LMFDB - Embedded newform 1160.4.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,4,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1160.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.4422156067$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3 x^{10} - 200 x^{9} + 418 x^{8} + 13211 x^{7} - 14353 x^{6} - 314463 x^{5} + 64817 x^{4} + \cdots + 712233 $ x^11 - 3x^10 - 200x^9 + 418x^8 + 13211x^7 - 14353x^6 - 314463x^5 + 64817x^4 + 1676824x^3 - 292536x^2 - 2502333x + 712233
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{3}\cdot 43 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.72995$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.72995 q^{3} -5.00000 q^{5} +32.6348 q^{7} -19.5474 q^{9} -33.7286 q^{11} +23.4245 q^{13} -13.6498 q^{15} +17.6536 q^{17} +86.2042 q^{19} +89.0915 q^{21} -22.6763 q^{23} +25.0000 q^{25} -127.072 q^{27} +29.0000 q^{29} +28.2772 q^{31} -92.0775 q^{33} -163.174 q^{35} +119.089 q^{37} +63.9476 q^{39} -112.751 q^{41} +223.122 q^{43} +97.7369 q^{45} +190.126 q^{47} +722.033 q^{49} +48.1936 q^{51} +755.034 q^{53} +168.643 q^{55} +235.333 q^{57} -718.755 q^{59} +371.250 q^{61} -637.925 q^{63} -117.122 q^{65} -437.836 q^{67} -61.9052 q^{69} +198.752 q^{71} -89.4919 q^{73} +68.2488 q^{75} -1100.73 q^{77} +878.612 q^{79} +180.879 q^{81} -224.981 q^{83} -88.2682 q^{85} +79.1686 q^{87} +692.859 q^{89} +764.453 q^{91} +77.1954 q^{93} -431.021 q^{95} -1208.80 q^{97} +659.306 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) $ 11 * q + 8 * q^3 - 55 * q^5 + 16 * q^7 + 117 * q^9 + 26 * q^11 + 52 * q^13 - 40 * q^15 + 14 * q^17 + 146 * q^19 - 288 * q^21 - 80 * q^23 + 275 * q^25 + 224 * q^27 + 319 * q^29 - 38 * q^31 - 152 * q^33 - 80 * q^35 + 376 * q^37 + 108 * q^39 - 146 * q^41 + 644 * q^43 - 585 * q^45 - 464 * q^47 + 393 * q^49 - 576 * q^51 - 188 * q^53 - 130 * q^55 + 1380 * q^57 + 672 * q^59 + 472 * q^61 + 628 * q^63 - 260 * q^65 + 1688 * q^67 + 138 * q^69 + 92 * q^71 + 1514 * q^73 + 200 * q^75 + 2884 * q^77 + 2130 * q^79 + 4067 * q^81 + 2584 * q^83 - 70 * q^85 + 232 * q^87 + 1606 * q^89 + 3208 * q^91 + 3540 * q^93 - 730 * q^95 + 3322 * q^97 + 4690 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.72995	0.525379	0.262690	−	0.964880i	$-0.415390\pi$
$3$	2.72995	0.525379	0.262690	+	0.964880i	$0.415390\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	32.6348	1.76212	0.881058	−	0.473008i	$-0.156832\pi$
$7$	32.6348	1.76212	0.881058	+	0.473008i	$0.156832\pi$
$8$	0	0
$9$	−19.5474	−0.723977
$10$	0	0
$11$	−33.7286	−0.924506	−0.462253	−	0.886748i	$-0.652959\pi$
$11$	−33.7286	−0.924506	−0.462253	+	0.886748i	$0.652959\pi$
$12$	0	0
$13$	23.4245	0.499752	0.249876	−	0.968278i	$-0.419610\pi$
$13$	23.4245	0.499752	0.249876	+	0.968278i	$0.419610\pi$
$14$	0	0
$15$	−13.6498	−0.234957
$16$	0	0
$17$	17.6536	0.251861	0.125930	−	0.992039i	$-0.459808\pi$
$17$	17.6536	0.251861	0.125930	+	0.992039i	$0.459808\pi$
$18$	0	0
$19$	86.2042	1.04087	0.520437	−	0.853900i	$-0.325769\pi$
$19$	86.2042	1.04087	0.520437	+	0.853900i	$0.325769\pi$
$20$	0	0
$21$	89.0915	0.925779
$22$	0	0
$23$	−22.6763	−0.205580	−0.102790	−	0.994703i	$-0.532777\pi$
$23$	−22.6763	−0.205580	−0.102790	+	0.994703i	$0.532777\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−127.072	−0.905741
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	28.2772	0.163830	0.0819152	−	0.996639i	$-0.473896\pi$
$31$	28.2772	0.163830	0.0819152	+	0.996639i	$0.473896\pi$
$32$	0	0
$33$	−92.0775	−0.485716
$34$	0	0
$35$	−163.174	−0.788042
$36$	0	0
$37$	119.089	0.529136	0.264568	−	0.964367i	$-0.414771\pi$
$37$	119.089	0.529136	0.264568	+	0.964367i	$0.414771\pi$
$38$	0	0
$39$	63.9476	0.262559
$40$	0	0
$41$	−112.751	−0.429483	−0.214741	−	0.976671i	$-0.568891\pi$
$41$	−112.751	−0.429483	−0.214741	+	0.976671i	$0.568891\pi$
$42$	0	0
$43$	223.122	0.791299	0.395649	−	0.918402i	$-0.370520\pi$
$43$	223.122	0.791299	0.395649	+	0.918402i	$0.370520\pi$
$44$	0	0
$45$	97.7369	0.323772
$46$	0	0
$47$	190.126	0.590058	0.295029	−	0.955488i	$-0.404671\pi$
$47$	190.126	0.590058	0.295029	+	0.955488i	$0.404671\pi$
$48$	0	0
$49$	722.033	2.10505
$50$	0	0
$51$	48.1936	0.132323
$52$	0	0
$53$	755.034	1.95683	0.978415	−	0.206652i	$-0.0662567\pi$
$53$	755.034	1.95683	0.978415	+	0.206652i	$0.0662567\pi$
$54$	0	0
$55$	168.643	0.413452
$56$	0	0
$57$	235.333	0.546854
$58$	0	0
$59$	−718.755	−1.58600	−0.792999	−	0.609223i	$-0.791482\pi$
$59$	−718.755	−1.58600	−0.792999	+	0.609223i	$0.791482\pi$
$60$	0	0
$61$	371.250	0.779240	0.389620	−	0.920976i	$-0.372606\pi$
$61$	371.250	0.779240	0.389620	+	0.920976i	$0.372606\pi$
$62$	0	0
$63$	−637.925	−1.27573
$64$	0	0
$65$	−117.122	−0.223496
$66$	0	0
