004
$52$	−18327.6	−0.939935
$53$	30457.8	1.48939	0.744697	−	0.667403i	$-0.232594\pi$
$53$	30457.8	1.48939	0.744697	+	0.667403i	$0.232594\pi$
$54$	−849.517	−0.0396449
$55$	0	0
$56$	3846.72	0.163916
$57$	9458.07	0.385581
$58$	−3946.19	−0.154031
$59$	−45236.1	−1.69182	−0.845912	−	0.533322i	$-0.820943\pi$
$59$	−45236.1	−1.69182	−0.845912	+	0.533322i	$0.820943\pi$
$60$	0	0
$61$	−7251.18	−0.249508	−0.124754	−	0.992188i	$-0.539814\pi$
$61$	−7251.18	−0.249508	−0.124754	+	0.992188i	$0.539814\pi$
$62$	−3650.40	−0.120604
$63$	−24226.8	−0.769032
$64$	−30699.9	−0.936887
$65$	0	0
$66$	−282.176	−0.00797372
$67$	−19685.7	−0.535751	−0.267876	−	0.963454i	$-0.586322\pi$
$67$	−19685.7	−0.535751	−0.267876	+	0.963454i	$0.586322\pi$
$68$	8037.67	0.210794
$69$	12733.4	0.321975
$70$	0	0
$71$	−48132.2	−1.13316	−0.566579	−	0.824008i	$-0.691733\pi$
$71$	−48132.2	−1.13316	−0.566579	+	0.824008i	$0.691733\pi$
$72$	−8667.87	−0.197052
$73$	−42502.1	−0.933475	−0.466738	−	0.884396i	$-0.654571\pi$
$73$	−42502.1	−0.933475	−0.466738	+	0.884396i	$0.654571\pi$
$74$	−1920.07	−0.0407603
$75$	0	0
$76$	97910.0	1.94443
$77$	−16416.5	−0.315540
$78$	1031.73	0.0192012
$79$	−65151.9	−1.17452	−0.587258	−	0.809400i	$-0.699793\pi$
$79$	−65151.9	−1.17452	−0.587258	+	0.809400i	$0.699793\pi$
$80$	0	0
$81$	52317.6	0.886004
$82$	3584.42	0.0588686
$83$	−70403.6	−1.12176	−0.560880	−	0.827897i	$-0.689537\pi$
$83$	−70403.6	−1.12176	−0.560880	+	0.827897i	$0.689537\pi$
$84$	10040.3	0.155255
$85$	0	0
$86$	−1077.51	−0.0157100
$87$	−20710.2	−0.293350
$88$	−5873.52	−0.0808522
$89$	29645.1	0.396714	0.198357	−	0.980130i	$-0.436439\pi$
$89$	29645.1	0.396714	0.198357	+	0.980130i	$0.436439\pi$
$90$	0	0
$91$	60024.2	0.759842
$92$	131816.	1.62368
$93$	−19157.9	−0.229689
$94$	4753.61	0.0554887
$95$	0	0
$96$	5397.90	0.0597787
$97$	89440.4	0.965171	0.482585	−	0.875849i	$-0.339698\pi$
$97$	89440.4	0.965171	0.482585	+	0.875849i	$0.339698\pi$
$98$	3528.79	0.0371160
$99$	36991.6	0.379329
$100$	0	0
$101$	−28633.0	−0.279295	−0.139647	−	0.990201i	$-0.544597\pi$
$101$	−28633.0	−0.279295	−0.139647	+	0.990201i	$0.544597\pi$
$102$	−452.470	−0.00430615
$103$	−30124.9	−0.279790	−0.139895	−	0.990166i	$-0.544676\pi$
$103$	−30124.9	−0.279790	−0.139895	+	0.990166i	$0.544676\pi$
$104$	21475.5	0.194697
$105$	0	0
$106$	−17749.4	−0.153433
$107$	−83073.3	−0.701459	−0.350729	−	0.936477i	$-0.614066\pi$
$107$	−83073.3	−0.701459	−0.350729	+	0.936477i	$0.614066\pi$
$108$	−46153.5	−0.380754
$109$	58783.5	0.473903	0.236951	−	0.971522i	$-0.423852\pi$
$109$	58783.5	0.473903	0.236951	+	0.971522i	$0.423852\pi$
$110$	0	0
$111$	−10076.8	−0.0776276
$112$	102810.	0.774445
$113$	232647.	1.71396	0.856981	−	0.515347i	$-0.172337\pi$
$113$	232647.	1.71396	0.856981	+	0.515347i	$0.172337\pi$
$114$	−5511.72	−0.0397214
$115$	0	0
$116$	−214392.	−1.47933
$117$	−135254.	−0.913449
$118$	26361.5	0.174287
$119$	−26323.9	−0.170405
$120$	0	0
$121$	−135985.	−0.844358
$122$	4225.64	0.0257036
$123$	18811.6	0.112115
$124$	−198323.	−1.15829
$125$	0	0
$126$	14118.2	0.0792234
$127$	−109116.	−0.600313	−0.300156	−	0.953890i	$-0.597039\pi$
$127$	−109116.	−0.600313	−0.300156	+	0.953890i	$0.597039\pi$
$128$	74368.9	0.401205
$129$	−5654.95	−0.0299195
$130$	0	0
$131$	139424.	0.709839	0.354920	−	0.934897i	$-0.384508\pi$
$131$	139424.	0.709839	0.354920	+	0.934897i	$0.384508\pi$
$132$	−15330.4	−0.0765805
$133$	−320662.	−1.57188
$134$	11471.9	0.0551915
$135$	0	0
$136$	−9418.19	−0.0436637
$137$	64194.6	0.292211	0.146106	−	0.989269i	$-0.453326\pi$
$137$	64194.6	0.292211	0.146106	+	0.989269i	$0.453326\pi$
$138$	−7420.43	−0.0331689
$139$	−281632.	−1.23636	−0.618181	−	0.786036i	$-0.712130\pi$
$139$	−281632.	−1.23636	−0.618181	+	0.786036i	$0.712130\pi$
$140$	0	0
$141$	24947.7	0.105678
$142$	28049.2	0.116735
$143$	−91650.4	−0.374796
$144$	−231664.	−0.931004
$145$	0	0
$146$	24768.2	0.0961639
$147$	18519.7	0.0706871
$148$	−104315.	−0.391466
$149$	225391.	0.831710	0.415855	−	0.909431i	$-0.363482\pi$
$149$	225391.	0.831710	0.415855	+	0.909431i	$0.363482\pi$
$150$	0	0
$151$	388390.	1.38620	0.693099	−	0.720842i	$-0.256245\pi$
$151$	388390.	1.38620	0.693099	+	0.720842i	$0.256245\pi$
$152$	−114727.	−0.402768
$153$	59316.1	0.204854
$154$	9566.78	0.0325060
$155$	0	0
$156$	56052.9	0.184411
$157$	369612.	1.19673	0.598366	−	0.801223i	$-0.295817\pi$
$157$	369612.	1.19673	0.598366	+	0.801223i	$0.295817\pi$
$158$	37967.4	0.120995
$159$	−93151.7	−0.292212
$160$	0	0
$161$	−431708.	−1.31258
$162$	−30488.2	−0.0912735
$163$	−522204.	−1.53947	−0.769735	−	0.638363i	$-0.779612\pi$
$163$	−522204.	−1.53947	−0.769735	+	0.638363i	$0.779612\pi$
$164$	194738.	0.565381
$165$	0	0
$166$	41027.9	0.115560
$167$	17232.5	0.0478141	0.0239071	−	0.999714i	$-0.492389\pi$
$167$	17232.5	0.0478141	0.0239071	+	0.999714i	$0.492389\pi$
