LMFDB - Embedded newform 1088.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.6420.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 21x - 15 $ x^3 - x^2 - 21*x - 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.763190$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.76319 q^{3} -3.36478 q^{5} -10.1807 q^{7} +6.21436 q^{9} +O(q^{10})$	$q-5.76319 q^{3} -3.36478 q^{5} -10.1807 q^{7} +6.21436 q^{9} +19.6577 q^{11} +68.5613 q^{13} +19.3919 q^{15} +17.0000 q^{17} -28.5502 q^{19} +58.6735 q^{21} -201.822 q^{23} -113.678 q^{25} +119.792 q^{27} -158.023 q^{29} -72.2347 q^{31} -113.291 q^{33} +34.2559 q^{35} +109.048 q^{37} -395.132 q^{39} +12.8061 q^{41} -431.101 q^{43} -20.9100 q^{45} +119.260 q^{47} -239.353 q^{49} -97.9742 q^{51} +341.030 q^{53} -66.1438 q^{55} +164.540 q^{57} -390.813 q^{59} +700.829 q^{61} -63.2667 q^{63} -230.694 q^{65} +113.307 q^{67} +1163.14 q^{69} +273.472 q^{71} +229.812 q^{73} +655.149 q^{75} -200.129 q^{77} +628.935 q^{79} -858.169 q^{81} -853.016 q^{83} -57.2013 q^{85} +910.717 q^{87} +529.007 q^{89} -698.004 q^{91} +416.302 q^{93} +96.0651 q^{95} +1236.68 q^{97} +122.160 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9	$ 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9} + 82 q^{11} + 6 q^{13} - 160 q^{15} + 51 q^{17} + 172 q^{19} + 72 q^{21} - 158 q^{23} + 69 q^{25} - 80 q^{27} + 330 q^{29} - 34 q^{31} - 84 q^{33} + 40 q^{35} + 378 q^{37} - 416 q^{39} - 282 q^{41} - 536 q^{43} + 954 q^{45} - 564 q^{47} - 321 q^{49} - 238 q^{51} + 894 q^{53} - 40 q^{55} - 984 q^{57} - 1216 q^{59} + 978 q^{61} - 742 q^{63} - 276 q^{65} + 272 q^{67} + 816 q^{69} - 158 q^{71} - 498 q^{73} - 2050 q^{75} + 1344 q^{77} + 1410 q^{79} + 111 q^{81} + 1624 q^{83} + 306 q^{85} - 1240 q^{87} + 42 q^{89} - 2224 q^{91} + 2088 q^{93} + 2960 q^{95} + 822 q^{97} - 1586 q^{99}+O(q^{100}) $ 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9 + 82 * q^11 + 6 * q^13 - 160 * q^15 + 51 * q^17 + 172 * q^19 + 72 * q^21 - 158 * q^23 + 69 * q^25 - 80 * q^27 + 330 * q^29 - 34 * q^31 - 84 * q^33 + 40 * q^35 + 378 * q^37 - 416 * q^39 - 282 * q^41 - 536 * q^43 + 954 * q^45 - 564 * q^47 - 321 * q^49 - 238 * q^51 + 894 * q^53 - 40 * q^55 - 984 * q^57 - 1216 * q^59 + 978 * q^61 - 742 * q^63 - 276 * q^65 + 272 * q^67 + 816 * q^69 - 158 * q^71 - 498 * q^73 - 2050 * q^75 + 1344 * q^77 + 1410 * q^79 + 111 * q^81 + 1624 * q^83 + 306 * q^85 - 1240 * q^87 + 42 * q^89 - 2224 * q^91 + 2088 * q^93 + 2960 * q^95 + 822 * q^97 - 1586 * 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
$3$	−5.76319	−1.10913	−0.554563	−	0.832142i	$-0.687115\pi$
$3$	−5.76319	−1.10913	−0.554563	+	0.832142i	$0.687115\pi$
$4$	0	0
$5$	−3.36478	−0.300955	−0.150478	−	0.988613i	$-0.548081\pi$
$5$	−3.36478	−0.300955	−0.150478	+	0.988613i	$0.548081\pi$
$6$	0	0
$7$	−10.1807	−0.549708	−0.274854	−	0.961486i	$-0.588629\pi$
$7$	−10.1807	−0.549708	−0.274854	+	0.961486i	$0.588629\pi$
$8$	0	0
$9$	6.21436	0.230161
$10$	0	0
$11$	19.6577	0.538819	0.269410	−	0.963026i	$-0.413171\pi$
$11$	19.6577	0.538819	0.269410	+	0.963026i	$0.413171\pi$
$12$	0	0
$13$	68.5613	1.46273	0.731365	−	0.681986i	$-0.238884\pi$
$13$	68.5613	1.46273	0.731365	+	0.681986i	$0.238884\pi$
$14$	0	0
$15$	19.3919	0.333797
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−28.5502	−0.344729	−0.172365	−	0.985033i	$-0.555141\pi$
$19$	−28.5502	−0.344729	−0.172365	+	0.985033i	$0.555141\pi$
$20$	0	0
$21$	58.6735	0.609695
$22$	0	0
$23$	−201.822	−1.82969	−0.914843	−	0.403810i	$-0.867686\pi$
$23$	−201.822	−1.82969	−0.914843	+	0.403810i	$0.867686\pi$
$24$	0	0
$25$	−113.678	−0.909426
$26$	0	0
$27$	119.792	0.853848
$28$	0	0
$29$	−158.023	−1.01187	−0.505934	−	0.862572i	$-0.668852\pi$
$29$	−158.023	−1.01187	−0.505934	+	0.862572i	$0.668852\pi$
$30$	0	0
$31$	−72.2347	−0.418508	−0.209254	−	0.977861i	$-0.567103\pi$
$31$	−72.2347	−0.418508	−0.209254	+	0.977861i	$0.567103\pi$
$32$	0	0
$33$	−113.291	−0.597619
$34$	0	0
$35$	34.2559	0.165437
$36$	0	0
$37$	109.048	0.484525	0.242262	−	0.970211i	$-0.422111\pi$
$37$	109.048	0.484525	0.242262	+	0.970211i	$0.422111\pi$
$38$	0	0
$39$	−395.132	−1.62235
$40$	0	0
$41$	12.8061	0.0487799	0.0243900	−	0.999703i	$-0.492236\pi$