$67$	−437.836	−0.798362	−0.399181	−	0.916872i	$-0.630705\pi$
$67$	−437.836	−0.798362	−0.399181	+	0.916872i	$0.630705\pi$
$68$	0	0
$69$	−61.9052	−0.108007
$70$	0	0
$71$	198.752	0.332218	0.166109	−	0.986107i	$-0.446880\pi$
$71$	198.752	0.332218	0.166109	+	0.986107i	$0.446880\pi$
$72$	0	0
$73$	−89.4919	−0.143483	−0.0717414	−	0.997423i	$-0.522856\pi$
$73$	−89.4919	−0.143483	−0.0717414	+	0.997423i	$0.522856\pi$
$74$	0	0
$75$	68.2488	0.105076
$76$	0	0
$77$	−1100.73	−1.62909
$78$	0	0
$79$	878.612	1.25129	0.625643	−	0.780109i	$-0.284837\pi$
$79$	878.612	1.25129	0.625643	+	0.780109i	$0.284837\pi$
$80$	0	0
$81$	180.879	0.248119
$82$	0	0
$83$	−224.981	−0.297529	−0.148765	−	0.988873i	$-0.547530\pi$
$83$	−224.981	−0.297529	−0.148765	+	0.988873i	$0.547530\pi$
$84$	0	0
$85$	−88.2682	−0.112636
$86$	0	0
$87$	79.1686	0.0975605
$88$	0	0
$89$	692.859	0.825202	0.412601	−	0.910912i	$-0.364620\pi$
$89$	692.859	0.825202	0.412601	+	0.910912i	$0.364620\pi$
$90$	0	0
$91$	764.453	0.880621
$92$	0	0
$93$	77.1954	0.0860730
$94$	0	0
$95$	−431.021	−0.465493
$96$	0	0
$97$	−1208.80	−1.26531	−0.632657	−	0.774432i	$-0.718036\pi$
$97$	−1208.80	−1.26531	−0.632657	+	0.774432i	$0.718036\pi$
$98$	0	0
$99$	659.306	0.669321
$100$	0	0
$101$	913.612	0.900077	0.450039	−	0.893009i	$-0.351410\pi$
$101$	913.612	0.900077	0.450039	+	0.893009i	$0.351410\pi$
$102$	0	0
$103$	−1052.21	−1.00657	−0.503286	−	0.864120i	$-0.667876\pi$
$103$	−1052.21	−1.00657	−0.503286	+	0.864120i	$0.667876\pi$
$104$	0	0
$105$	−445.458	−0.414021
$106$	0	0
$107$	−646.549	−0.584152	−0.292076	−	0.956395i	$-0.594346\pi$
$107$	−646.549	−0.584152	−0.292076	+	0.956395i	$0.594346\pi$
$108$	0	0
$109$	2032.35	1.78591	0.892955	−	0.450145i	$-0.148628\pi$
$109$	2032.35	1.78591	0.892955	+	0.450145i	$0.148628\pi$
$110$	0	0
$111$	325.106	0.277997
$112$	0	0
$113$	477.038	0.397133	0.198566	−	0.980087i	$-0.436372\pi$
$113$	477.038	0.397133	0.198566	+	0.980087i	$0.436372\pi$
$114$	0	0
$115$	113.382	0.0919382
$116$	0	0
$117$	−457.887	−0.361809
$118$	0	0
$119$	576.124	0.443808
$120$	0	0
$121$	−193.379	−0.145288
$122$	0	0
$123$	−307.806	−0.225641
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1773.25	1.23898	0.619490	−	0.785005i	$-0.287339\pi$
$127$	1773.25	1.23898	0.619490	+	0.785005i	$0.287339\pi$
$128$	0	0
$129$	609.113	0.415732
$130$	0	0
$131$	1351.25	0.901213	0.450607	−	0.892723i	$-0.351208\pi$
$131$	1351.25	0.901213	0.450607	+	0.892723i	$0.351208\pi$
$132$	0	0
$133$	2813.26	1.83414
$134$	0	0
$135$	635.360	0.405060
$136$	0	0
$137$	−617.025	−0.384788	−0.192394	−	0.981318i	$-0.561625\pi$
$137$	−617.025	−0.384788	−0.192394	+	0.981318i	$0.561625\pi$
$138$	0	0
$139$	2756.09	1.68179	0.840894	−	0.541200i	$-0.182030\pi$
$139$	2756.09	1.68179	0.840894	+	0.541200i	$0.182030\pi$
$140$	0	0
$141$	519.035	0.310004
$142$	0	0
$143$	−790.075	−0.462024
$144$	0	0
$145$	−145.000	−0.0830455
$146$	0	0
$147$	1971.11	1.10595
$148$	0	0
$149$	−1893.20	−1.04092	−0.520460	−	0.853886i	$-0.674240\pi$
$149$	−1893.20	−1.04092	−0.520460	+	0.853886i	$0.674240\pi$
$150$	0	0
$151$	2412.29	1.30006	0.650031	−	0.759907i	$-0.274756\pi$
$151$	2412.29	1.30006	0.650031	+	0.759907i	$0.274756\pi$
$152$	0	0
$153$	−345.082	−0.182341
$154$	0	0
$155$	−141.386	−0.0732671
$156$	0	0
$157$	−2563.89	−1.30332	−0.651659	−	0.758512i	$-0.725927\pi$
$157$	−2563.89	−1.30332	−0.651659	+	0.758512i	$0.725927\pi$
$158$	0	0
$159$	2061.21	1.02808
$160$	0	0
$161$	−740.038	−0.362256
$162$	0	0
$163$	2798.75	1.34488	0.672438	−	0.740154i	$-0.265247\pi$
$163$	2798.75	1.34488	0.672438	+	0.740154i	$0.265247\pi$
$164$	0	0
$165$	460.388	0.217219
$166$	0	0
$167$	459.740	0.213029	0.106514	−	0.994311i	$-0.466031\pi$
$167$	459.740	0.213029	0.106514	+	0.994311i	$0.466031\pi$
$168$	0	0
$169$	−1648.29	−0.750248
$170$	0	0
$171$	−1685.07	−0.753569
$172$	0	0
$173$	1683.90	0.740028	0.370014	−	0.929026i	$-0.379353\pi$
$173$	1683.90	0.740028	0.370014	+	0.929026i	$0.379353\pi$
$174$	0	0
$175$	815.871	0.352423
$176$	0	0
$177$	−1962.16	−0.833251
$178$	0	0
$179$	−797.438	−0.332979	−0.166490	−	0.986043i	$-0.553243\pi$
$179$	−797.438	−0.332979	−0.166490	+	0.986043i	$0.553243\pi$
$180$	0	0
$181$	3746.24	1.53843	0.769215	−	0.638990i	$-0.220648\pi$
$181$	3746.24	1.53843	0.769215	+	0.638990i	$0.220648\pi$
$182$	0	0
$183$	1013.49	0.409397
$184$	0	0
$185$	−595.443	−0.236637
$186$	0	0
$187$	−595.433	−0.232847
$188$	0	0
$189$	−4146.98	−1.59602
$190$	0	0
$191$	296.879	0.112468	0.0562340	−	0.998418i	$-0.482091\pi$
$191$	296.879	0.112468	0.0562340	+	0.998418i	$0.482091\pi$
$192$	0	0
$193$	15.7419	0.00587113	0.00293556	−	0.999996i	$-0.499066\pi$
$193$	15.7419	0.00587113	0.00293556	+	0.999996i	$0.499066\pi$
$194$	0	0