$168$	−11764.7	−0.0321595
$169$	−36189.0	−0.0974674
$170$	0	0
$171$	722553.	1.88964
$172$	−58540.1	−0.150880
$173$	571643.	1.45214	0.726072	−	0.687618i	$-0.241344\pi$
$173$	571643.	1.45214	0.726072	+	0.687618i	$0.241344\pi$
$174$	12068.9	0.0302201
$175$	0	0
$176$	−156980.	−0.381999
$177$	138349.	0.331928
$178$	−17275.7	−0.0408683
$179$	−191963.	−0.447802	−0.223901	−	0.974612i	$-0.571879\pi$
$179$	−191963.	−0.447802	−0.223901	+	0.974612i	$0.571879\pi$
$180$	0	0
$181$	110804.	0.251396	0.125698	−	0.992069i	$-0.459883\pi$
$181$	110804.	0.251396	0.125698	+	0.992069i	$0.459883\pi$
$182$	−34979.3	−0.0782767
$183$	22176.9	0.0489522
$184$	−154457.	−0.336327
$185$	0	0
$186$	11164.3	0.0236619
$187$	40193.8	0.0840533
$188$	258259.	0.532919
$189$	151156.	0.307801
$190$	0	0
$191$	295668.	0.586437	0.293218	−	0.956046i	$-0.405274\pi$
$191$	295668.	0.586437	0.293218	+	0.956046i	$0.405274\pi$
$192$	93892.0	0.183813
$193$	375944.	0.726491	0.363246	−	0.931693i	$-0.381669\pi$
$193$	375944.	0.726491	0.363246	+	0.931693i	$0.381669\pi$
$194$	−52121.6	−0.0994291
$195$	0	0
$196$	191716.	0.356466
$197$	732825.	1.34535	0.672674	−	0.739939i	$-0.265146\pi$
$197$	732825.	1.34535	0.672674	+	0.739939i	$0.265146\pi$
$198$	−21557.0	−0.0390773
$199$	589939.	1.05603	0.528013	−	0.849236i	$-0.322937\pi$
$199$	589939.	1.05603	0.528013	+	0.849236i	$0.322937\pi$
$200$	0	0
$201$	60206.3	0.105112
$202$	16685.9	0.0287722
$203$	702150.	1.19589
$204$	−24582.3	−0.0413568
$205$	0	0
$206$	17555.3	0.0288231
$207$	972774.	1.57792
$208$	573969.	0.919878
$209$	489616.	0.775336
$210$	0	0
$211$	−371450.	−0.574374	−0.287187	−	0.957875i	$-0.592720\pi$
$211$	−371450.	−0.574374	−0.287187	+	0.957875i	$0.592720\pi$
$212$	−964307.	−1.47359
$213$	147207.	0.222320
$214$	48411.2	0.0722622
$215$	0	0
$216$	54080.6	0.0788692
$217$	649520.	0.936361
$218$	−34256.2	−0.0488201
$219$	129988.	0.183143
$220$	0	0
$221$	−146962.	−0.202406
$222$	5872.30	0.00799697
$223$	835552.	1.12515	0.562576	−	0.826746i	$-0.309810\pi$
$223$	835552.	1.12515	0.562576	+	0.826746i	$0.309810\pi$
$224$	−183008.	−0.243697
$225$	0	0
$226$	−135576.	−0.176567
$227$	363342.	0.468006	0.234003	−	0.972236i	$-0.424818\pi$
$227$	363342.	0.468006	0.234003	+	0.972236i	$0.424818\pi$
$228$	−299446.	−0.381489
$229$	−106091.	−0.133688	−0.0668438	−	0.997763i	$-0.521293\pi$
$229$	−106091.	−0.133688	−0.0668438	+	0.997763i	$0.521293\pi$
$230$	0	0
$231$	50208.0	0.0619075
$232$	251216.	0.306427
$233$	−713760.	−0.861315	−0.430658	−	0.902515i	$-0.641718\pi$
$233$	−713760.	−0.861315	−0.430658	+	0.902515i	$0.641718\pi$
$234$	78819.4	0.0941008
$235$	0	0
$236$	1.43219e6	1.67387
$237$	199259.	0.230435
$238$	15340.3	0.0175547
$239$	−70400.0	−0.0797220	−0.0398610	−	0.999205i	$-0.512692\pi$
$239$	−70400.0	−0.0797220	−0.0398610	+	0.999205i	$0.512692\pi$
$240$	0	0
$241$	44600.6	0.0494650	0.0247325	−	0.999694i	$-0.492127\pi$
$241$	44600.6	0.0494650	0.0247325	+	0.999694i	$0.492127\pi$
$242$	79245.5	0.0869833
$243$	−514245.	−0.558668
$244$	229575.	0.246860
$245$	0	0
$246$	−10962.5	−0.0115497
$247$	−1.79020e6	−1.86706
$248$	232386.	0.239928
$249$	215321.	0.220084
$250$	0	0
$251$	1.05989e6	1.06188	0.530942	−	0.847408i	$-0.321838\pi$
$251$	1.05989e6	1.06188	0.530942	+	0.847408i	$0.321838\pi$
$252$	767030.	0.760871
$253$	659170.	0.647435
$254$	63587.4	0.0618425
$255$	0	0
$256$	939058.	0.895556
$257$	966091.	0.912400	0.456200	−	0.889877i	$-0.349210\pi$
$257$	966091.	0.912400	0.456200	+	0.889877i	$0.349210\pi$
$258$	3295.44	0.00308222
$259$	341640.	0.316460
$260$	0	0
$261$	−1.58217e6	−1.43764
$262$	−81249.8	−0.0731255
$263$	−855270.	−0.762454	−0.381227	−	0.924481i	$-0.624498\pi$
$263$	−855270.	−0.762454	−0.381227	+	0.924481i	$0.624498\pi$
$264$	17963.5	0.0158628
$265$	0	0
$266$	186867.	0.161930
$267$	−90665.9	−0.0778334
$268$	623256.	0.530065
$269$	1.55224e6	1.30791	0.653955	−	0.756533i	$-0.273108\pi$
$269$	1.55224e6	1.30791	0.653955	+	0.756533i	$0.273108\pi$
$270$	0	0
$271$	−4177.31	−0.00345520	−0.00172760	−	0.999999i	$-0.500550\pi$
$271$	−4177.31	−0.00345520	−0.00172760	+	0.999999i	$0.500550\pi$
$272$	−251717.	−0.206296
$273$	−183577.	−0.149077
$274$	−37409.5	−0.0301027
$275$	0	0
$276$	−403145.	−0.318558
$277$	1.59749e6	1.25095	0.625475	−	0.780244i	$-0.284905\pi$
$277$	1.59749e6	1.25095	0.625475	+	0.780244i	$0.284905\pi$
$278$	164122.	0.127366
$279$	−1.46357e6	−1.12565
$280$	0	0
$281$	1.60304e6	1.21110	0.605549	−	0.795808i	$-0.292954\pi$
$281$	1.60304e6	1.21110	0.605549	+	0.795808i	$0.292954\pi$
$282$	−14538.4	−0.0108866
$283$	−2.04664e6	−1.51906	−0.759530	−	0.650472i	$-0.774571\pi$
$283$	−2.04664e6	−1.51906	−0.759530	+	0.650472i	$0.774571\pi$
$284$	1.52389e6	1.12113
$285$	0	0
$286$	53409.5	0.0386103
$287$	−637780.	−0.457053
$288$	412374.	0.292961