$41$	12.8061	0.0487799	0.0243900	+	0.999703i	$0.492236\pi$
$42$	0	0
$43$	−431.101	−1.52889	−0.764444	−	0.644690i	$-0.776987\pi$
$43$	−431.101	−1.52889	−0.764444	+	0.644690i	$0.776987\pi$
$44$	0	0
$45$	−20.9100	−0.0692683
$46$	0	0
$47$	119.260	0.370125	0.185062	−	0.982727i	$-0.440751\pi$
$47$	119.260	0.370125	0.185062	+	0.982727i	$0.440751\pi$
$48$	0	0
$49$	−239.353	−0.697821
$50$	0	0
$51$	−97.9742	−0.269003
$52$	0	0
$53$	341.030	0.883852	0.441926	−	0.897052i	$-0.354295\pi$
$53$	341.030	0.883852	0.441926	+	0.897052i	$0.354295\pi$
$54$	0	0
$55$	−66.1438	−0.162160
$56$	0	0
$57$	164.540	0.382348
$58$	0	0
$59$	−390.813	−0.862364	−0.431182	−	0.902265i	$-0.641903\pi$
$59$	−390.813	−0.862364	−0.431182	+	0.902265i	$0.641903\pi$
$60$	0	0
$61$	700.829	1.47102	0.735508	−	0.677516i	$-0.236944\pi$
$61$	700.829	1.47102	0.735508	+	0.677516i	$0.236944\pi$
$62$	0	0
$63$	−63.2667	−0.126521
$64$	0	0
$65$	−230.694	−0.440216
$66$	0	0
$67$	113.307	0.206606	0.103303	−	0.994650i	$-0.467059\pi$
$67$	113.307	0.206606	0.103303	+	0.994650i	$0.467059\pi$
$68$	0	0
$69$	1163.14	2.02935
$70$	0	0
$71$	273.472	0.457115	0.228558	−	0.973530i	$-0.426599\pi$
$71$	273.472	0.457115	0.228558	+	0.973530i	$0.426599\pi$
$72$	0	0
$73$	229.812	0.368459	0.184229	−	0.982883i	$-0.441021\pi$
$73$	229.812	0.368459	0.184229	+	0.982883i	$0.441021\pi$
$74$	0	0
$75$	655.149	1.00867
$76$	0	0
$77$	−200.129	−0.296193
$78$	0	0
$79$	628.935	0.895706	0.447853	−	0.894107i	$-0.352189\pi$
$79$	628.935	0.895706	0.447853	+	0.894107i	$0.352189\pi$
$80$	0	0
$81$	−858.169	−1.17719
$82$	0	0
$83$	−853.016	−1.12808	−0.564040	−	0.825747i	$-0.690754\pi$
$83$	−853.016	−1.12808	−0.564040	+	0.825747i	$0.690754\pi$
$84$	0	0
$85$	−57.2013	−0.0729924
$86$	0	0
$87$	910.717	1.12229
$88$	0	0
$89$	529.007	0.630052	0.315026	−	0.949083i	$-0.397987\pi$
$89$	529.007	0.630052	0.315026	+	0.949083i	$0.397987\pi$
$90$	0	0
$91$	−698.004	−0.804074
$92$	0	0
$93$	416.302	0.464178
$94$	0	0
$95$	96.0651	0.103748
$96$	0	0
$97$	1236.68	1.29450	0.647248	−	0.762280i	$-0.275920\pi$
$97$	1236.68	1.29450	0.647248	+	0.762280i	$0.275920\pi$
$98$	0	0
$99$	122.160	0.124015
$100$	0	0
$101$	975.723	0.961268	0.480634	−	0.876921i	$-0.340407\pi$
$101$	975.723	0.961268	0.480634	+	0.876921i	$0.340407\pi$
$102$	0	0
$103$	−1537.93	−1.47123	−0.735613	−	0.677402i	$-0.763106\pi$
$103$	−1537.93	−1.47123	−0.735613	+	0.677402i	$0.763106\pi$
$104$	0	0
$105$	−197.424	−0.183491
$106$	0	0
$107$	−786.818	−0.710883	−0.355442	−	0.934698i	$-0.615670\pi$
$107$	−786.818	−0.710883	−0.355442	+	0.934698i	$0.615670\pi$
$108$	0	0
$109$	2157.72	1.89608	0.948038	−	0.318156i	$-0.103064\pi$
$109$	2157.72	1.89608	0.948038	+	0.318156i	$0.103064\pi$
$110$	0	0
$111$	−628.466	−0.537399
$112$	0	0
$113$	−1514.46	−1.26078	−0.630390	−	0.776279i	$-0.717105\pi$
$113$	−1514.46	−1.26078	−0.630390	+	0.776279i	$0.717105\pi$
$114$	0	0
$115$	679.087	0.550654
$116$	0	0
$117$	426.065	0.336664
$118$	0	0
$119$	−173.072	−0.133324
$120$	0	0
$121$	−944.576	−0.709674
$122$	0	0
$123$	−73.8040	−0.0541031
$124$	0	0
$125$	803.100	0.574652
$126$	0	0
$127$	1566.55	1.09456	0.547279	−	0.836950i	$-0.315664\pi$
$127$	1566.55	1.09456	0.547279	+	0.836950i	$0.315664\pi$
$128$	0	0
$129$	2484.51	1.69573
$130$	0	0
$131$	1665.47	1.11078	0.555392	−	0.831589i	$-0.312568\pi$
$131$	1665.47	1.11078	0.555392	+	0.831589i	$0.312568\pi$
$132$	0	0
$133$	290.661	0.189500
$134$	0	0
$135$	−403.073	−0.256970
$136$	0	0
$137$	−349.267	−0.217809	−0.108905	−	0.994052i	$-0.534734\pi$
$137$	−349.267	−0.217809	−0.108905	+	0.994052i	$0.534734\pi$
$138$	0	0
$139$	1096.12	0.668863	0.334432	−	0.942420i	$-0.391456\pi$
$139$	1096.12	0.668863	0.334432	+	0.942420i	$0.391456\pi$
$140$	0	0
$141$	−687.318	−0.410515
$142$	0	0
$143$	1347.76	0.788147
$144$	0	0
$145$	531.713	0.304527
$146$	0	0
$147$	1379.44	0.773972
$148$	0	0
$149$	−306.663	−0.168609	−0.0843047	−	0.996440i	$-0.526867\pi$
$149$	−306.663	−0.168609	−0.0843047	+	0.996440i	$0.526867\pi$
$150$	0	0
$151$	−1475.20	−0.795036	−0.397518	−	0.917594i	$-0.630128\pi$
$151$	−1475.20	−0.795036	−0.397518	+	0.917594i	$0.630128\pi$
$152$	0	0
$153$	105.644	0.0558223
$154$	0	0
$155$	243.054	0.125952
$156$	0	0