$195$	−319.738	−0.117420
$196$	0	0
$197$	2936.29	1.06194	0.530970	−	0.847391i	$-0.321828\pi$
$197$	2936.29	1.06194	0.530970	+	0.847391i	$0.321828\pi$
$198$	0	0
$199$	3514.81	1.25205	0.626026	−	0.779802i	$-0.284680\pi$
$199$	3514.81	1.25205	0.626026	+	0.779802i	$0.284680\pi$
$200$	0	0
$201$	−1195.27	−0.419443
$202$	0	0
$203$	946.410	0.327217
$204$	0	0
$205$	563.757	0.192071
$206$	0	0
$207$	443.262	0.148835
$208$	0	0
$209$	−2907.55	−0.962295
$210$	0	0
$211$	3702.46	1.20800	0.604000	−	0.796984i	$-0.293573\pi$
$211$	3702.46	1.20800	0.604000	+	0.796984i	$0.293573\pi$
$212$	0	0
$213$	542.582	0.174540
$214$	0	0
$215$	−1115.61	−0.353880
$216$	0	0
$217$	922.823	0.288688
$218$	0	0
$219$	−244.309	−0.0753828
$220$	0	0
$221$	413.527	0.125868
$222$	0	0
$223$	−3026.88	−0.908945	−0.454472	−	0.890761i	$-0.650172\pi$
$223$	−3026.88	−0.908945	−0.454472	+	0.890761i	$0.650172\pi$
$224$	0	0
$225$	−488.684	−0.144795
$226$	0	0
$227$	−1610.29	−0.470833	−0.235416	−	0.971895i	$-0.575645\pi$
$227$	−1610.29	−0.470833	−0.235416	+	0.971895i	$0.575645\pi$
$228$	0	0
$229$	−3605.34	−1.04038	−0.520191	−	0.854050i	$-0.674139\pi$
$229$	−3605.34	−1.04038	−0.520191	+	0.854050i	$0.674139\pi$
$230$	0	0
$231$	−3004.94	−0.855888
$232$	0	0
$233$	1434.83	0.403428	0.201714	−	0.979444i	$-0.435349\pi$
$233$	1434.83	0.403428	0.201714	+	0.979444i	$0.435349\pi$
$234$	0	0
$235$	−950.630	−0.263882
$236$	0	0
$237$	2398.57	0.657400
$238$	0	0
$239$	−3834.69	−1.03785	−0.518924	−	0.854821i	$-0.673667\pi$
$239$	−3834.69	−1.03785	−0.518924	+	0.854821i	$0.673667\pi$
$240$	0	0
$241$	−2395.09	−0.640171	−0.320086	−	0.947389i	$-0.603712\pi$
$241$	−2395.09	−0.640171	−0.320086	+	0.947389i	$0.603712\pi$
$242$	0	0
$243$	3924.73	1.03610
$244$	0	0
$245$	−3610.17	−0.941408
$246$	0	0
$247$	2019.29	0.520179
$248$	0	0
$249$	−614.188	−0.156316
$250$	0	0
$251$	965.099	0.242695	0.121348	−	0.992610i	$-0.461278\pi$
$251$	965.099	0.242695	0.121348	+	0.992610i	$0.461278\pi$
$252$	0	0
$253$	764.842	0.190060
$254$	0	0
$255$	−240.968	−0.0591764
$256$	0	0
$257$	563.104	0.136675	0.0683374	−	0.997662i	$-0.478231\pi$
$257$	563.104	0.136675	0.0683374	+	0.997662i	$0.478231\pi$
$258$	0	0
$259$	3886.43	0.932399
$260$	0	0
$261$	−566.874	−0.134439
$262$	0	0
$263$	−4475.58	−1.04934	−0.524670	−	0.851306i	$-0.675811\pi$
$263$	−4475.58	−1.04934	−0.524670	+	0.851306i	$0.675811\pi$
$264$	0	0
$265$	−3775.17	−0.875121
$266$	0	0
$267$	1891.47	0.433544
$268$	0	0
$269$	4577.98	1.03764	0.518819	−	0.854884i	$-0.326372\pi$
$269$	4577.98	1.03764	0.518819	+	0.854884i	$0.326372\pi$
$270$	0	0
$271$	1874.58	0.420194	0.210097	−	0.977681i	$-0.432622\pi$
$271$	1874.58	0.420194	0.210097	+	0.977681i	$0.432622\pi$
$272$	0	0
$273$	2086.92	0.462660
$274$	0	0
$275$	−843.216	−0.184901
$276$	0	0
$277$	8294.65	1.79920	0.899598	−	0.436720i	$-0.143860\pi$
$277$	8294.65	1.79920	0.899598	+	0.436720i	$0.143860\pi$
$278$	0	0
$279$	−552.745	−0.118609
$280$	0	0
$281$	−4941.27	−1.04901	−0.524504	−	0.851408i	$-0.675749\pi$
$281$	−4941.27	−1.04901	−0.524504	+	0.851408i	$0.675749\pi$
$282$	0	0
$283$	−7227.68	−1.51817	−0.759083	−	0.650994i	$-0.774352\pi$
$283$	−7227.68	−1.51817	−0.759083	+	0.650994i	$0.774352\pi$
$284$	0	0
$285$	−1176.67	−0.244560
$286$	0	0
$287$	−3679.62	−0.756799
$288$	0	0
$289$	−4601.35	−0.936566
$290$	0	0
$291$	−3299.98	−0.664770
$292$	0	0
$293$	−1597.35	−0.318492	−0.159246	−	0.987239i	$-0.550906\pi$
$293$	−1597.35	−0.318492	−0.159246	+	0.987239i	$0.550906\pi$
$294$	0	0
$295$	3593.77	0.709280
$296$	0	0
$297$	4285.97	0.837364
$298$	0	0
$299$	−531.181	−0.102739
$300$	0	0
$301$	7281.56	1.39436
$302$	0	0
$303$	2494.12	0.472882
$304$	0	0
$305$	−1856.25	−0.348487
$306$	0	0
$307$	5345.70	0.993796	0.496898	−	0.867809i	$-0.334472\pi$
$307$	5345.70	0.993796	0.496898	+	0.867809i	$0.334472\pi$
$308$	0	0
$309$	−2872.47	−0.528832
$310$	0	0
$311$	2835.51	0.517000	0.258500	−	0.966011i	$-0.416772\pi$
$311$	2835.51	0.517000	0.258500	+	0.966011i	$0.416772\pi$
$312$	0	0
$313$	5953.26	1.07507	0.537537	−	0.843240i	$-0.319355\pi$
$313$	5953.26	1.07507	0.537537	+	0.843240i	$0.319355\pi$
$314$	0	0
$315$	3189.63	0.570524
$316$	0	0
$317$	−567.775	−0.100598	−0.0502988	−	0.998734i	$-0.516017\pi$
$317$	−567.775	−0.100598	−0.0502988	+	0.998734i	$0.516017\pi$
$318$	0	0
$319$	−978.131	−0.171676
$320$	0	0
$321$	−1765.05	−0.306901
$322$	0	0
$323$	1521.82	0.262156
$324$	0	0
$325$	585.611	0.0999504
$326$	0	0
$327$	5548.23	0.938280
$328$	0	0
$329$	6204.73	1.03975
$330$	0	0
$331$	3185.67	0.529005	0.264502	−	0.964385i	$-0.414792\pi$
$331$	3185.67	0.529005	0.264502	+	0.964385i	$0.414792\pi$
$332$	0	0
$333$	−2327.87	−0.383082
$334$	0	0
$335$	2189.18	0.357038
$336$	0	0