$289$	−1.35541e6	−0.954608
$290$	0	0
$291$	−273543.	−0.189362
$292$	1.34563e6	0.923569
$293$	−1.35028e6	−0.918869	−0.459434	−	0.888212i	$-0.651948\pi$
$293$	−1.35028e6	−0.918869	−0.459434	+	0.888212i	$0.651948\pi$
$294$	−10792.4	−0.00728198
$295$	0	0
$296$	122232.	0.0810880
$297$	−230798.	−0.151824
$298$	−131347.	−0.0856803
$299$	−2.41014e6	−1.55907
$300$	0	0
$301$	191723.	0.121971
$302$	−226335.	−0.142802
$303$	87570.6	0.0547964
$304$	−3.06626e6	−1.90294
$305$	0	0
$306$	−34566.6	−0.0211035
$307$	361103.	0.218668	0.109334	−	0.994005i	$-0.465128\pi$
$307$	361103.	0.218668	0.109334	+	0.994005i	$0.465128\pi$
$308$	519754.	0.312192
$309$	92133.4	0.0548935
$310$	0	0
$311$	2.12425e6	1.24539	0.622694	−	0.782466i	$-0.286038\pi$
$311$	2.12425e6	1.24539	0.622694	+	0.782466i	$0.286038\pi$
$312$	−65680.3	−0.0381987
$313$	541717.	0.312545	0.156272	−	0.987714i	$-0.450052\pi$
$313$	541717.	0.312545	0.156272	+	0.987714i	$0.450052\pi$
$314$	−215392.	−0.123284
$315$	0	0
$316$	2.06273e6	1.16205
$317$	3.06921e6	1.71545	0.857725	−	0.514110i	$-0.171878\pi$
$317$	3.06921e6	1.71545	0.857725	+	0.514110i	$0.171878\pi$
$318$	54284.4	0.0301028
$319$	−1.07211e6	−0.589877
$320$	0	0
$321$	254070.	0.137623
$322$	251579.	0.135218
$323$	785100.	0.418715
$324$	−1.65640e6	−0.876601
$325$	0	0
$326$	304316.	0.158592
$327$	−179782.	−0.0929775
$328$	−228185.	−0.117112
$329$	−845817.	−0.430811
$330$	0	0
$331$	−282184.	−0.141567	−0.0707836	−	0.997492i	$-0.522550\pi$
$331$	−282184.	−0.141567	−0.0707836	+	0.997492i	$0.522550\pi$
$332$	2.22901e6	1.10986
$333$	−769823.	−0.380435
$334$	−10042.3	−0.00492567
$335$	0	0
$336$	−314432.	−0.151942
$337$	−1.04584e6	−0.501640	−0.250820	−	0.968034i	$-0.580700\pi$
$337$	−1.04584e6	−0.501640	−0.250820	+	0.968034i	$0.580700\pi$
$338$	21089.2	0.0100408
$339$	−711523.	−0.336271
$340$	0	0
$341$	−991746.	−0.461864
$342$	−421070.	−0.194665
$343$	−2.37060e6	−1.08799
$344$	68594.7	0.0312532
$345$	0	0
$346$	−333127.	−0.149596
$347$	−1.90831e6	−0.850796	−0.425398	−	0.905006i	$-0.639866\pi$
$347$	−1.90831e6	−0.850796	−0.425398	+	0.905006i	$0.639866\pi$
$348$	655694.	0.290237
$349$	1.46350e6	0.643175	0.321588	−	0.946880i	$-0.395784\pi$
$349$	1.46350e6	0.643175	0.321588	+	0.946880i	$0.395784\pi$
$350$	0	0
$351$	843874.	0.365603
$352$	279433.	0.120205
$353$	−867374.	−0.370484	−0.185242	−	0.982693i	$-0.559307\pi$
$353$	−867374.	−0.370484	−0.185242	+	0.982693i	$0.559307\pi$
$354$	−80623.4	−0.0341942
$355$	0	0
$356$	−938575.	−0.392504
$357$	80508.6	0.0334327
$358$	111867.	0.0461313
$359$	3.45466e6	1.41472	0.707359	−	0.706855i	$-0.249887\pi$
$359$	3.45466e6	1.41472	0.707359	+	0.706855i	$0.249887\pi$
$360$	0	0
$361$	7.08751e6	2.86237
$362$	−64571.2	−0.0258981
$363$	415893.	0.165659
$364$	−1.90039e6	−0.751778
$365$	0	0
$366$	−12923.6	−0.00504292
$367$	2.78797e6	1.08050	0.540248	−	0.841506i	$-0.318330\pi$
$367$	2.78797e6	1.08050	0.540248	+	0.841506i	$0.318330\pi$
$368$	−4.12811e6	−1.58903
$369$	1.43712e6	0.549449
$370$	0	0
$371$	3.15817e6	1.19125
$372$	606546.	0.227251
$373$	1.65578e6	0.616212	0.308106	−	0.951352i	$-0.400305\pi$
$373$	1.65578e6	0.616212	0.308106	+	0.951352i	$0.400305\pi$
$374$	−23423.0	−0.00865892
$375$	0	0
$376$	−302617.	−0.110388
$377$	3.91997e6	1.42046
$378$	−88086.4	−0.0317088
$379$	−2.35538e6	−0.842291	−0.421146	−	0.906993i	$-0.638372\pi$
$379$	−2.35538e6	−0.842291	−0.421146	+	0.906993i	$0.638372\pi$
$380$	0	0
$381$	333717.	0.117779
$382$	−172301.	−0.0604130
$383$	4.13864e6	1.44165	0.720827	−	0.693115i	$-0.243762\pi$
$383$	4.13864e6	1.44165	0.720827	+	0.693115i	$0.243762\pi$
$384$	−227449.	−0.0787146
$385$	0	0
$386$	−219083.	−0.0748410
$387$	−432012.	−0.146629
$388$	−2.83172e6	−0.954928
$389$	−5.27781e6	−1.76840	−0.884198	−	0.467112i	$-0.845294\pi$
$389$	−5.27781e6	−1.76840	−0.884198	+	0.467112i	$0.845294\pi$
$390$	0	0
$391$	1.05698e6	0.349643
$392$	−224644.	−0.0738380
$393$	−426412.	−0.139267
$394$	−427056.	−0.138594
$395$	0	0
$396$	−1.17117e6	−0.375303
$397$	−3.71439e6	−1.18280	−0.591401	−	0.806378i	$-0.701425\pi$
$397$	−3.71439e6	−1.18280	−0.591401	+	0.806378i	$0.701425\pi$
$398$	−343789.	−0.108789
$399$	980707.	0.308395
$400$	0	0
$401$	3.60034e6	1.11810	0.559052	−	0.829132i	$-0.311165\pi$
$401$	3.60034e6	1.11810	0.559052	+	0.829132i	$0.311165\pi$
$402$	−35085.4	−0.0108283
$403$	3.62615e6	1.11220
$404$	906532.	0.276331
$405$	0	0
$406$	−409180.	−0.123197
$407$	−521647.	−0.156096
$408$	28804.4	0.00856661
$409$	1.65305e6	0.488627	0.244314	−	0.969696i	$-0.421437\pi$
$409$	1.65305e6	0.488627	0.244314	+	0.969696i	$0.421437\pi$
$410$	0	0
$411$	−196331.	−0.0573304
$412$	953765.	0.276821
$413$	−4.69053e6	−1.35315
$414$	−566887.	−0.162553
$415$	0	0
$416$	−1.02170e6	−0.289461
$417$	861339.	0.242568
$418$	−285325.	−0.0798728