$157$	−432.051	−0.219627	−0.109813	−	0.993952i	$-0.535025\pi$
$157$	−432.051	−0.219627	−0.109813	+	0.993952i	$0.535025\pi$
$158$	0	0
$159$	−1965.42	−0.980303
$160$	0	0
$161$	2054.69	1.00579
$162$	0	0
$163$	−547.470	−0.263075	−0.131537	−	0.991311i	$-0.541991\pi$
$163$	−547.470	−0.263075	−0.131537	+	0.991311i	$0.541991\pi$
$164$	0	0
$165$	381.199	0.179856
$166$	0	0
$167$	2249.45	1.04232	0.521161	−	0.853458i	$-0.325499\pi$
$167$	2249.45	1.04232	0.521161	+	0.853458i	$0.325499\pi$
$168$	0	0
$169$	2503.66	1.13958
$170$	0	0
$171$	−177.421	−0.0793433
$172$	0	0
$173$	832.169	0.365715	0.182857	−	0.983139i	$-0.441465\pi$
$173$	832.169	0.365715	0.182857	+	0.983139i	$0.441465\pi$
$174$	0	0
$175$	1157.33	0.499919
$176$	0	0
$177$	2252.33	0.956470
$178$	0	0
$179$	2067.66	0.863374	0.431687	−	0.902024i	$-0.357919\pi$
$179$	2067.66	0.863374	0.431687	+	0.902024i	$0.357919\pi$
$180$	0	0
$181$	1897.96	0.779416	0.389708	−	0.920939i	$-0.372576\pi$
$181$	1897.96	0.779416	0.389708	+	0.920939i	$0.372576\pi$
$182$	0	0
$183$	−4039.01	−1.63154
$184$	0	0
$185$	−366.924	−0.145820
$186$	0	0
$187$	334.180	0.130683
$188$	0	0
$189$	−1219.57	−0.469367
$190$	0	0
$191$	4583.23	1.73629	0.868143	−	0.496315i	$-0.165314\pi$
$191$	4583.23	1.73629	0.868143	+	0.496315i	$0.165314\pi$
$192$	0	0
$193$	3465.26	1.29241	0.646205	−	0.763164i	$-0.276355\pi$
$193$	3465.26	1.29241	0.646205	+	0.763164i	$0.276355\pi$
$194$	0	0
$195$	1329.53	0.488256
$196$	0	0
$197$	1233.82	0.446225	0.223112	−	0.974793i	$-0.428378\pi$
$197$	1233.82	0.446225	0.223112	+	0.974793i	$0.428378\pi$
$198$	0	0
$199$	268.509	0.0956488	0.0478244	−	0.998856i	$-0.484771\pi$
$199$	268.509	0.0956488	0.0478244	+	0.998856i	$0.484771\pi$
$200$	0	0
$201$	−653.007	−0.229152
$202$	0	0
$203$	1608.79	0.556231
$204$	0	0
$205$	−43.0897	−0.0146806
$206$	0	0
$207$	−1254.19	−0.421123
$208$	0	0
$209$	−561.230	−0.185747
$210$	0	0
$211$	−4345.16	−1.41769	−0.708846	−	0.705363i	$-0.750784\pi$
$211$	−4345.16	−1.41769	−0.708846	+	0.705363i	$0.750784\pi$
$212$	0	0
$213$	−1576.07	−0.506998
$214$	0	0
$215$	1450.56	0.460127
$216$	0	0
$217$	735.402	0.230057
$218$	0	0
$219$	−1324.45	−0.408668
$220$	0	0
$221$	1165.54	0.354764
$222$	0	0
$223$	3233.54	0.971005	0.485502	−	0.874235i	$-0.338637\pi$
$223$	3233.54	0.971005	0.485502	+	0.874235i	$0.338637\pi$
$224$	0	0
$225$	−706.437	−0.209315
$226$	0	0
$227$	5155.20	1.50732	0.753662	−	0.657262i	$-0.228285\pi$
$227$	5155.20	1.50732	0.753662	+	0.657262i	$0.228285\pi$
$228$	0	0
$229$	−234.000	−0.0675247	−0.0337623	−	0.999430i	$-0.510749\pi$
$229$	−234.000	−0.0675247	−0.0337623	+	0.999430i	$0.510749\pi$
$230$	0	0
$231$	1153.38	0.328516
$232$	0	0
$233$	1623.64	0.456515	0.228257	−	0.973601i	$-0.426697\pi$
$233$	1623.64	0.456515	0.228257	+	0.973601i	$0.426697\pi$
$234$	0	0
$235$	−401.284	−0.111391
$236$	0	0
$237$	−3624.67	−0.993451
$238$	0	0
$239$	−2966.49	−0.802871	−0.401435	−	0.915887i	$-0.631489\pi$
$239$	−2966.49	−0.802871	−0.401435	+	0.915887i	$0.631489\pi$
$240$	0	0
$241$	3077.94	0.822686	0.411343	−	0.911481i	$-0.365060\pi$
$241$	3077.94	0.822686	0.411343	+	0.911481i	$0.365060\pi$
$242$	0	0
$243$	1711.42	0.451801
$244$	0	0
$245$	805.370	0.210013
$246$	0	0
$247$	−1957.44	−0.504246
$248$	0	0
$249$	4916.09	1.25118
$250$	0	0
$251$	−2028.35	−0.510074	−0.255037	−	0.966931i	$-0.582088\pi$
$251$	−2028.35	−0.510074	−0.255037	+	0.966931i	$0.582088\pi$
$252$	0	0
$253$	−3967.35	−0.985870
$254$	0	0
$255$	329.662	0.0809578
$256$	0	0
$257$	−14.3544	−0.00348406	−0.00174203	−	0.999998i	$-0.500555\pi$
$257$	−14.3544	−0.00348406	−0.00174203	+	0.999998i	$0.500555\pi$
$258$	0	0
$259$	−1110.19	−0.266347
$260$	0	0
$261$	−982.012	−0.232893
$262$	0	0
$263$	3670.15	0.860498	0.430249	−	0.902710i	$-0.358426\pi$
$263$	3670.15	0.860498	0.430249	+	0.902710i	$0.358426\pi$
$264$	0	0
$265$	−1147.49	−0.266000
$266$	0	0
$267$	−3048.77	−0.698807
$268$	0	0
$269$	−5203.62	−1.17944	−0.589722	−	0.807607i	$-0.700763\pi$
$269$	−5203.62	−1.17944	−0.589722	+	0.807607i	$0.700763\pi$
$270$	0	0
$271$	2246.49	0.503560	0.251780	−	0.967785i	$-0.418984\pi$
$271$	2246.49	0.503560	0.251780	+	0.967785i	$0.418984\pi$
$272$	0	0
$273$	4022.73	0.891820
$274$	0	0
$275$	−2234.65	−0.490016