$337$	−2751.11	−0.444696	−0.222348	−	0.974967i	$-0.571372\pi$
$337$	−2751.11	−0.444696	−0.222348	+	0.974967i	$0.571372\pi$
$338$	0	0
$339$	1302.29	0.208645
$340$	0	0
$341$	−953.752	−0.151462
$342$	0	0
$343$	12369.7	1.94723
$344$	0	0
$345$	309.526	0.0483024
$346$	0	0
$347$	8556.93	1.32380	0.661902	−	0.749590i	$-0.269749\pi$
$347$	8556.93	1.32380	0.661902	+	0.749590i	$0.269749\pi$
$348$	0	0
$349$	−3333.95	−0.511353	−0.255676	−	0.966762i	$-0.582298\pi$
$349$	−3333.95	−0.511353	−0.255676	+	0.966762i	$0.582298\pi$
$350$	0	0
$351$	−2976.59	−0.452646
$352$	0	0
$353$	8870.01	1.33740	0.668701	−	0.743531i	$-0.266851\pi$
$353$	8870.01	1.33740	0.668701	+	0.743531i	$0.266851\pi$
$354$	0	0
$355$	−993.758	−0.148572
$356$	0	0
$357$	1572.79	0.233168
$358$	0	0
$359$	−5849.66	−0.859981	−0.429991	−	0.902833i	$-0.641483\pi$
$359$	−5849.66	−0.859981	−0.429991	+	0.902833i	$0.641483\pi$
$360$	0	0
$361$	572.172	0.0834192
$362$	0	0
$363$	−527.914	−0.0763315
$364$	0	0
$365$	447.460	0.0641674
$366$	0	0
$367$	1155.43	0.164340	0.0821701	−	0.996618i	$-0.473815\pi$
$367$	1155.43	0.164340	0.0821701	+	0.996618i	$0.473815\pi$
$368$	0	0
$369$	2203.99	0.310936
$370$	0	0
$371$	24640.4	3.44816
$372$	0	0
$373$	−12296.8	−1.70698	−0.853490	−	0.521110i	$-0.825518\pi$
$373$	−12296.8	−1.70698	−0.853490	+	0.521110i	$0.825518\pi$
$374$	0	0
$375$	−341.244	−0.0469913
$376$	0	0
$377$	679.309	0.0928016
$378$	0	0
$379$	−2947.56	−0.399488	−0.199744	−	0.979848i	$-0.564011\pi$
$379$	−2947.56	−0.399488	−0.199744	+	0.979848i	$0.564011\pi$
$380$	0	0
$381$	4840.88	0.650934
$382$	0	0
$383$	−3141.44	−0.419112	−0.209556	−	0.977797i	$-0.567202\pi$
$383$	−3141.44	−0.419112	−0.209556	+	0.977797i	$0.567202\pi$
$384$	0	0
$385$	5503.64	0.728550
$386$	0	0
$387$	−4361.46	−0.572882
$388$	0	0
$389$	−14731.3	−1.92006	−0.960032	−	0.279890i	$-0.909702\pi$
$389$	−14731.3	−1.92006	−0.960032	+	0.279890i	$0.909702\pi$
$390$	0	0
$391$	−400.320	−0.0517776
$392$	0	0
$393$	3688.83	0.473479
$394$	0	0
$395$	−4393.06	−0.559592
$396$	0	0
$397$	−6458.07	−0.816426	−0.408213	−	0.912887i	$-0.633848\pi$
$397$	−6458.07	−0.816426	−0.408213	+	0.912887i	$0.633848\pi$
$398$	0	0
$399$	7680.07	0.963620
$400$	0	0
$401$	−3418.40	−0.425703	−0.212852	−	0.977085i	$-0.568275\pi$
$401$	−3418.40	−0.425703	−0.212852	+	0.977085i	$0.568275\pi$
$402$	0	0
$403$	662.379	0.0818745
$404$	0	0
$405$	−904.394	−0.110962
$406$	0	0
$407$	−4016.69	−0.489189
$408$	0	0
$409$	−13983.1	−1.69051	−0.845255	−	0.534362i	$-0.820552\pi$
$409$	−13983.1	−1.69051	−0.845255	+	0.534362i	$0.820552\pi$
$410$	0	0
$411$	−1684.45	−0.202160
$412$	0	0
$413$	−23456.5	−2.79471
$414$	0	0
$415$	1124.91	0.133059
$416$	0	0
$417$	7523.99	0.883576
$418$	0	0
$419$	14577.2	1.69962	0.849810	−	0.527089i	$-0.176716\pi$
$419$	14577.2	1.69962	0.849810	+	0.527089i	$0.176716\pi$
$420$	0	0
$421$	−10189.3	−1.17956	−0.589781	−	0.807563i	$-0.700786\pi$
$421$	−10189.3	−1.17956	−0.589781	+	0.807563i	$0.700786\pi$
$422$	0	0
$423$	−3716.46	−0.427188
$424$	0	0
$425$	441.341	0.0503722
$426$	0	0
$427$	12115.7	1.37311
$428$	0	0
$429$	−2156.87	−0.242738
$430$	0	0
$431$	1454.83	0.162591	0.0812954	−	0.996690i	$-0.474094\pi$
$431$	1454.83	0.162591	0.0812954	+	0.996690i	$0.474094\pi$
$432$	0	0
$433$	−5813.61	−0.645229	−0.322615	−	0.946530i	$-0.604562\pi$
$433$	−5813.61	−0.645229	−0.322615	+	0.946530i	$0.604562\pi$
$434$	0	0
$435$	−395.843	−0.0436304
$436$	0	0
$437$	−1954.80	−0.213983
$438$	0	0
$439$	−16307.4	−1.77291	−0.886456	−	0.462813i	$-0.846840\pi$
$439$	−16307.4	−1.77291	−0.886456	+	0.462813i	$0.846840\pi$
$440$	0	0
$441$	−14113.8	−1.52401
$442$	0	0
$443$	−14548.1	−1.56028	−0.780138	−	0.625607i	$-0.784851\pi$
$443$	−14548.1	−1.56028	−0.780138	+	0.625607i	$0.784851\pi$
$444$	0	0
$445$	−3464.30	−0.369041
$446$	0	0
$447$	−5168.35	−0.546878
$448$	0	0
$449$	−2528.57	−0.265769	−0.132885	−	0.991132i	$-0.542424\pi$
$449$	−2528.57	−0.265769	−0.132885	+	0.991132i	$0.542424\pi$
$450$	0	0
$451$	3802.95	0.397060
$452$	0	0
$453$	6585.44	0.683026
$454$	0	0
$455$	−3822.27	−0.393826
$456$	0	0
$457$	10161.9	1.04016	0.520082	−	0.854116i	$-0.325901\pi$
$457$	10161.9	1.04016	0.520082	+	0.854116i	$0.325901\pi$
$458$	0	0
$459$	−2243.28	−0.228121
$460$	0	0
$461$	−242.986	−0.0245488	−0.0122744	−	0.999925i	$-0.503907\pi$
$461$	−242.986	−0.0245488	−0.0122744	+	0.999925i	$0.503907\pi$
$462$	0	0
$463$	1821.44	0.182828	0.0914141	−	0.995813i	$-0.470861\pi$
$463$	1821.44	0.182828	0.0914141	+	0.995813i	$0.470861\pi$
$464$	0	0
$465$	−385.977	−0.0384930
$466$	0	0
$467$	5999.05	0.594439	0.297219	−	0.954809i	$-0.403941\pi$
$467$	5999.05	0.594439	0.297219	+	0.954809i	$0.403941\pi$
$468$	0	0
$469$	−14288.7	−1.40681
$470$	0	0
$471$	−6999.30	−0.684736
$472$	0	0