$419$	1.25123e6	0.348180	0.174090	−	0.984730i	$-0.444302\pi$
$419$	1.25123e6	0.348180	0.174090	+	0.984730i	$0.444302\pi$
$420$	0	0
$421$	3.12108e6	0.858221	0.429111	−	0.903252i	$-0.358827\pi$
$421$	3.12108e6	0.858221	0.429111	+	0.903252i	$0.358827\pi$
$422$	216464.	0.0591703
$423$	1.90589e6	0.517902
$424$	1.12993e6	0.305238
$425$	0	0
$426$	−85785.1	−0.0229028
$427$	−751875.	−0.199561
$428$	2.63013e6	0.694015
$429$	280302.	0.0735331
$430$	0	0
$431$	246469.	0.0639102	0.0319551	−	0.999489i	$-0.489827\pi$
$431$	246469.	0.0639102	0.0319551	+	0.999489i	$0.489827\pi$
$432$	1.44540e6	0.372630
$433$	7.56031e6	1.93785	0.968924	−	0.247357i	$-0.0795621\pi$
$433$	7.56031e6	1.93785	0.968924	+	0.247357i	$0.0795621\pi$
$434$	−378510.	−0.0964612
$435$	0	0
$436$	−1.86111e6	−0.468873
$437$	1.28755e7	3.22523
$438$	−75750.6	−0.0188669
$439$	−3.76145e6	−0.931525	−0.465763	−	0.884910i	$-0.654220\pi$
$439$	−3.76145e6	−0.931525	−0.465763	+	0.884910i	$0.654220\pi$
$440$	0	0
$441$	1.41482e6	0.346421
$442$	85642.2	0.0208513
$443$	1.02986e6	0.249327	0.124664	−	0.992199i	$-0.460215\pi$
$443$	1.02986e6	0.249327	0.124664	+	0.992199i	$0.460215\pi$
$444$	319036.	0.0768038
$445$	0	0
$446$	−486920.	−0.115910
$447$	−689333.	−0.163177
$448$	−3.18327e6	−0.749340
$449$	−4.29961e6	−1.00650	−0.503249	−	0.864141i	$-0.667862\pi$
$449$	−4.29961e6	−1.00650	−0.503249	+	0.864141i	$0.667862\pi$
$450$	0	0
$451$	973820.	0.225443
$452$	−7.36570e6	−1.69577
$453$	−1.18784e6	−0.271966
$454$	−211739.	−0.0482126
$455$	0	0
$456$	350878.	0.0790213
$457$	1.42831e6	0.319913	0.159957	−	0.987124i	$-0.448865\pi$
$457$	1.42831e6	0.319913	0.159957	+	0.987124i	$0.448865\pi$
$458$	61825.0	0.0137721
$459$	−370085.	−0.0819917
$460$	0	0
$461$	−3.06394e6	−0.671472	−0.335736	−	0.941956i	$-0.608985\pi$
$461$	−3.06394e6	−0.671472	−0.335736	+	0.941956i	$0.608985\pi$
$462$	−29258.9	−0.00637753
$463$	3.51890e6	0.762877	0.381438	−	0.924394i	$-0.375429\pi$
$463$	3.51890e6	0.762877	0.381438	+	0.924394i	$0.375429\pi$
$464$	6.71416e6	1.44776
$465$	0	0
$466$	415945.	0.0887302
$467$	6.90153e6	1.46438	0.732188	−	0.681102i	$-0.238499\pi$
$467$	6.90153e6	1.46438	0.732188	+	0.681102i	$0.238499\pi$
$468$	4.28218e6	0.903755
$469$	−2.04121e6	−0.428504
$470$	0	0
$471$	−1.13041e6	−0.234793
$472$	−1.67818e6	−0.346724
$473$	−292740.	−0.0601629
$474$	−116119.	−0.0237387
$475$	0	0
$476$	833426.	0.168597
$477$	−7.11636e6	−1.43206
$478$	41025.8	0.00821272
$479$	−4.37684e6	−0.871609	−0.435805	−	0.900041i	$-0.643536\pi$
$479$	−4.37684e6	−0.871609	−0.435805	+	0.900041i	$0.643536\pi$
$480$	0	0
$481$	1.90731e6	0.375889
$482$	−25991.1	−0.00509574
$483$	1.32033e6	0.257522
$484$	4.30533e6	0.835398
$485$	0	0
$486$	299677.	0.0575524
$487$	−4.08443e6	−0.780385	−0.390192	−	0.920733i	$-0.627591\pi$
$487$	−4.08443e6	−0.780385	−0.390192	+	0.920733i	$0.627591\pi$
$488$	−269006.	−0.0511343
$489$	1.59710e6	0.302037
$490$	0	0
$491$	−4.67807e6	−0.875716	−0.437858	−	0.899044i	$-0.644263\pi$
$491$	−4.67807e6	−0.875716	−0.437858	+	0.899044i	$0.644263\pi$
$492$	−595583.	−0.110925
$493$	−1.71912e6	−0.318559
$494$	1.04324e6	0.192339
$495$	0	0
$496$	6.21090e6	1.13357
$497$	−4.99083e6	−0.906321
$498$	−125479.	−0.0226724
$499$	9.66466e6	1.73754	0.868771	−	0.495214i	$-0.164910\pi$
$499$	9.66466e6	1.73754	0.868771	+	0.495214i	$0.164910\pi$
$500$	0	0
$501$	−52703.5	−0.00938091
$502$	−617655.	−0.109392
$503$	1.03319e7	1.82079	0.910396	−	0.413738i	$-0.135777\pi$
$503$	1.03319e7	1.82079	0.910396	+	0.413738i	$0.135777\pi$
$504$	−898772.	−0.157606
$505$	0	0
$506$	−384133.	−0.0666969
$507$	110680.	0.0191227
$508$	3.45464e6	0.593942
$509$	−4.20006e6	−0.718557	−0.359278	−	0.933230i	$-0.616977\pi$
$509$	−4.20006e6	−0.718557	−0.359278	+	0.933230i	$0.616977\pi$
$510$	0	0
$511$	−4.40704e6	−0.746611
$512$	−2.92704e6	−0.493463
$513$	−4.50815e6	−0.756320
$514$	−562992.	−0.0939928
$515$	0	0
$516$	179038.	0.0296020
$517$	1.29147e6	0.212500
$518$	−199092.	−0.0326008
$519$	−1.74830e6	−0.284904
$520$	0	0
$521$	6.46853e6	1.04403	0.522013	−	0.852938i	$-0.325181\pi$
$521$	6.46853e6	1.04403	0.522013	+	0.852938i	$0.325181\pi$
$522$	922012.	0.148102
$523$	3.00248e6	0.479983	0.239991	−	0.970775i	$-0.422855\pi$
$523$	3.00248e6	0.479983	0.239991	+	0.970775i	$0.422855\pi$
$524$	−4.41423e6	−0.702306
$525$	0	0
$526$	498411.	0.0785458
$527$	−1.59027e6	−0.249427
$528$	480104.	0.0749463
$529$	1.08979e7	1.69319
$530$	0	0
$531$	1.05692e7	1.62670
$532$	1.01523e7	1.55520
$533$	−3.56061e6	−0.542883
$534$	52835.8	0.00801817
$535$	0	0
$536$	−730304.	−0.109797
$537$	587098.	0.0878566
$538$	−904572.	−0.134737
$539$	958708.	0.142139
$540$	0	0
$541$	−5.21767e6	−0.766449	−0.383224	−	0.923655i	$-0.625186\pi$
$541$	−5.21767e6	−0.766449	−0.383224	+	0.923655i	$0.625186\pi$
$542$	2434.34	0.000355945 0
$543$	−338880.	−0.0493227