$276$	0	0
$277$	−201.069	−0.0436140	−0.0218070	−	0.999762i	$-0.506942\pi$
$277$	−201.069	−0.0436140	−0.0218070	+	0.999762i	$0.506942\pi$
$278$	0	0
$279$	−448.892	−0.0963243
$280$	0	0
$281$	−1773.44	−0.376493	−0.188247	−	0.982122i	$-0.560280\pi$
$281$	−1773.44	−0.376493	−0.188247	+	0.982122i	$0.560280\pi$
$282$	0	0
$283$	−3163.77	−0.664546	−0.332273	−	0.943183i	$-0.607815\pi$
$283$	−3163.77	−0.664546	−0.332273	+	0.943183i	$0.607815\pi$
$284$	0	0
$285$	−553.641	−0.115070
$286$	0	0
$287$	−130.375	−0.0268147
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−7127.24	−1.43576
$292$	0	0
$293$	4352.00	0.867735	0.433868	−	0.900977i	$-0.357149\pi$
$293$	4352.00	0.867735	0.433868	+	0.900977i	$0.357149\pi$
$294$	0	0
$295$	1315.00	0.259533
$296$	0	0
$297$	2354.82	0.460070
$298$	0	0
$299$	−13837.2	−2.67634
$300$	0	0
$301$	4388.92	0.840442
$302$	0	0
$303$	−5623.28	−1.06617
$304$	0	0
$305$	−2358.14	−0.442710
$306$	0	0
$307$	10663.5	1.98241	0.991204	−	0.132341i	$-0.0422493\pi$
$307$	10663.5	1.98241	0.991204	+	0.132341i	$0.0422493\pi$
$308$	0	0
$309$	8863.36	1.63178
$310$	0	0
$311$	−2781.58	−0.507168	−0.253584	−	0.967313i	$-0.581609\pi$
$311$	−2781.58	−0.507168	−0.253584	+	0.967313i	$0.581609\pi$
$312$	0	0
$313$	6526.97	1.17868	0.589339	−	0.807886i	$-0.299388\pi$
$313$	6526.97	1.17868	0.589339	+	0.807886i	$0.299388\pi$
$314$	0	0
$315$	212.879	0.0380773
$316$	0	0
$317$	−7426.86	−1.31588	−0.657940	−	0.753070i	$-0.728572\pi$
$317$	−7426.86	−1.31588	−0.657940	+	0.753070i	$0.728572\pi$
$318$	0	0
$319$	−3106.37	−0.545214
$320$	0	0
$321$	4534.58	0.788460
$322$	0	0
$323$	−485.353	−0.0836091
$324$	0	0
$325$	−7793.93	−1.33024
$326$	0	0
$327$	−12435.4	−2.10299
$328$	0	0
$329$	−1214.15	−0.203460
$330$	0	0
$331$	−9198.67	−1.52751	−0.763753	−	0.645508i	$-0.776646\pi$
$331$	−9198.67	−1.52751	−0.763753	+	0.645508i	$0.776646\pi$
$332$	0	0
$333$	677.665	0.111519
$334$	0	0
$335$	−381.252	−0.0621791
$336$	0	0
$337$	−3263.60	−0.527536	−0.263768	−	0.964586i	$-0.584965\pi$
$337$	−3263.60	−0.527536	−0.263768	+	0.964586i	$0.584965\pi$
$338$	0	0
$339$	8728.10	1.39836
$340$	0	0
$341$	−1419.97	−0.225500
$342$	0	0
$343$	5928.78	0.933306
$344$	0	0
$345$	−3913.71	−0.610745
$346$	0	0
$347$	6061.48	0.937744	0.468872	−	0.883266i	$-0.344661\pi$
$347$	6061.48	0.937744	0.468872	+	0.883266i	$0.344661\pi$
$348$	0	0
$349$	2324.35	0.356503	0.178252	−	0.983985i	$-0.442956\pi$
$349$	2324.35	0.356503	0.178252	+	0.983985i	$0.442956\pi$
$350$	0	0
$351$	8213.07	1.24895
$352$	0	0
$353$	6446.24	0.971951	0.485976	−	0.873972i	$-0.338464\pi$
$353$	6446.24	0.971951	0.485976	+	0.873972i	$0.338464\pi$
$354$	0	0
$355$	−920.174	−0.137571
$356$	0	0
$357$	997.449	0.147873
$358$	0	0
$359$	−6633.00	−0.975143	−0.487571	−	0.873083i	$-0.662117\pi$
$359$	−6633.00	−0.975143	−0.487571	+	0.873083i	$0.662117\pi$
$360$	0	0
$361$	−6043.89	−0.881162
$362$	0	0
$363$	5443.77	0.787118
$364$	0	0
$365$	−773.269	−0.110890
$366$	0	0
$367$	−2326.39	−0.330890	−0.165445	−	0.986219i	$-0.552906\pi$
$367$	−2326.39	−0.330890	−0.165445	+	0.986219i	$0.552906\pi$
$368$	0	0
$369$	79.5817	0.0112273
$370$	0	0
$371$	−3471.94	−0.485860
$372$	0	0
$373$	−2307.33	−0.320292	−0.160146	−	0.987093i	$-0.551196\pi$
$373$	−2307.33	−0.320292	−0.160146	+	0.987093i	$0.551196\pi$
$374$	0	0
$375$	−4628.42	−0.637361
$376$	0	0
$377$	−10834.3	−1.48009
$378$	0	0
$379$	13893.6	1.88302	0.941512	−	0.336981i	$-0.109406\pi$
$379$	13893.6	1.88302	0.941512	+	0.336981i	$0.109406\pi$
$380$	0	0
$381$	−9028.33	−1.21400
$382$	0	0
$383$	−2166.16	−0.288997	−0.144498	−	0.989505i	$-0.546157\pi$
$383$	−2166.16	−0.288997	−0.144498	+	0.989505i	$0.546157\pi$
$384$	0	0
$385$	673.392	0.0891409
$386$	0	0
$387$	−2679.01	−0.351891
$388$	0	0
$389$	7628.37	0.994277	0.497139	−	0.867671i	$-0.334384\pi$
$389$	7628.37	0.994277	0.497139	+	0.867671i	$0.334384\pi$
$390$	0	0
$391$	−3430.97	−0.443764
$392$	0	0
$393$	−9598.41	−1.23200
$394$	0	0
$395$	−2116.23	−0.269567
$396$	0	0
$397$	−4881.93	−0.617171	−0.308586	−	0.951197i	$-0.599856\pi$
$397$	−4881.93	−0.617171	−0.308586	+	0.951197i	$0.599856\pi$
$398$	0	0
$399$	−1675.14	−0.210180
$400$	0	0
$401$	−1947.16	−0.242485	−0.121242	−	0.992623i	$-0.538688\pi$