$473$	−7525.61	−0.731560
$474$	0	0
$475$	2155.11	0.208175
$476$	0	0
$477$	−14758.9	−1.41670
$478$	0	0
$479$	−19125.1	−1.82431	−0.912156	−	0.409842i	$-0.865584\pi$
$479$	−19125.1	−1.82431	−0.912156	+	0.409842i	$0.865584\pi$
$480$	0	0
$481$	2789.58	0.264437
$482$	0	0
$483$	−2020.27	−0.190322
$484$	0	0
$485$	6044.02	0.565866
$486$	0	0
$487$	−2143.13	−0.199414	−0.0997071	−	0.995017i	$-0.531791\pi$
$487$	−2143.13	−0.199414	−0.0997071	+	0.995017i	$0.531791\pi$
$488$	0	0
$489$	7640.44	0.706570
$490$	0	0
$491$	−14070.1	−1.29323	−0.646615	−	0.762817i	$-0.723816\pi$
$491$	−14070.1	−1.29323	−0.646615	+	0.762817i	$0.723816\pi$
$492$	0	0
$493$	511.956	0.0467694
$494$	0	0
$495$	−3296.53	−0.299329
$496$	0	0
$497$	6486.23	0.585406
$498$	0	0
$499$	3549.57	0.318438	0.159219	−	0.987243i	$-0.449102\pi$
$499$	3549.57	0.318438	0.159219	+	0.987243i	$0.449102\pi$
$500$	0	0
$501$	1255.07	0.111921
$502$	0	0
$503$	2006.06	0.177824	0.0889122	−	0.996039i	$-0.471661\pi$
$503$	2006.06	0.177824	0.0889122	+	0.996039i	$0.471661\pi$
$504$	0	0
$505$	−4568.06	−0.402527
$506$	0	0
$507$	−4499.76	−0.394165
$508$	0	0
$509$	−16187.3	−1.40961	−0.704803	−	0.709403i	$-0.748965\pi$
$509$	−16187.3	−1.40961	−0.704803	+	0.709403i	$0.748965\pi$
$510$	0	0
$511$	−2920.56	−0.252833
$512$	0	0
$513$	−10954.1	−0.942763
$514$	0	0
$515$	5261.03	0.450153
$516$	0	0
$517$	−6412.69	−0.545512
$518$	0	0
$519$	4596.97	0.388795
$520$	0	0
$521$	9262.91	0.778916	0.389458	−	0.921044i	$-0.372662\pi$
$521$	9262.91	0.778916	0.389458	+	0.921044i	$0.372662\pi$
$522$	0	0
$523$	−1666.98	−0.139373	−0.0696865	−	0.997569i	$-0.522200\pi$
$523$	−1666.98	−0.139373	−0.0696865	+	0.997569i	$0.522200\pi$
$524$	0	0
$525$	2227.29	0.185156
$526$	0	0
$527$	499.196	0.0412625
$528$	0	0
$529$	−11652.8	−0.957737
$530$	0	0
$531$	14049.8	1.14823
$532$	0	0
$533$	−2641.14	−0.214635
$534$	0	0
$535$	3232.74	0.261241
$536$	0	0
$537$	−2176.97	−0.174940
$538$	0	0
$539$	−24353.2	−1.94613
$540$	0	0
$541$	−18376.9	−1.46042	−0.730208	−	0.683225i	$-0.760577\pi$
$541$	−18376.9	−1.46042	−0.730208	+	0.683225i	$0.760577\pi$
$542$	0	0
$543$	10227.0	0.808259
$544$	0	0
$545$	−10161.8	−0.798684
$546$	0	0
$547$	−19500.6	−1.52428	−0.762142	−	0.647409i	$-0.775852\pi$
$547$	−19500.6	−1.52428	−0.762142	+	0.647409i	$0.775852\pi$
$548$	0	0
$549$	−7256.95	−0.564152
$550$	0	0
$551$	2499.92	0.193285
$552$	0	0
$553$	28673.4	2.20491
$554$	0	0
$555$	−1625.53	−0.124324
$556$	0	0
$557$	−10162.4	−0.773058	−0.386529	−	0.922277i	$-0.626326\pi$
$557$	−10162.4	−0.773058	−0.386529	+	0.922277i	$0.626326\pi$
$558$	0	0
$559$	5226.52	0.395453
$560$	0	0
$561$	−1625.50	−0.122333
$562$	0	0
$563$	11988.9	0.897465	0.448732	−	0.893666i	$-0.351876\pi$
$563$	11988.9	0.897465	0.448732	+	0.893666i	$0.351876\pi$
$564$	0	0
$565$	−2385.19	−0.177603
$566$	0	0
$567$	5902.95	0.437214
$568$	0	0
$569$	15529.6	1.14418	0.572088	−	0.820192i	$-0.306133\pi$
$569$	15529.6	1.14418	0.572088	+	0.820192i	$0.306133\pi$
$570$	0	0
$571$	22574.2	1.65447	0.827233	−	0.561858i	$-0.189913\pi$
$571$	22574.2	1.65447	0.827233	+	0.561858i	$0.189913\pi$
$572$	0	0
$573$	810.464	0.0590883
$574$	0	0
$575$	−566.908	−0.0411160
$576$	0	0
$577$	−19424.8	−1.40150	−0.700750	−	0.713407i	$-0.747151\pi$
$577$	−19424.8	−1.40150	−0.700750	+	0.713407i	$0.747151\pi$
$578$	0	0
$579$	42.9746	0.00308457
$580$	0	0
$581$	−7342.23	−0.524281
$582$	0	0
$583$	−25466.3	−1.80910
$584$	0	0
$585$	2289.43	0.161806
$586$	0	0
$587$	−15690.8	−1.10329	−0.551643	−	0.834081i	$-0.685999\pi$
$587$	−15690.8	−1.10329	−0.551643	+	0.834081i	$0.685999\pi$
$588$	0	0
$589$	2437.62	0.170527
$590$	0	0
$591$	8015.93	0.557921
$592$	0	0
$593$	17116.3	1.18530	0.592648	−	0.805461i	$-0.298082\pi$
$593$	17116.3	1.18530	0.592648	+	0.805461i	$0.298082\pi$
$594$	0	0
$595$	−2880.62	−0.198477
$596$	0	0
$597$	9595.26	0.657802
$598$	0	0
$599$	320.621	0.0218701	0.0109351	−	0.999940i	$-0.496519\pi$
$599$	320.621	0.0218701	0.0109351	+	0.999940i	$0.496519\pi$
$600$	0	0
$601$	12826.3	0.870541	0.435270	−	0.900300i	$-0.356653\pi$
$601$	12826.3	0.870541	0.435270	+	0.900300i	$0.356653\pi$
$602$	0	0
$603$	8558.55	0.577995
$604$	0	0
$605$	966.894	0.0649749
$606$	0	0
$607$	−18810.9	−1.25784	−0.628922	−	0.777468i	$-0.716503\pi$
$607$	−18810.9	−1.25784	−0.628922	+	0.777468i	$0.716503\pi$
$608$	0	0
$609$	2583.65	0.171913
$610$	0	0
$611$	4453.60	0.294883
$612$	0	0
$613$	5563.73	0.366586	0.183293	−	0.983058i	$-0.441324\pi$
$613$	5563.73	0.366586	0.183293	+	0.983058i	$0.441324\pi$
$614$	0	0
$615$	1539.03	0.100910
$616$	0	0
$617$	−13275.1	−0.866186	−0.433093	−	0.901349i	$-0.642578\pi$
$617$	−13275.1	−0.866186	−0.433093	+	0.901349i	$0.642578\pi$
$618$	0	0