$544$	448071.	0.0649157
$545$	0	0
$546$	106980.	0.0153575
$547$	7.86187e6	1.12346	0.561730	−	0.827321i	$-0.310136\pi$
$547$	7.86187e6	1.12346	0.561730	+	0.827321i	$0.310136\pi$
$548$	−2.03243e6	−0.289110
$549$	1.69421e6	0.239903
$550$	0	0
$551$	−2.09413e7	−2.93850
$552$	472387.	0.0659858
$553$	−6.75560e6	−0.939401
$554$	−930944.	−0.128869
$555$	0	0
$556$	8.91659e6	1.22324
$557$	−1.21531e7	−1.65978	−0.829889	−	0.557929i	$-0.811596\pi$
$557$	−1.21531e7	−1.65978	−0.829889	+	0.557929i	$0.811596\pi$
$558$	852902.	0.115961
$559$	1.07035e6	0.144876
$560$	0	0
$561$	−122928.	−0.0164909
$562$	−934177.	−0.124764
$563$	−5.15518e6	−0.685446	−0.342723	−	0.939437i	$-0.611349\pi$
$563$	−5.15518e6	−0.685446	−0.342723	+	0.939437i	$0.611349\pi$
$564$	−789856.	−0.104556
$565$	0	0
$566$	1.19268e6	0.156489
$567$	5.42482e6	0.708643
$568$	−1.78562e6	−0.232230
$569$	−1.42079e6	−0.183972	−0.0919858	−	0.995760i	$-0.529321\pi$
$569$	−1.42079e6	−0.183972	−0.0919858	+	0.995760i	$0.529321\pi$
$570$	0	0
$571$	−1.14786e7	−1.47332	−0.736661	−	0.676262i	$-0.763599\pi$
$571$	−1.14786e7	−1.47332	−0.736661	+	0.676262i	$0.763599\pi$
$572$	2.90169e6	0.370818
$573$	−904266.	−0.115056
$574$	371668.	0.0470842
$575$	0	0
$576$	7.17292e6	0.900824
$577$	−1.00683e7	−1.25897	−0.629484	−	0.777014i	$-0.716734\pi$
$577$	−1.00683e7	−1.25897	−0.629484	+	0.777014i	$0.716734\pi$
$578$	789867.	0.0983409
$579$	−1.14978e6	−0.142534
$580$	0	0
$581$	−7.30015e6	−0.897205
$582$	159408.	0.0195075
$583$	−4.82218e6	−0.587587
$584$	−1.57675e6	−0.191307
$585$	0	0
$586$	786877.	0.0946592
$587$	1.33560e7	1.59985	0.799927	−	0.600097i	$-0.204871\pi$
$587$	1.33560e7	1.59985	0.799927	+	0.600097i	$0.204871\pi$
$588$	−586340.	−0.0699369
$589$	−1.93716e7	−2.30080
$590$	0	0
$591$	−2.24126e6	−0.263951
$592$	3.26686e6	0.383113
$593$	−1.20725e7	−1.40981	−0.704906	−	0.709301i	$-0.749011\pi$
$593$	−1.20725e7	−1.40981	−0.704906	+	0.709301i	$0.749011\pi$
$594$	134498.	0.0156405
$595$	0	0
$596$	−7.13598e6	−0.822883
$597$	−1.80426e6	−0.207187
$598$	1.40452e6	0.160611
$599$	1.59735e7	1.81900	0.909502	−	0.415700i	$-0.136463\pi$
$599$	1.59735e7	1.81900	0.909502	+	0.415700i	$0.136463\pi$
$600$	0	0
$601$	7.82358e6	0.883526	0.441763	−	0.897132i	$-0.354353\pi$
$601$	7.82358e6	0.883526	0.441763	+	0.897132i	$0.354353\pi$
$602$	−111727.	−0.0125651
$603$	4.59948e6	0.515129
$604$	−1.22966e7	−1.37149
$605$	0	0
$606$	−51032.0	−0.00564496
$607$	−3.67967e6	−0.405356	−0.202678	−	0.979245i	$-0.564964\pi$
$607$	−3.67967e6	−0.405356	−0.202678	+	0.979245i	$0.564964\pi$
$608$	5.45813e6	0.598804
$609$	−2.14744e6	−0.234627
$610$	0	0
$611$	−4.72204e6	−0.511713
$612$	−1.87797e6	−0.202680
$613$	−4.33193e6	−0.465619	−0.232810	−	0.972522i	$-0.574792\pi$
$613$	−4.33193e6	−0.465619	−0.232810	+	0.972522i	$0.574792\pi$
$614$	−210434.	−0.0225265
$615$	0	0
$616$	−609025.	−0.0646671
$617$	6.45241e6	0.682353	0.341176	−	0.939999i	$-0.389175\pi$
$617$	6.45241e6	0.682353	0.341176	+	0.939999i	$0.389175\pi$
$618$	−53691.0	−0.00565497
$619$	7.05387e6	0.739947	0.369974	−	0.929042i	$-0.379367\pi$
$619$	7.05387e6	0.739947	0.369974	+	0.929042i	$0.379367\pi$
$620$	0	0
$621$	−6.06933e6	−0.631556
$622$	−1.23791e6	−0.128296
$623$	3.07390e6	0.317299
$624$	−1.75542e6	−0.180476
$625$	0	0
$626$	−315687.	−0.0321974
$627$	−1.49743e6	−0.152117
$628$	−1.17021e7	−1.18403
$629$	−836461.	−0.0842984
$630$	0	0
$631$	1.46818e7	1.46793	0.733966	−	0.679186i	$-0.237667\pi$
$631$	1.46818e7	1.46793	0.733966	+	0.679186i	$0.237667\pi$
$632$	−2.41702e6	−0.240706
$633$	1.13604e6	0.112689
$634$	−1.78859e6	−0.176721
$635$	0	0
$636$	2.94922e6	0.289111
$637$	−3.50535e6	−0.342281
$638$	624773.	0.0607674
$639$	1.12459e7	1.08954
$640$	0	0
$641$	3.24675e6	0.312107	0.156054	−	0.987749i	$-0.450123\pi$
$641$	3.24675e6	0.312107	0.156054	+	0.987749i	$0.450123\pi$
$642$	−148060.	−0.0141775
$643$	7.57964e6	0.722971	0.361486	−	0.932378i	$-0.382270\pi$
$643$	7.57964e6	0.722971	0.361486	+	0.932378i	$0.382270\pi$
$644$	1.36680e7	1.29865
$645$	0	0
$646$	−457519.	−0.0431348
$647$	−6.13290e6	−0.575977	−0.287988	−	0.957634i	$-0.592986\pi$
$647$	−6.13290e6	−0.575977	−0.287988	+	0.957634i	$0.592986\pi$
$648$	1.94089e6	0.181578
$649$	7.16193e6	0.667449
$650$	0	0
$651$	−1.98648e6	−0.183710
$652$	1.65332e7	1.52313
$653$	1.13174e6	0.103864	0.0519318	−	0.998651i	$-0.483462\pi$
$653$	1.13174e6	0.103864	0.0519318	+	0.998651i	$0.483462\pi$
$654$	104769.	0.00957827
$655$	0	0
$656$	−6.09864e6	−0.553316
$657$	9.93045e6	0.897544
$658$	492902.	0.0443809
$659$	1.03411e7	0.927586	0.463793	−	0.885944i	$-0.346488\pi$
$659$	1.03411e7	0.927586	0.463793	+	0.885944i	$0.346488\pi$
$660$	0	0
$661$	−8.30708e6	−0.739511	−0.369756	−	0.929129i	$-0.620559\pi$
$661$	−8.30708e6	−0.739511	−0.369756	+	0.929129i	$0.620559\pi$
$662$	164444.	0.0145838