$401$	−1947.16	−0.242485	−0.121242	+	0.992623i	$0.538688\pi$
$402$	0	0
$403$	−4952.51	−0.612164
$404$	0	0
$405$	2887.55	0.354281
$406$	0	0
$407$	2143.63	0.261071
$408$	0	0
$409$	−12932.4	−1.56348	−0.781741	−	0.623603i	$-0.785668\pi$
$409$	−12932.4	−1.56348	−0.781741	+	0.623603i	$0.785668\pi$
$410$	0	0
$411$	2012.89	0.241578
$412$	0	0
$413$	3978.76	0.474048
$414$	0	0
$415$	2870.21	0.339502
$416$	0	0
$417$	−6317.17	−0.741854
$418$	0	0
$419$	3744.77	0.436620	0.218310	−	0.975879i	$-0.429946\pi$
$419$	3744.77	0.436620	0.218310	+	0.975879i	$0.429946\pi$
$420$	0	0
$421$	−8169.66	−0.945759	−0.472880	−	0.881127i	$-0.656785\pi$
$421$	−8169.66	−0.945759	−0.472880	+	0.881127i	$0.656785\pi$
$422$	0	0
$423$	741.124	0.0851884
$424$	0	0
$425$	−1932.53	−0.220568
$426$	0	0
$427$	−7134.95	−0.808629
$428$	0	0
$429$	−7767.37	−0.874155
$430$	0	0
$431$	−13425.8	−1.50046	−0.750229	−	0.661178i	$-0.770057\pi$
$431$	−13425.8	−1.50046	−0.750229	+	0.661178i	$0.770057\pi$
$432$	0	0
$433$	2852.32	0.316567	0.158284	−	0.987394i	$-0.449404\pi$
$433$	2852.32	0.316567	0.158284	+	0.987394i	$0.449404\pi$
$434$	0	0
$435$	−3064.36	−0.337759
$436$	0	0
$437$	5762.05	0.630746
$438$	0	0
$439$	3601.91	0.391594	0.195797	−	0.980644i	$-0.437271\pi$
$439$	3601.91	0.391594	0.195797	+	0.980644i	$0.437271\pi$
$440$	0	0
$441$	−1487.42	−0.160611
$442$	0	0
$443$	16280.9	1.74611	0.873057	−	0.487619i	$-0.162134\pi$
$443$	16280.9	1.74611	0.873057	+	0.487619i	$0.162134\pi$
$444$	0	0
$445$	−1779.99	−0.189617
$446$	0	0
$447$	1767.36	0.187009
$448$	0	0
$449$	−14667.3	−1.54163	−0.770815	−	0.637059i	$-0.780151\pi$
$449$	−14667.3	−1.54163	−0.770815	+	0.637059i	$0.780151\pi$
$450$	0	0
$451$	251.738	0.0262836
$452$	0	0
$453$	8501.88	0.881795
$454$	0	0
$455$	2348.63	0.241990
$456$	0	0
$457$	−8863.01	−0.907208	−0.453604	−	0.891203i	$-0.649862\pi$
$457$	−8863.01	−0.907208	−0.453604	+	0.891203i	$0.649862\pi$
$458$	0	0
$459$	2036.46	0.207089
$460$	0	0
$461$	15528.5	1.56884	0.784421	−	0.620229i	$-0.212960\pi$
$461$	15528.5	1.56884	0.784421	+	0.620229i	$0.212960\pi$
$462$	0	0
$463$	17523.1	1.75889	0.879444	−	0.476002i	$-0.157914\pi$
$463$	17523.1	1.75889	0.879444	+	0.476002i	$0.157914\pi$
$464$	0	0
$465$	−1400.77	−0.139697
$466$	0	0
$467$	33.9325	0.00336233	0.00168117	−	0.999999i	$-0.499465\pi$
$467$	33.9325	0.00336233	0.00168117	+	0.999999i	$0.499465\pi$
$468$	0	0
$469$	−1153.54	−0.113573
$470$	0	0
$471$	2489.99	0.243594
$472$	0	0
$473$	−8474.43	−0.823795
$474$	0	0
$475$	3245.53	0.313506
$476$	0	0
$477$	2119.28	0.203428
$478$	0	0
$479$	15082.3	1.43868	0.719340	−	0.694658i	$-0.244444\pi$
$479$	15082.3	1.43868	0.719340	+	0.694658i	$0.244444\pi$
$480$	0	0
$481$	7476.49	0.708729
$482$	0	0
$483$	−11841.6	−1.11555
$484$	0	0
$485$	−4161.17	−0.389585
$486$	0	0
$487$	14901.9	1.38659	0.693294	−	0.720655i	$-0.256159\pi$
$487$	14901.9	1.38659	0.693294	+	0.720655i	$0.256159\pi$
$488$	0	0
$489$	3155.17	0.291783
$490$	0	0
$491$	1355.85	0.124620	0.0623102	−	0.998057i	$-0.480153\pi$
$491$	1355.85	0.124620	0.0623102	+	0.998057i	$0.480153\pi$
$492$	0	0
$493$	−2686.39	−0.245414
$494$	0	0
$495$	−411.041	−0.0373231
$496$	0	0
$497$	−2784.15	−0.251280
$498$	0	0
$499$	10242.2	0.918850	0.459425	−	0.888217i	$-0.348056\pi$
$499$	10242.2	0.918850	0.459425	+	0.888217i	$0.348056\pi$
$500$	0	0
$501$	−12964.0	−1.15607
$502$	0	0
$503$	8547.13	0.757650	0.378825	−	0.925468i	$-0.376328\pi$
$503$	8547.13	0.757650	0.378825	+	0.925468i	$0.376328\pi$
$504$	0	0
$505$	−3283.10	−0.289299
$506$	0	0
$507$	−14429.0	−1.26394
$508$	0	0
$509$	−677.699	−0.0590147	−0.0295074	−	0.999565i	$-0.509394\pi$
$509$	−677.699	−0.0590147	−0.0295074	+	0.999565i	$0.509394\pi$
$510$	0	0
$511$	−2339.66	−0.202545
$512$	0	0
$513$	−3420.07	−0.294346
$514$	0	0
$515$	5174.79	0.442774
$516$	0	0
$517$	2344.37	0.199430
$518$	0	0
$519$	−4795.95	−0.405624
$520$	0	0
$521$	−20159.6	−1.69522	−0.847610	−	0.530619i	$-0.821959\pi$
$521$	−20159.6	−1.69522	−0.847610	+	0.530619i	$0.821959\pi$
$522$	0	0
$523$	8189.54	0.684710	0.342355	−	0.939571i	$-0.388775\pi$
$523$	8189.54	0.684710	0.342355	+	0.939571i	$0.388775\pi$
$524$	0	0
$525$	−6669.90	−0.554473
$526$	0	0
$527$	−1227.99	−0.101503
$528$	0	0
$529$	28565.1	2.34775
$530$	0	0