$619$	−28067.6	−1.82251	−0.911254	−	0.411846i	$-0.864884\pi$
$619$	−28067.6	−1.82251	−0.911254	+	0.411846i	$0.864884\pi$
$620$	0	0
$621$	2881.53	0.186202
$622$	0	0
$623$	22611.4	1.45410
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−7937.47	−0.505570
$628$	0	0
$629$	2102.35	0.133269
$630$	0	0
$631$	−6859.00	−0.432730	−0.216365	−	0.976313i	$-0.569420\pi$
$631$	−6859.00	−0.432730	−0.216365	+	0.976313i	$0.569420\pi$
$632$	0	0
$633$	10107.5	0.634658
$634$	0	0
$635$	−8866.25	−0.554089
$636$	0	0
$637$	16913.2	1.05200
$638$	0	0
$639$	−3885.07	−0.240518
$640$	0	0
$641$	−19465.6	−1.19945	−0.599724	−	0.800207i	$-0.704723\pi$
$641$	−19465.6	−1.19945	−0.599724	+	0.800207i	$0.704723\pi$
$642$	0	0
$643$	26696.9	1.63736	0.818682	−	0.574247i	$-0.194705\pi$
$643$	26696.9	1.63736	0.818682	+	0.574247i	$0.194705\pi$
$644$	0	0
$645$	−3045.56	−0.185921
$646$	0	0
$647$	8412.89	0.511197	0.255599	−	0.966783i	$-0.417727\pi$
$647$	8412.89	0.511197	0.255599	+	0.966783i	$0.417727\pi$
$648$	0	0
$649$	24242.6	1.46627
$650$	0	0
$651$	2519.26	0.151671
$652$	0	0
$653$	7334.94	0.439569	0.219784	−	0.975548i	$-0.429465\pi$
$653$	7334.94	0.439569	0.219784	+	0.975548i	$0.429465\pi$
$654$	0	0
$655$	−6756.23	−0.403035
$656$	0	0
$657$	1749.33	0.103878
$658$	0	0
$659$	−4799.57	−0.283709	−0.141855	−	0.989887i	$-0.545307\pi$
$659$	−4799.57	−0.283709	−0.141855	+	0.989887i	$0.545307\pi$
$660$	0	0
$661$	1557.26	0.0916343	0.0458171	−	0.998950i	$-0.485411\pi$
$661$	1557.26	0.0916343	0.0458171	+	0.998950i	$0.485411\pi$
$662$	0	0
$663$	1128.91	0.0661284
$664$	0	0
$665$	−14066.3	−0.820253
$666$	0	0
$667$	−657.613	−0.0381752
$668$	0	0
$669$	−8263.22	−0.477541
$670$	0	0
$671$	−12521.7	−0.720412
$672$	0	0
$673$	−11532.0	−0.660515	−0.330257	−	0.943891i	$-0.607136\pi$
$673$	−11532.0	−0.660515	−0.330257	+	0.943891i	$0.607136\pi$
$674$	0	0
$675$	−3176.80	−0.181148
$676$	0	0
$677$	5681.03	0.322511	0.161255	−	0.986913i	$-0.448446\pi$
$677$	5681.03	0.322511	0.161255	+	0.986913i	$0.448446\pi$
$678$	0	0
$679$	−39449.2	−2.22963
$680$	0	0
$681$	−4396.02	−0.247366
$682$	0	0
$683$	−10410.0	−0.583203	−0.291601	−	0.956540i	$-0.594188\pi$
$683$	−10410.0	−0.583203	−0.291601	+	0.956540i	$0.594188\pi$
$684$	0	0
$685$	3085.12	0.172082
$686$	0	0
$687$	−9842.40	−0.546595
$688$	0	0
$689$	17686.3	0.977929
$690$	0	0
$691$	23408.8	1.28873	0.644366	−	0.764717i	$-0.277121\pi$
$691$	23408.8	1.28873	0.644366	+	0.764717i	$0.277121\pi$
$692$	0	0
$693$	21516.4	1.17942
$694$	0	0
$695$	−13780.4	−0.752118
$696$	0	0
$697$	−1990.47	−0.108170
$698$	0	0
$699$	3917.01	0.211953
$700$	0	0
$701$	25111.5	1.35299	0.676497	−	0.736445i	$-0.263497\pi$
$701$	25111.5	1.35299	0.676497	+	0.736445i	$0.263497\pi$
$702$	0	0
$703$	10265.9	0.550764
$704$	0	0
$705$	−2595.17	−0.138638
$706$	0	0
$707$	29815.6	1.58604
$708$	0	0
$709$	13621.7	0.721543	0.360771	−	0.932654i	$-0.382513\pi$
$709$	13621.7	0.721543	0.360771	+	0.932654i	$0.382513\pi$
$710$	0	0
$711$	−17174.6	−0.905902
$712$	0	0
$713$	−641.224	−0.0336802
$714$	0	0
$715$	3950.38	0.206623
$716$	0	0
$717$	−10468.5	−0.545263
$718$	0	0
$719$	21520.2	1.11623	0.558115	−	0.829764i	$-0.311525\pi$
$719$	21520.2	1.11623	0.558115	+	0.829764i	$0.311525\pi$
$720$	0	0
$721$	−34338.6	−1.77370
$722$	0	0
$723$	−6538.48	−0.336333
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	−13731.8	−0.700530	−0.350265	−	0.936651i	$-0.613908\pi$
$727$	−13731.8	−0.700530	−0.350265	+	0.936651i	$0.613908\pi$
$728$	0	0
$729$	5830.60	0.296225
$730$	0	0
$731$	3938.92	0.199297
$732$	0	0
$733$	14047.1	0.707833	0.353916	−	0.935277i	$-0.384850\pi$
$733$	14047.1	0.707833	0.353916	+	0.935277i	$0.384850\pi$
$734$	0	0
$735$	−9855.57	−0.494596
$736$	0	0
$737$	14767.6	0.738090
$738$	0	0
$739$	−17818.9	−0.886982	−0.443491	−	0.896279i	$-0.646260\pi$
$739$	−17818.9	−0.886982	−0.443491	+	0.896279i	$0.646260\pi$
$740$	0	0
$741$	5512.55	0.273291
$742$	0	0
$743$	3761.53	0.185730	0.0928649	−	0.995679i	$-0.470398\pi$
$743$	3761.53	0.185730	0.0928649	+	0.995679i	$0.470398\pi$
$744$	0	0
$745$	9466.01	0.465514
$746$	0	0
$747$	4397.79	0.215404
$748$	0	0
$749$	−21100.0	−1.02934
$750$	0	0
$751$	9764.67	0.474458	0.237229	−	0.971454i	$-0.423761\pi$
$751$	9764.67	0.474458	0.237229	+	0.971454i	$0.423761\pi$
$752$	0	0
$753$	2634.67	0.127507
$754$	0	0
$755$	−12061.5	−0.581406
$756$	0	0
$757$	−23780.7	−1.14178	−0.570888	−	0.821028i	$-0.693401\pi$
$757$	−23780.7	−1.14178	−0.570888	+	0.821028i	$0.693401\pi$
$758$	0	0
$759$	2087.98	0.0998536
$760$	0	0
$761$	6342.11	0.302104	0.151052	−	0.988526i	$-0.451734\pi$
$761$	6342.11	0.302104	0.151052	+	0.988526i	$0.451734\pi$
$762$	0	0
$763$	66325.6	3.14698
$764$	0	0
$765$	1725.41	0.0815456
$766$	0	0
$767$	−16836.4	−0.792606
$768$	0	0