$663$	449465.	0.0397111
$664$	−2.61185e6	−0.229894
$665$	0	0
$666$	448616.	0.0391913
$667$	−2.81933e7	−2.45376
$668$	−545587.	−0.0473067
$669$	−2.55544e6	−0.220750
$670$	0	0
$671$	1.14803e6	0.0984344
$672$	559708.	0.0478121
$673$	1.13225e7	0.963617	0.481809	−	0.876276i	$-0.339980\pi$
$673$	1.13225e7	0.963617	0.481809	+	0.876276i	$0.339980\pi$
$674$	609468.	0.0516775
$675$	0	0
$676$	1.14576e6	0.0964331
$677$	−2.12894e7	−1.78522	−0.892609	−	0.450831i	$-0.851128\pi$
$677$	−2.12894e7	−1.78522	−0.892609	+	0.450831i	$0.851128\pi$
$678$	414642.	0.0346417
$679$	9.27407e6	0.771962
$680$	0	0
$681$	−1.11124e6	−0.0918205
$682$	577943.	0.0475799
$683$	1.18724e7	0.973841	0.486920	−	0.873446i	$-0.338120\pi$
$683$	1.18724e7	0.973841	0.486920	+	0.873446i	$0.338120\pi$
$684$	−2.28763e7	−1.86959
$685$	0	0
$686$	1.38147e6	0.112081
$687$	324468.	0.0262289
$688$	1.83331e6	0.147661
$689$	1.76315e7	1.41495
$690$	0	0
$691$	−1.96927e7	−1.56896	−0.784478	−	0.620157i	$-0.787069\pi$
$691$	−1.96927e7	−1.56896	−0.784478	+	0.620157i	$0.787069\pi$
$692$	−1.80985e7	−1.43673
$693$	3.83566e6	0.303394
$694$	1.11207e6	0.0876465
$695$	0	0
$696$	−768314.	−0.0601195
$697$	1.56152e6	0.121749
$698$	−852859.	−0.0662581
$699$	2.18295e6	0.168986
$700$	0	0
$701$	−1.36097e7	−1.04605	−0.523027	−	0.852316i	$-0.675197\pi$
$701$	−1.36097e7	−1.04605	−0.523027	+	0.852316i	$0.675197\pi$
$702$	−491770.	−0.0376634
$703$	−1.01893e7	−0.777597
$704$	4.86051e6	0.369615
$705$	0	0
$706$	505464.	0.0381662
$707$	−2.96895e6	−0.223385
$708$	−4.38019e6	−0.328405
$709$	1.30274e7	0.973290	0.486645	−	0.873600i	$-0.338220\pi$
$709$	1.30274e7	0.973290	0.486645	+	0.873600i	$0.338220\pi$
$710$	0	0
$711$	1.52225e7	1.12931
$712$	1.09978e6	0.0813029
$713$	−2.60801e7	−1.92125
$714$	−46916.6	−0.00344414
$715$	0	0
$716$	6.07764e6	0.443050
$717$	215310.	0.0156411
$718$	−2.01321e6	−0.145740
$719$	−9.11723e6	−0.657719	−0.328860	−	0.944379i	$-0.606664\pi$
$719$	−9.11723e6	−0.657719	−0.328860	+	0.944379i	$0.606664\pi$
$720$	0	0
$721$	−3.12365e6	−0.223781
$722$	−4.13026e6	−0.294873
$723$	−136406.	−0.00970480
$724$	−3.50809e6	−0.248728
$725$	0	0
$726$	−242363.	−0.0170657
$727$	1.52424e7	1.06959	0.534797	−	0.844981i	$-0.320388\pi$
$727$	1.52424e7	1.06959	0.534797	+	0.844981i	$0.320388\pi$
$728$	2.22680e6	0.155723
$729$	−1.11404e7	−0.776396
$730$	0	0
$731$	−469408.	−0.0324906
$732$	−702128.	−0.0484327
$733$	1.61394e7	1.10950	0.554751	−	0.832016i	$-0.312813\pi$
$733$	1.61394e7	1.10950	0.554751	+	0.832016i	$0.312813\pi$
$734$	−1.62470e6	−0.111310
$735$	0	0
$736$	7.34828e6	0.500025
$737$	3.11670e6	0.211362
$738$	−837486.	−0.0566026
$739$	−9.24471e6	−0.622704	−0.311352	−	0.950295i	$-0.600782\pi$
$739$	−9.24471e6	−0.622704	−0.311352	+	0.950295i	$0.600782\pi$
$740$	0	0
$741$	5.47510e6	0.366308
$742$	−1.84043e6	−0.122719
$743$	3.89612e6	0.258917	0.129459	−	0.991585i	$-0.458676\pi$
$743$	3.89612e6	0.258917	0.129459	+	0.991585i	$0.458676\pi$
$744$	−710724.	−0.0470727
$745$	0	0
$746$	−964909.	−0.0634804
$747$	1.64495e7	1.07858
$748$	−1.27255e6	−0.0831613
$749$	−8.61387e6	−0.561040
$750$	0	0
$751$	−2.36303e7	−1.52887	−0.764433	−	0.644704i	$-0.776981\pi$
$751$	−2.36303e7	−1.52887	−0.764433	+	0.644704i	$0.776981\pi$
$752$	−8.08795e6	−0.521547
$753$	−3.24155e6	−0.208337
$754$	−2.28437e6	−0.146332
$755$	0	0
$756$	−4.78565e6	−0.304535
$757$	−2.86009e6	−0.181401	−0.0907005	−	0.995878i	$-0.528911\pi$
$757$	−2.86009e6	−0.181401	−0.0907005	+	0.995878i	$0.528911\pi$
$758$	1.37260e6	0.0867704
$759$	−2.01599e6	−0.127024
$760$	0	0
$761$	−2.90585e7	−1.81891	−0.909456	−	0.415800i	$-0.863502\pi$
$761$	−2.90585e7	−1.81891	−0.909456	+	0.415800i	$0.863502\pi$
$762$	−194475.	−0.0121332
$763$	6.09526e6	0.379036
$764$	−9.36097e6	−0.580213
$765$	0	0
$766$	−2.41180e6	−0.148515
$767$	−2.61864e7	−1.60726
$768$	−2.87200e6	−0.175704
$769$	−1.85731e7	−1.13258	−0.566290	−	0.824206i	$-0.691622\pi$
$769$	−1.85731e7	−1.13258	−0.566290	+	0.824206i	$0.691622\pi$
$770$	0	0
$771$	−2.95468e6	−0.179009
$772$	−1.19026e7	−0.718781
$773$	−1.38698e7	−0.834872	−0.417436	−	0.908706i	$-0.637071\pi$
$773$	−1.38698e7	−0.834872	−0.417436	+	0.908706i	$0.637071\pi$
$774$	251756.	0.0151052
$775$	0	0
$776$	3.31808e6	0.197803
$777$	−1.04487e6	−0.0620881
$778$	3.07566e6	0.182175
$779$	1.90215e7	1.12306
$780$	0	0
$781$	7.62045e6	0.447047
$782$	−615958.	−0.0360192
$783$	9.87145e6	0.575409
$784$	−6.00400e6	−0.348859
$785$	0	0
$786$	248493.	0.0143469
$787$	1.95803e7	1.12689	0.563445	−	0.826154i	$-0.309476\pi$
$787$	1.95803e7	1.12689	0.563445	+	0.826154i	$0.309476\pi$
$788$	−2.32015e7	−1.33107
$789$	2.61574e6	0.149590
$790$	0	0
$791$	2.41232e7	1.37086
$792$	1.37233e6	0.0777400
$793$	−4.19757e6	−0.237037
$794$	2.16457e6	0.121849
$795$	0	0
$796$	−1.86777e7	−1.04482