$531$	−2428.65	−0.198483
$532$	0	0
$533$	878.003	0.0713519
$534$	0	0
$535$	2647.47	0.213944
$536$	0	0
$537$	−11916.3	−0.957591
$538$	0	0
$539$	−4705.12	−0.376000
$540$	0	0
$541$	4321.54	0.343434	0.171717	−	0.985146i	$-0.445069\pi$
$541$	4321.54	0.343434	0.171717	+	0.985146i	$0.445069\pi$
$542$	0	0
$543$	−10938.3	−0.864470
$544$	0	0
$545$	−7260.27	−0.570634
$546$	0	0
$547$	8390.66	0.655866	0.327933	−	0.944701i	$-0.393648\pi$
$547$	8390.66	0.655866	0.327933	+	0.944701i	$0.393648\pi$
$548$	0	0
$549$	4355.20	0.338571
$550$	0	0
$551$	4511.58	0.348820
$552$	0	0
$553$	−6403.02	−0.492376
$554$	0	0
$555$	2114.65	0.161733
$556$	0	0
$557$	6763.71	0.514520	0.257260	−	0.966342i	$-0.417180\pi$
$557$	6763.71	0.514520	0.257260	+	0.966342i	$0.417180\pi$
$558$	0	0
$559$	−29556.8	−2.23635
$560$	0	0
$561$	−1925.95	−0.144944
$562$	0	0
$563$	−13019.6	−0.974617	−0.487308	−	0.873230i	$-0.662021\pi$
$563$	−13019.6	−0.974617	−0.487308	+	0.873230i	$0.662021\pi$
$564$	0	0
$565$	5095.82	0.379438
$566$	0	0
$567$	8736.79	0.647109
$568$	0	0
$569$	−24033.3	−1.77070	−0.885348	−	0.464928i	$-0.846080\pi$
$569$	−24033.3	−1.77070	−0.885348	+	0.464928i	$0.846080\pi$
$570$	0	0
$571$	−15702.5	−1.15084	−0.575420	−	0.817858i	$-0.695162\pi$
$571$	−15702.5	−1.15084	−0.575420	+	0.817858i	$0.695162\pi$
$572$	0	0
$573$	−26414.0	−1.92576
$574$	0	0
$575$	22942.8	1.66396
$576$	0	0
$577$	7868.98	0.567747	0.283873	−	0.958862i	$-0.408380\pi$
$577$	7868.98	0.567747	0.283873	+	0.958862i	$0.408380\pi$
$578$	0	0
$579$	−19971.0	−1.43345
$580$	0	0
$581$	8684.32	0.620114
$582$	0	0
$583$	6703.86	0.476236
$584$	0	0
$585$	−1433.61	−0.101321
$586$	0	0
$587$	22627.2	1.59101	0.795507	−	0.605944i	$-0.207204\pi$
$587$	22627.2	1.59101	0.795507	+	0.605944i	$0.207204\pi$
$588$	0	0
$589$	2062.31	0.144272
$590$	0	0
$591$	−7110.76	−0.494920
$592$	0	0
$593$	−26200.2	−1.81436	−0.907179	−	0.420744i	$-0.861769\pi$
$593$	−26200.2	−1.81436	−0.907179	+	0.420744i	$0.861769\pi$
$594$	0	0
$595$	582.351	0.0401245
$596$	0	0
$597$	−1547.47	−0.106087
$598$	0	0
$599$	19161.1	1.30702	0.653508	−	0.756920i	$-0.273297\pi$
$599$	19161.1	1.30702	0.653508	+	0.756920i	$0.273297\pi$
$600$	0	0
$601$	22757.5	1.54459	0.772293	−	0.635266i	$-0.219110\pi$
$601$	22757.5	1.54459	0.772293	+	0.635266i	$0.219110\pi$
$602$	0	0
$603$	704.127	0.0475527
$604$	0	0
$605$	3178.29	0.213580
$606$	0	0
$607$	−5303.32	−0.354621	−0.177311	−	0.984155i	$-0.556740\pi$
$607$	−5303.32	−0.354621	−0.177311	+	0.984155i	$0.556740\pi$
$608$	0	0
$609$	−9271.77	−0.616931
$610$	0	0
$611$	8176.63	0.541393
$612$	0	0
$613$	14456.9	0.952542	0.476271	−	0.879299i	$-0.341988\pi$
$613$	14456.9	0.952542	0.476271	+	0.879299i	$0.341988\pi$
$614$	0	0
$615$	248.334	0.0162826
$616$	0	0
$617$	−6798.33	−0.443583	−0.221791	−	0.975094i	$-0.571190\pi$
$617$	−6798.33	−0.443583	−0.221791	+	0.975094i	$0.571190\pi$
$618$	0	0
$619$	23246.6	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$619$	23246.6	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$620$	0	0
$621$	−24176.6	−1.56227
$622$	0	0
$623$	−5385.68	−0.346344
$624$	0	0
$625$	11507.5	0.736481
$626$	0	0
$627$	3234.47	0.206017
$628$	0	0
$629$	1853.82	0.117515
$630$	0	0
$631$	9913.80	0.625455	0.312728	−	0.949843i	$-0.398757\pi$
$631$	9913.80	0.625455	0.312728	+	0.949843i	$0.398757\pi$
$632$	0	0
$633$	25042.0	1.57240
$634$	0	0
$635$	−5271.10	−0.329413
$636$	0	0
$637$	−16410.3	−1.02072
$638$	0	0
$639$	1699.45	0.105210
$640$	0	0
$641$	5770.50	0.355571	0.177786	−	0.984069i	$-0.443107\pi$
$641$	5770.50	0.355571	0.177786	+	0.984069i	$0.443107\pi$
$642$	0	0
$643$	5779.58	0.354471	0.177235	−	0.984169i	$-0.443285\pi$
$643$	5779.58	0.354471	0.177235	+	0.984169i	$0.443285\pi$
$644$	0	0
$645$	−8359.85	−0.510339
$646$	0	0
$647$	−28542.8	−1.73437	−0.867183	−	0.497990i	$-0.834072\pi$
$647$	−28542.8	−1.73437	−0.867183	+	0.497990i	$0.834072\pi$
$648$	0	0
$649$	−7682.46	−0.464658
$650$	0	0
$651$	−4238.26	−0.255162
$652$	0	0
$653$	−24351.1	−1.45931	−0.729657	−	0.683814i	$-0.760320\pi$
$653$	−24351.1	−1.45931	−0.729657	+	0.683814i	$0.760320\pi$
$654$	0	0
$655$	−5603.94	−0.334296
$656$	0	0
$657$	1428.14	0.0848050
$658$	0	0
$659$	−21188.5	−1.25248	−0.626242	−	0.779628i	$-0.715408\pi$