$769$	4968.25	0.232977	0.116489	−	0.993192i	$-0.462836\pi$
$769$	4968.25	0.232977	0.116489	+	0.993192i	$0.462836\pi$
$770$	0	0
$771$	1537.25	0.0718061
$772$	0	0
$773$	22348.5	1.03987	0.519934	−	0.854206i	$-0.325957\pi$
$773$	22348.5	1.03987	0.519934	+	0.854206i	$0.325957\pi$
$774$	0	0
$775$	706.931	0.0327661
$776$	0	0
$777$	10609.8	0.489863
$778$	0	0
$779$	−9719.64	−0.447038
$780$	0	0
$781$	−6703.62	−0.307137
$782$	0	0
$783$	−3685.09	−0.168192
$784$	0	0
$785$	12819.5	0.582862
$786$	0	0
$787$	−6726.01	−0.304646	−0.152323	−	0.988331i	$-0.548675\pi$
$787$	−6726.01	−0.304646	−0.152323	+	0.988331i	$0.548675\pi$
$788$	0	0
$789$	−12218.1	−0.551301
$790$	0	0
$791$	15568.1	0.699794
$792$	0	0
$793$	8696.32	0.389427
$794$	0	0
$795$	−10306.0	−0.459770
$796$	0	0
$797$	−25298.1	−1.12435	−0.562174	−	0.827019i	$-0.690035\pi$
$797$	−25298.1	−1.12435	−0.562174	+	0.827019i	$0.690035\pi$
$798$	0	0
$799$	3356.42	0.148613
$800$	0	0
$801$	−13543.6	−0.597427
$802$	0	0
$803$	3018.44	0.132651
$804$	0	0
$805$	3700.19	0.162006
$806$	0	0
$807$	12497.7	0.545153
$808$	0	0
$809$	−25540.5	−1.10996	−0.554979	−	0.831865i	$-0.687274\pi$
$809$	−25540.5	−1.10996	−0.554979	+	0.831865i	$0.687274\pi$
$810$	0	0
$811$	−20640.1	−0.893677	−0.446839	−	0.894615i	$-0.647450\pi$
$811$	−20640.1	−0.893677	−0.446839	+	0.894615i	$0.647450\pi$
$812$	0	0
$813$	5117.50	0.220761
$814$	0	0
$815$	−13993.7	−0.601447
$816$	0	0
$817$	19234.1	0.823642
$818$	0	0
$819$	−14943.1	−0.637549
$820$	0	0
$821$	21932.8	0.932349	0.466175	−	0.884693i	$-0.345632\pi$
$821$	21932.8	0.932349	0.466175	+	0.884693i	$0.345632\pi$
$822$	0	0
$823$	−9930.66	−0.420609	−0.210304	−	0.977636i	$-0.567445\pi$
$823$	−9930.66	−0.420609	−0.210304	+	0.977636i	$0.567445\pi$
$824$	0	0
$825$	−2301.94	−0.0971433
$826$	0	0
$827$	−24065.8	−1.01191	−0.505956	−	0.862559i	$-0.668860\pi$
$827$	−24065.8	−1.01191	−0.505956	+	0.862559i	$0.668860\pi$
$828$	0	0
$829$	29885.6	1.25208	0.626038	−	0.779793i	$-0.284676\pi$
$829$	29885.6	1.25208	0.626038	+	0.779793i	$0.284676\pi$
$830$	0	0
$831$	22644.0	0.945260
$832$	0	0
$833$	12746.5	0.530181
$834$	0	0
$835$	−2298.70	−0.0952693
$836$	0	0
$837$	−3593.24	−0.148388
$838$	0	0
$839$	3525.59	0.145074	0.0725370	−	0.997366i	$-0.476890\pi$
$839$	3525.59	0.145074	0.0725370	+	0.997366i	$0.476890\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−13489.4	−0.551127
$844$	0	0
$845$	8241.47	0.335521
$846$	0	0
$847$	−6310.89	−0.256015
$848$	0	0
$849$	−19731.2	−0.797613
$850$	0	0
$851$	−2700.49	−0.108780
$852$	0	0
$853$	−32471.5	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$853$	−32471.5	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$854$	0	0
$855$	8425.33	0.337006
$856$	0	0
$857$	17181.3	0.684833	0.342417	−	0.939548i	$-0.388755\pi$
$857$	17181.3	0.684833	0.342417	+	0.939548i	$0.388755\pi$
$858$	0	0
$859$	−4979.08	−0.197769	−0.0988847	−	0.995099i	$-0.531528\pi$
$859$	−4979.08	−0.197769	−0.0988847	+	0.995099i	$0.531528\pi$
$860$	0	0
$861$	−10045.2	−0.397606
$862$	0	0
$863$	4517.03	0.178171	0.0890854	−	0.996024i	$-0.471606\pi$
$863$	4517.03	0.178171	0.0890854	+	0.996024i	$0.471606\pi$
$864$	0	0
$865$	−8419.52	−0.330950
$866$	0	0
$867$	−12561.5	−0.492052
$868$	0	0
$869$	−29634.4	−1.15682
$870$	0	0
$871$	−10256.1	−0.398983
$872$	0	0
$873$	23629.0	0.916058
$874$	0	0
$875$	−4079.36	−0.157608
$876$	0	0
$877$	−6452.68	−0.248451	−0.124225	−	0.992254i	$-0.539645\pi$
$877$	−6452.68	−0.248451	−0.124225	+	0.992254i	$0.539645\pi$
$878$	0	0
$879$	−4360.69	−0.167329
$880$	0	0
$881$	2837.29	0.108502	0.0542512	−	0.998527i	$-0.482723\pi$
$881$	2837.29	0.108502	0.0542512	+	0.998527i	$0.482723\pi$
$882$	0	0
$883$	−11073.2	−0.422018	−0.211009	−	0.977484i	$-0.567675\pi$
$883$	−11073.2	−0.422018	−0.211009	+	0.977484i	$0.567675\pi$
$884$	0	0
$885$	9810.82	0.372641
$886$	0	0
$887$	−30442.7	−1.15238	−0.576192	−	0.817314i	$-0.695462\pi$
$887$	−30442.7	−1.15238	−0.576192	+	0.817314i	$0.695462\pi$
$888$	0	0
$889$	57869.7	2.18323
$890$	0	0
$891$	−6100.79	−0.229388
$892$	0	0
$893$	16389.7	0.614176
$894$	0	0
$895$	3987.19	0.148913
$896$	0	0
$897$	−1450.10	−0.0539769
$898$	0	0
$899$	820.040	0.0304225
$900$	0	0
$901$	13329.1	0.492849
$902$	0	0
$903$	19878.3	0.732568
$904$	0	0
$905$	−18731.2	−0.688007
$906$	0	0
$907$	33276.0	1.21820	0.609102	−	0.793092i	$-0.291530\pi$
$907$	33276.0	1.21820	0.609102	+	0.793092i	$0.291530\pi$
$908$	0	0
$909$	−17858.7	−0.651635
$910$	0	0
$911$	31871.2	1.15910	0.579550	−	0.814936i	$-0.303228\pi$
$911$	31871.2	1.15910	0.579550	+	0.814936i	$0.303228\pi$
$912$	0	0
$913$	7588.32	0.275067
$914$	0	0
$915$	−5067.47	−0.183088
$916$	0	0
$917$	44097.7	1.58804
$918$	0	0
$919$	−14502.6	−0.520562	−0.260281	−	0.965533i	$-0.583815\pi$