$797$	2.66479e7	1.48600	0.742998	−	0.669293i	$-0.233403\pi$
$797$	2.66479e7	1.48600	0.742998	+	0.669293i	$0.233403\pi$
$798$	−571510.	−0.0317699
$799$	2.07087e6	0.114759
$800$	0	0
$801$	−6.92646e6	−0.381443
$802$	−2.09811e6	−0.115184
$803$	6.72907e6	0.368270
$804$	−1.90615e6	−0.103996
$805$	0	0
$806$	−2.11315e6	−0.114576
$807$	−4.74734e6	−0.256606
$808$	−1.06223e6	−0.0572390
$809$	4.31569e6	0.231835	0.115917	−	0.993259i	$-0.463019\pi$
$809$	4.31569e6	0.231835	0.115917	+	0.993259i	$0.463019\pi$
$810$	0	0
$811$	1.07890e7	0.576007	0.288003	−	0.957629i	$-0.407009\pi$
$811$	1.07890e7	0.576007	0.288003	+	0.957629i	$0.407009\pi$
$812$	−2.22304e7	−1.18320
$813$	12775.8	0.000677894 0
$814$	303991.	0.0160805
$815$	0	0
$816$	769847.	0.0404743
$817$	−5.71805e6	−0.299704
$818$	−963319.	−0.0503369
$819$	−1.40244e7	−0.730594
$820$	0	0
$821$	−2.12759e7	−1.10162	−0.550808	−	0.834632i	$-0.685680\pi$
$821$	−2.12759e7	−1.10162	−0.550808	+	0.834632i	$0.685680\pi$
$822$	114413.	0.00590602
$823$	−7.04803e6	−0.362717	−0.181359	−	0.983417i	$-0.558050\pi$
$823$	−7.04803e6	−0.362717	−0.181359	+	0.983417i	$0.558050\pi$
$824$	−1.11758e6	−0.0573404
$825$	0	0
$826$	2.73342e6	0.139398
$827$	−2.00838e7	−1.02113	−0.510566	−	0.859839i	$-0.670564\pi$
$827$	−2.00838e7	−1.02113	−0.510566	+	0.859839i	$0.670564\pi$
$828$	−3.07984e7	−1.56118
$829$	2.04193e7	1.03194	0.515970	−	0.856606i	$-0.327431\pi$
$829$	2.04193e7	1.03194	0.515970	+	0.856606i	$0.327431\pi$
$830$	0	0
$831$	−4.88575e6	−0.245431
$832$	−1.77716e7	−0.890059
$833$	1.53729e6	0.0767614
$834$	−501948.	−0.0249887
$835$	0	0
$836$	−1.55014e7	−0.767108
$837$	9.13153e6	0.450537
$838$	−729160.	−0.0358685
$839$	−4.92445e6	−0.241520	−0.120760	−	0.992682i	$-0.538533\pi$
$839$	−4.92445e6	−0.241520	−0.120760	+	0.992682i	$0.538533\pi$
$840$	0	0
$841$	2.53438e7	1.23561
$842$	−1.81882e6	−0.0884114
$843$	−4.90271e6	−0.237612
$844$	1.17603e7	0.568278
$845$	0	0
$846$	−1.11066e6	−0.0533528
$847$	−1.41003e7	−0.675334
$848$	3.01994e7	1.44214
$849$	6.25940e6	0.298032
$850$	0	0
$851$	−1.37178e7	−0.649324
$852$	−4.66062e6	−0.219961
$853$	1.25088e7	0.588632	0.294316	−	0.955708i	$-0.404908\pi$
$853$	1.25088e7	0.588632	0.294316	+	0.955708i	$0.404908\pi$
$854$	438157.	0.0205582
$855$	0	0
$856$	−3.08188e6	−0.143758
$857$	−2.21879e7	−1.03196	−0.515982	−	0.856600i	$-0.672573\pi$
$857$	−2.21879e7	−1.03196	−0.515982	+	0.856600i	$0.672573\pi$
$858$	−163347.	−0.00757517
$859$	−1.40835e7	−0.651218	−0.325609	−	0.945505i	$-0.605569\pi$
$859$	−1.40835e7	−0.651218	−0.325609	+	0.945505i	$0.605569\pi$
$860$	0	0
$861$	1.95058e6	0.0896716
$862$	−143631.	−0.00658384
$863$	7.94491e6	0.363130	0.181565	−	0.983379i	$-0.441884\pi$
$863$	7.94491e6	0.363130	0.181565	+	0.983379i	$0.441884\pi$
$864$	−2.57289e6	−0.117256
$865$	0	0
$866$	−4.40579e6	−0.199632
$867$	4.14535e6	0.187290
$868$	−2.05641e7	−0.926424
$869$	1.03151e7	0.463364
$870$	0	0
$871$	−1.13957e7	−0.508973
$872$	2.18077e6	0.0971221
$873$	−2.08974e7	−0.928019
$874$	−7.50323e6	−0.332253
$875$	0	0
$876$	−4.11546e6	−0.181200
$877$	4.03970e6	0.177358	0.0886788	−	0.996060i	$-0.471736\pi$
$877$	4.03970e6	0.177358	0.0886788	+	0.996060i	$0.471736\pi$
$878$	2.19200e6	0.0959630
$879$	4.12966e6	0.180278
$880$	0	0
$881$	−1.99840e7	−0.867447	−0.433724	−	0.901046i	$-0.642801\pi$
$881$	−1.99840e7	−0.867447	−0.433724	+	0.901046i	$0.642801\pi$
$882$	−824489.	−0.0356873
$883$	−1.66868e7	−0.720230	−0.360115	−	0.932908i	$-0.617263\pi$
$883$	−1.66868e7	−0.720230	−0.360115	+	0.932908i	$0.617263\pi$
$884$	4.65286e6	0.200258
$885$	0	0
$886$	−600155.	−0.0256850
$887$	1.79964e6	0.0768027	0.0384014	−	0.999262i	$-0.487773\pi$
$887$	1.79964e6	0.0768027	0.0384014	+	0.999262i	$0.487773\pi$
$888$	−373833.	−0.0159091
$889$	−1.13142e7	−0.480141
$890$	0	0
$891$	−8.28310e6	−0.349541
$892$	−2.64539e7	−1.11321
$893$	2.52261e7	1.05858
$894$	401711.	0.0168101
$895$	0	0
$896$	7.71131e6	0.320892
$897$	7.37114e6	0.305882
$898$	2.50561e6	0.103687
$899$	4.24179e7	1.75045
$900$	0	0
$901$	−7.73238e6	−0.317323
$902$	−567496.	−0.0232245
$903$	−586362.	−0.0239302
$904$	8.63080e6	0.351261
$905$	0	0
$906$	692219.	0.0280171
$907$	3.64748e6	0.147223	0.0736113	−	0.997287i	$-0.476548\pi$
$907$	3.64748e6	0.147223	0.0736113	+	0.997287i	$0.476548\pi$
$908$	−1.15036e7	−0.463039
$909$	6.68999e6	0.268544
$910$	0	0
$911$	4.92223e6	0.196502	0.0982508	−	0.995162i	$-0.468675\pi$
$911$	4.92223e6	0.196502	0.0982508	+	0.995162i	$0.468675\pi$
$912$	9.37781e6	0.373348
$913$	1.11465e7	0.442551
$914$	−832352.	−0.0329565
$915$	0	0
$916$	3.35890e6	0.132269
$917$	1.44569e7	0.567743
$918$	215668.	0.00844655
$919$	1.52960e7	0.597432	0.298716	−	0.954342i	$-0.403442\pi$
$919$	1.52960e7	0.597432	0.298716	+	0.954342i	$0.403442\pi$
$920$	0	0
$921$	−1.10439e6	−0.0429016
$922$	1.78552e6	0.0691731