$659$	−21188.5	−1.25248	−0.626242	+	0.779628i	$0.715408\pi$
$660$	0	0
$661$	26464.3	1.55725	0.778625	−	0.627489i	$-0.215917\pi$
$661$	26464.3	1.55725	0.778625	+	0.627489i	$0.215917\pi$
$662$	0	0
$663$	−6717.24	−0.393478
$664$	0	0
$665$	−978.013	−0.0570311
$666$	0	0
$667$	31892.5	1.85140
$668$	0	0
$669$	−18635.5	−1.07697
$670$	0	0
$671$	13776.7	0.792612
$672$	0	0
$673$	11012.2	0.630744	0.315372	−	0.948968i	$-0.397871\pi$
$673$	11012.2	0.630744	0.315372	+	0.948968i	$0.397871\pi$
$674$	0	0
$675$	−13617.7	−0.776512
$676$	0	0
$677$	17672.5	1.00326	0.501632	−	0.865081i	$-0.332733\pi$
$677$	17672.5	1.00326	0.501632	+	0.865081i	$0.332733\pi$
$678$	0	0
$679$	−12590.3	−0.711594
$680$	0	0
$681$	−29710.4	−1.67181
$682$	0	0
$683$	27300.2	1.52945	0.764724	−	0.644357i	$-0.222875\pi$
$683$	27300.2	1.52945	0.764724	+	0.644357i	$0.222875\pi$
$684$	0	0
$685$	1175.21	0.0655509
$686$	0	0
$687$	1348.59	0.0748934
$688$	0	0
$689$	23381.5	1.29284
$690$	0	0
$691$	−10434.7	−0.574466	−0.287233	−	0.957861i	$-0.592735\pi$
$691$	−10434.7	−0.574466	−0.287233	+	0.957861i	$0.592735\pi$
$692$	0	0
$693$	−1243.68	−0.0681722
$694$	0	0
$695$	−3688.22	−0.201298
$696$	0	0
$697$	217.704	0.0118309
$698$	0	0
$699$	−9357.32	−0.506333
$700$	0	0
$701$	−22633.5	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$701$	−22633.5	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$702$	0	0
$703$	−3113.34	−0.167030
$704$	0	0
$705$	2312.68	0.123547
$706$	0	0
$707$	−9933.57	−0.528416
$708$	0	0
$709$	−16677.8	−0.883422	−0.441711	−	0.897157i	$-0.645628\pi$
$709$	−16677.8	−0.883422	−0.441711	+	0.897157i	$0.645628\pi$
$710$	0	0
$711$	3908.43	0.206157
$712$	0	0
$713$	14578.5	0.765737
$714$	0	0
$715$	−4534.91	−0.237197
$716$	0	0
$717$	17096.4	0.890485
$718$	0	0
$719$	−22171.6	−1.15002	−0.575008	−	0.818147i	$-0.695001\pi$
$719$	−22171.6	−1.15002	−0.575008	+	0.818147i	$0.695001\pi$
$720$	0	0
$721$	15657.2	0.808745
$722$	0	0
$723$	−17738.7	−0.912462
$724$	0	0
$725$	17963.8	0.920218
$726$	0	0
$727$	−33187.7	−1.69307	−0.846535	−	0.532333i	$-0.821316\pi$
$727$	−33187.7	−1.69307	−0.846535	+	0.532333i	$0.821316\pi$
$728$	0	0
$729$	13307.3	0.676083
$730$	0	0
$731$	−7328.71	−0.370810
$732$	0	0
$733$	−1323.59	−0.0666955	−0.0333478	−	0.999444i	$-0.510617\pi$
$733$	−1323.59	−0.0666955	−0.0333478	+	0.999444i	$0.510617\pi$
$734$	0	0
$735$	−4641.50	−0.232931
$736$	0	0
$737$	2227.34	0.111323
$738$	0	0
$739$	7455.85	0.371134	0.185567	−	0.982632i	$-0.440588\pi$
$739$	7455.85	0.371134	0.185567	+	0.982632i	$0.440588\pi$
$740$	0	0
$741$	11281.1	0.559272
$742$	0	0
$743$	−20516.3	−1.01302	−0.506508	−	0.862235i	$-0.669064\pi$
$743$	−20516.3	−1.01302	−0.506508	+	0.862235i	$0.669064\pi$
$744$	0	0
$745$	1031.85	0.0507439
$746$	0	0
$747$	−5300.94	−0.259640
$748$	0	0
$749$	8010.38	0.390778
$750$	0	0
$751$	6375.86	0.309798	0.154899	−	0.987930i	$-0.450495\pi$
$751$	6375.86	0.309798	0.154899	+	0.987930i	$0.450495\pi$
$752$	0	0
$753$	11689.8	0.565736
$754$	0	0
$755$	4963.74	0.239270
$756$	0	0
$757$	15936.4	0.765151	0.382576	−	0.923924i	$-0.375037\pi$
$757$	15936.4	0.765151	0.382576	+	0.923924i	$0.375037\pi$
$758$	0	0
$759$	22864.6	1.09345
$760$	0	0
$761$	3719.92	0.177197	0.0885987	−	0.996067i	$-0.471761\pi$
$761$	3719.92	0.177197	0.0885987	+	0.996067i	$0.471761\pi$
$762$	0	0
$763$	−21967.2	−1.04229
$764$	0	0
$765$	−355.469	−0.0168000
$766$	0	0
$767$	−26794.6	−1.26141
$768$	0	0
$769$	8762.78	0.410916	0.205458	−	0.978666i	$-0.434132\pi$
$769$	8762.78	0.410916	0.205458	+	0.978666i	$0.434132\pi$
$770$	0	0
$771$	82.7271	0.00386426
$772$	0	0
$773$	−20385.4	−0.948529	−0.474265	−	0.880382i	$-0.657286\pi$
$773$	−20385.4	−0.948529	−0.474265	+	0.880382i	$0.657286\pi$
$774$	0	0
$775$	8211.51	0.380602
$776$	0	0
$777$	6398.24	0.295413
$778$	0	0
$779$	−365.616	−0.0168159
$780$	0	0
$781$	5375.83	0.246302
$782$	0	0
$783$	−18929.8	−0.863981
$784$	0	0
$785$	1453.76	0.0660979
$786$	0	0
$787$	27141.4	1.22934	0.614668	−	0.788786i	$-0.289290\pi$
$787$	27141.4	1.22934	0.614668	+	0.788786i	$0.289290\pi$
$788$	0	0
$789$	−21151.8	−0.954401
$790$	0	0
$791$	15418.3	0.693061
$792$	0	0
$793$	48049.8	2.15170
$794$	0	0
$795$	6613.22	0.295027
$796$	0	0