$919$	−14502.6	−0.520562	−0.260281	+	0.965533i	$0.583815\pi$
$920$	0	0
$921$	14593.5	0.522119
$922$	0	0
$923$	4655.65	0.166027
$924$	0	0
$925$	2977.21	0.105827
$926$	0	0
$927$	20567.9	0.728734
$928$	0	0
$929$	−30601.2	−1.08072	−0.540362	−	0.841433i	$-0.681713\pi$
$929$	−30601.2	−1.08072	−0.540362	+	0.841433i	$0.681713\pi$
$930$	0	0
$931$	62242.3	2.19110
$932$	0	0
$933$	7740.80	0.271621
$934$	0	0
$935$	2977.17	0.104132
$936$	0	0
$937$	−16692.8	−0.581995	−0.290998	−	0.956724i	$-0.593987\pi$
$937$	−16692.8	−0.581995	−0.290998	+	0.956724i	$0.593987\pi$
$938$	0	0
$939$	16252.1	0.564822
$940$	0	0
$941$	32223.3	1.11631	0.558155	−	0.829737i	$-0.311509\pi$
$941$	32223.3	1.11631	0.558155	+	0.829737i	$0.311509\pi$
$942$	0	0
$943$	2556.79	0.0882931
$944$	0	0
$945$	20734.9	0.713763
$946$	0	0
$947$	39876.1	1.36832	0.684159	−	0.729333i	$-0.260169\pi$
$947$	39876.1	1.36832	0.684159	+	0.729333i	$0.260169\pi$
$948$	0	0
$949$	−2096.30	−0.0717058
$950$	0	0
$951$	−1550.00	−0.0528519
$952$	0	0
$953$	−23616.0	−0.802726	−0.401363	−	0.915919i	$-0.631463\pi$
$953$	−23616.0	−0.802726	−0.401363	+	0.915919i	$0.631463\pi$
$954$	0	0
$955$	−1484.39	−0.0502972
$956$	0	0
$957$	−2670.25	−0.0901953
$958$	0	0
$959$	−20136.5	−0.678041
$960$	0	0
$961$	−28991.4	−0.973160
$962$	0	0
$963$	12638.3	0.422912
$964$	0	0
$965$	−78.7096	−0.00262565
$966$	0	0
$967$	−38203.8	−1.27048	−0.635238	−	0.772316i	$-0.719098\pi$
$967$	−38203.8	−1.27048	−0.635238	+	0.772316i	$0.719098\pi$
$968$	0	0
$969$	4154.49	0.137731
$970$	0	0
$971$	490.958	0.0162262	0.00811308	−	0.999967i	$-0.497417\pi$
$971$	490.958	0.0162262	0.00811308	+	0.999967i	$0.497417\pi$
$972$	0	0
$973$	89944.5	2.96350
$974$	0	0
$975$	1598.69	0.0525119
$976$	0	0
$977$	−1628.73	−0.0533342	−0.0266671	−	0.999644i	$-0.508489\pi$
$977$	−1628.73	−0.0533342	−0.0266671	+	0.999644i	$0.508489\pi$
$978$	0	0
$979$	−23369.2	−0.762904
$980$	0	0
$981$	−39727.2	−1.29296
$982$	0	0
$983$	−9795.97	−0.317846	−0.158923	−	0.987291i	$-0.550802\pi$
$983$	−9795.97	−0.317846	−0.158923	+	0.987291i	$0.550802\pi$
$984$	0	0
$985$	−14681.5	−0.474914
$986$	0	0
$987$	16938.6	0.546263
$988$	0	0
$989$	−5059.59	−0.162675
$990$	0	0
$991$	37700.6	1.20847	0.604237	−	0.796805i	$-0.293478\pi$
$991$	37700.6	1.20847	0.604237	+	0.796805i	$0.293478\pi$
$992$	0	0
$993$	8696.73	0.277928
$994$	0	0
$995$	−17574.1	−0.559935
$996$	0	0
$997$	7451.87	0.236713	0.118357	−	0.992971i	$-0.462237\pi$
$997$	7451.87	0.236713	0.118357	+	0.992971i	$0.462237\pi$
$998$	0	0
$999$	−15132.8	−0.479260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.h.1.7	✓	11
4.3	odd	2		2320.4.a.z.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.h.1.7	✓	11	1.1	even	1	trivial
2320.4.a.z.1.5		11	4.3	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:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.72995 q^{3} -5.00000 q^{5} +32.6348 q^{7} -19.5474 q^{9} -33.7286 q^{11} +23.4245 q^{13} -13.6498 q^{15} +17.6536 q^{17} +86.2042 q^{19} +89.0915 q^{21} -22.6763 q^{23} +25.0000 q^{25} -127.072 q^{27} +29.0000 q^{29} +28.2772 q^{31} -92.0775 q^{33} -163.174 q^{35} +119.089 q^{37} +63.9476 q^{39} -112.751 q^{41} +223.122 q^{43} +97.7369 q^{45} +190.126 q^{47} +722.033 q^{49} +48.1936 q^{51} +755.034 q^{53} +168.643 q^{55} +235.333 q^{57} -718.755 q^{59} +371.250 q^{61} -637.925 q^{63} -117.122 q^{65} -437.836 q^{67} -61.9052 q^{69} +198.752 q^{71} -89.4919 q^{73} +68.2488 q^{75} -1100.73 q^{77} +878.612 q^{79} +180.879 q^{81} -224.981 q^{83} -88.2682 q^{85} +79.1686 q^{87} +692.859 q^{89} +764.453 q^{91} +77.1954 q^{93} -431.021 q^{95} -1208.80 q^{97} +659.306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 8 q^{3} - 55 q^{5} + 16 q^{7} + 117 q^{9} + 26 q^{11} + 52 q^{13} - 40 q^{15} + 14 q^{17} + 146 q^{19} - 288 q^{21} - 80 q^{23} + 275 q^{25} + 224 q^{27} + 319 q^{29} - 38 q^{31} - 152 q^{33} - 80 q^{35}+ \cdots + 4690 q^{99}+O(q^{100}) \) 11 * q + 8 * q^3 - 55 * q^5 + 16 * q^7 + 117 * q^9 + 26 * q^11 + 52 * q^13 - 40 * q^15 + 14 * q^17 + 146 * q^19 - 288 * q^21 - 80 * q^23 + 275 * q^25 + 224 * q^27 + 319 * q^29 - 38 * q^31 - 152 * q^33 - 80 * q^35 + 376 * q^37 + 108 * q^39 - 146 * q^41 + 644 * q^43 - 585 * q^45 - 464 * q^47 + 393 * q^49 - 576 * q^51 - 188 * q^53 - 130 * q^55 + 1380 * q^57 + 672 * q^59 + 472 * q^61 + 628 * q^63 - 260 * q^65 + 1688 * q^67 + 138 * q^69 + 92 * q^71 + 1514 * q^73 + 200 * q^75 + 2884 * q^77 + 2130 * q^79 + 4067 * q^81 + 2584 * q^83 - 70 * q^85 + 232 * q^87 + 1606 * q^89 + 3208 * q^91 + 3540 * q^93 - 730 * q^95 + 3322 * q^97 + 4690 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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