$923$	−2.78629e7	−1.07652
$924$	−1.58961e6	−0.0612505
$925$	0	0
$926$	−2.05065e6	−0.0785893
$927$	7.03856e6	0.269020
$928$	−1.19516e7	−0.455571
$929$	3.10755e7	1.18135	0.590674	−	0.806910i	$-0.298862\pi$
$929$	3.10755e7	1.18135	0.590674	+	0.806910i	$0.298862\pi$
$930$	0	0
$931$	1.87263e7	0.708073
$932$	2.25979e7	0.852175
$933$	−6.49676e6	−0.244339
$934$	−4.02188e6	−0.150856
$935$	0	0
$936$	−5.01767e6	−0.187203
$937$	3.19233e7	1.18784	0.593922	−	0.804523i	$-0.297579\pi$
$937$	3.19233e7	1.18784	0.593922	+	0.804523i	$0.297579\pi$
$938$	1.18952e6	0.0441432
$939$	−1.65678e6	−0.0613198
$940$	0	0
$941$	−4.89415e7	−1.80179	−0.900894	−	0.434040i	$-0.857088\pi$
$941$	−4.89415e7	−1.80179	−0.900894	+	0.434040i	$0.857088\pi$
$942$	658752.	0.0241877
$943$	2.56087e7	0.937795
$944$	−4.48522e7	−1.63815
$945$	0	0
$946$	170595.	0.00619781
$947$	1.45580e7	0.527505	0.263753	−	0.964590i	$-0.415040\pi$
$947$	1.45580e7	0.527505	0.263753	+	0.964590i	$0.415040\pi$
$948$	−6.30863e6	−0.227989
$949$	−2.46037e7	−0.886818
$950$	0	0
$951$	−9.38680e6	−0.336563
$952$	−976572.	−0.0349230
$953$	2.71191e7	0.967261	0.483631	−	0.875272i	$-0.339318\pi$
$953$	2.71191e7	0.967261	0.483631	+	0.875272i	$0.339318\pi$
$954$	4.14708e6	0.147527
$955$	0	0
$956$	2.22889e6	0.0788759
$957$	3.27891e6	0.115731
$958$	2.55062e6	0.0897907
$959$	6.65633e6	0.233716
$960$	0	0
$961$	1.06093e7	0.370576
$962$	−1.11149e6	−0.0387229
$963$	1.94098e7	0.674458
$964$	−1.41207e6	−0.0489401
$965$	0	0
$966$	−769424.	−0.0265291
$967$	−1.16883e7	−0.401963	−0.200982	−	0.979595i	$-0.564413\pi$
$967$	−1.16883e7	−0.401963	−0.200982	+	0.979595i	$0.564413\pi$
$968$	−5.04480e6	−0.173044
$969$	−2.40113e6	−0.0821499
$970$	0	0
$971$	3.07622e6	0.104706	0.0523528	−	0.998629i	$-0.483328\pi$
$971$	3.07622e6	0.104706	0.0523528	+	0.998629i	$0.483328\pi$
$972$	1.62812e7	0.552739
$973$	−2.92024e7	−0.988865
$974$	2.38021e6	0.0803930
$975$	0	0
$976$	−7.18964e6	−0.241592
$977$	−3.81186e7	−1.27762	−0.638809	−	0.769365i	$-0.720573\pi$
$977$	−3.81186e7	−1.27762	−0.638809	+	0.769365i	$0.720573\pi$
$978$	−930714.	−0.0311150
$979$	−4.69350e6	−0.156509
$980$	0	0
$981$	−1.37345e7	−0.455661
$982$	2.72616e6	0.0902137
$983$	3.14497e7	1.03809	0.519043	−	0.854748i	$-0.326289\pi$
$983$	3.14497e7	1.03809	0.519043	+	0.854748i	$0.326289\pi$
$984$	697878.	0.0229769
$985$	0	0
$986$	1.00182e6	0.0328170
$987$	2.58683e6	0.0845231
$988$	5.66783e7	1.84725
$989$	−7.69821e6	−0.250264
$990$	0	0
$991$	1.08378e7	0.350555	0.175277	−	0.984519i	$-0.443918\pi$
$991$	1.08378e7	0.350555	0.175277	+	0.984519i	$0.443918\pi$
$992$	−1.10558e7	−0.356705
$993$	863027.	0.0277748
$994$	2.90842e6	0.0933665
$995$	0	0
$996$	−6.81715e6	−0.217748
$997$	1.37214e7	0.437179	0.218590	−	0.975817i	$-0.429854\pi$
$997$	1.37214e7	0.437179	0.218590	+	0.975817i	$0.429854\pi$
$998$	−5.63211e6	−0.178997
$999$	4.80308e6	0.152267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.a.1.4		8
5.4	even	2		43.6.a.a.1.5	✓	8
15.14	odd	2		387.6.a.c.1.4		8
20.19	odd	2		688.6.a.e.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.5	✓	8	5.4	even	2
387.6.a.c.1.4		8	15.14	odd	2
688.6.a.e.1.4		8	20.19	odd	2
1075.6.a.a.1.4		8	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.582753 q^{2} -3.05838 q^{3} -31.6604 q^{4} +1.78228 q^{6} +103.690 q^{7} +37.0983 q^{8} -233.646 q^{9} +O(q^{10})\)	\(q-0.582753 q^{2} -3.05838 q^{3} -31.6604 q^{4} +1.78228 q^{6} +103.690 q^{7} +37.0983 q^{8} -233.646 q^{9} -158.323 q^{11} +96.8296 q^{12} +578.882 q^{13} -60.4256 q^{14} +991.514 q^{16} -253.871 q^{17} +136.158 q^{18} -3092.51 q^{19} -317.124 q^{21} +92.2633 q^{22} -4163.45 q^{23} -113.461 q^{24} -337.345 q^{26} +1457.77 q^{27} -3282.87 q^{28} +6771.63 q^{29} +6264.06 q^{31} -1764.95 q^{32} +484.213 q^{33} +147.944 q^{34} +7397.34 q^{36} +3294.82 q^{37} +1802.17 q^{38} -1770.44 q^{39} -6150.84 q^{41} +184.805 q^{42} +1849.00 q^{43} +5012.58 q^{44} +2426.26 q^{46} -8157.17 q^{47} -3032.43 q^{48} -6055.38 q^{49} +776.436 q^{51} -18327.6 q^{52} +30457.8 q^{53} -849.517 q^{54} +3846.72 q^{56} +9458.07 q^{57} -3946.19 q^{58} -45236.1 q^{59} -7251.18 q^{61} -3650.40 q^{62} -24226.8 q^{63} -30699.9 q^{64} -282.176 q^{66} -19685.7 q^{67} +8037.67 q^{68} +12733.4 q^{69} -48132.2 q^{71} -8667.87 q^{72} -42502.1 q^{73} -1920.07 q^{74} +97910.0 q^{76} -16416.5 q^{77} +1031.73 q^{78} -65151.9 q^{79} +52317.6 q^{81} +3584.42 q^{82} -70403.6 q^{83} +10040.3 q^{84} -1077.51 q^{86} -20710.2 q^{87} -5873.52 q^{88} +29645.1 q^{89} +60024.2 q^{91} +131816. q^{92} -19157.9 q^{93} +4753.61 q^{94} +5397.90 q^{96} +89440.4 q^{97} +3528.79 q^{98} +36991.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} + O(q^{10}) \)	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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