$797$	−11346.2	−0.504272	−0.252136	−	0.967692i	$-0.581133\pi$
$797$	−11346.2	−0.504272	−0.252136	+	0.967692i	$0.581133\pi$
$798$	0	0
$799$	2027.42	0.0897684
$800$	0	0
$801$	3287.44	0.145014
$802$	0	0
$803$	4517.58	0.198533
$804$	0	0
$805$	−6913.60	−0.302699
$806$	0	0
$807$	29989.4	1.30815
$808$	0	0
$809$	−14559.5	−0.632737	−0.316369	−	0.948636i	$-0.602464\pi$
$809$	−14559.5	−0.632737	−0.316369	+	0.948636i	$0.602464\pi$
$810$	0	0
$811$	−19107.2	−0.827304	−0.413652	−	0.910435i	$-0.635747\pi$
$811$	−19107.2	−0.827304	−0.413652	+	0.910435i	$0.635747\pi$
$812$	0	0
$813$	−12947.0	−0.558512
$814$	0	0
$815$	1842.12	0.0791737
$816$	0	0
$817$	12308.0	0.527053
$818$	0	0
$819$	−4337.65	−0.185067
$820$	0	0
$821$	21154.4	0.899261	0.449631	−	0.893215i	$-0.351556\pi$
$821$	21154.4	0.899261	0.449631	+	0.893215i	$0.351556\pi$
$822$	0	0
$823$	14242.9	0.603250	0.301625	−	0.953427i	$-0.402471\pi$
$823$	14242.9	0.603250	0.301625	+	0.953427i	$0.402471\pi$
$824$	0	0
$825$	12878.7	0.543490
$826$	0	0
$827$	25221.9	1.06052	0.530260	−	0.847835i	$-0.322094\pi$
$827$	25221.9	1.06052	0.530260	+	0.847835i	$0.322094\pi$
$828$	0	0
$829$	−33301.0	−1.39516	−0.697582	−	0.716505i	$-0.745741\pi$
$829$	−33301.0	−1.39516	−0.697582	+	0.716505i	$0.745741\pi$
$830$	0	0
$831$	1158.80	0.0483734
$832$	0	0
$833$	−4069.00	−0.169247
$834$	0	0
$835$	−7568.92	−0.313692
$836$	0	0
$837$	−8653.11	−0.357342
$838$	0	0
$839$	−20312.8	−0.835846	−0.417923	−	0.908482i	$-0.637242\pi$
$839$	−20312.8	−0.835846	−0.417923	+	0.908482i	$0.637242\pi$
$840$	0	0
$841$	582.301	0.0238756
$842$	0	0
$843$	10220.7	0.417579
$844$	0	0
$845$	−8424.26	−0.342962
$846$	0	0
$847$	9616.47	0.390113
$848$	0	0
$849$	18233.4	0.737065
$850$	0	0
$851$	−22008.3	−0.886528
$852$	0	0
$853$	9537.61	0.382839	0.191419	−	0.981508i	$-0.438691\pi$
$853$	9537.61	0.382839	0.191419	+	0.981508i	$0.438691\pi$
$854$	0	0
$855$	596.983	0.0238788
$856$	0	0
$857$	4483.32	0.178701	0.0893507	−	0.996000i	$-0.471521\pi$
$857$	4483.32	0.178701	0.0893507	+	0.996000i	$0.471521\pi$
$858$	0	0
$859$	−43697.7	−1.73568	−0.867839	−	0.496845i	$-0.834492\pi$
$859$	−43697.7	−1.73568	−0.867839	+	0.496845i	$0.834492\pi$
$860$	0	0
$861$	751.379	0.0297409
$862$	0	0
$863$	42055.1	1.65883	0.829415	−	0.558632i	$-0.188674\pi$
$863$	42055.1	1.65883	0.829415	+	0.558632i	$0.188674\pi$
$864$	0	0
$865$	−2800.07	−0.110064
$866$	0	0
$867$	−1665.56	−0.0652427
$868$	0	0
$869$	12363.4	0.482623
$870$	0	0
$871$	7768.44	0.302209
$872$	0	0
$873$	7685.18	0.297943
$874$	0	0
$875$	−8176.15	−0.315891
$876$	0	0
$877$	15681.8	0.603805	0.301902	−	0.953339i	$-0.402378\pi$
$877$	15681.8	0.603805	0.301902	+	0.953339i	$0.402378\pi$
$878$	0	0
$879$	−25081.4	−0.962428
$880$	0	0
$881$	−12511.3	−0.478453	−0.239227	−	0.970964i	$-0.576894\pi$
$881$	−12511.3	−0.478453	−0.239227	+	0.970964i	$0.576894\pi$
$882$	0	0
$883$	7968.76	0.303703	0.151852	−	0.988403i	$-0.451476\pi$
$883$	7968.76	0.303703	0.151852	+	0.988403i	$0.451476\pi$
$884$	0	0
$885$	−7578.59	−0.287855
$886$	0	0
$887$	35933.1	1.36022	0.680111	−	0.733109i	$-0.261932\pi$
$887$	35933.1	1.36022	0.680111	+	0.733109i	$0.261932\pi$
$888$	0	0
$889$	−15948.6	−0.601688
$890$	0	0
$891$	−16869.6	−0.634291
$892$	0	0
$893$	−3404.89	−0.127593
$894$	0	0
$895$	−6957.21	−0.259837
$896$	0	0
$897$	79746.3	2.96840
$898$	0	0
$899$	11414.8	0.423474
$900$	0	0
$901$	5797.52	0.214365
$902$	0	0
$903$	−25294.2	−0.932156
$904$	0	0
$905$	−6386.22	−0.234569
$906$	0	0
$907$	5476.30	0.200482	0.100241	−	0.994963i	$-0.468039\pi$
$907$	5476.30	0.200482	0.100241	+	0.994963i	$0.468039\pi$
$908$	0	0
$909$	6063.49	0.221247
$910$	0	0
$911$	−33544.2	−1.21995	−0.609973	−	0.792422i	$-0.708820\pi$
$911$	−33544.2	−1.21995	−0.609973	+	0.792422i	$0.708820\pi$
$912$	0	0
$913$	−16768.3	−0.607831
$914$	0	0
$915$	13590.4	0.491021
$916$	0	0
$917$	−16955.7	−0.610606
$918$	0	0
$919$	5353.49	0.192160	0.0960802	−	0.995374i	$-0.469369\pi$
$919$	5353.49	0.192160	0.0960802	+	0.995374i	$0.469369\pi$
$920$	0	0
$921$	−61455.9	−2.19874
$922$	0	0
$923$	18749.6	0.668636
$924$	0	0
$925$	−12396.4	−0.440639
$926$	0	0
$927$	−9557.22	−0.338620
$928$	0	0
$929$	−22930.6	−0.809828	−0.404914	−	0.914355i	$-0.632699\pi$
$929$	−22930.6	−0.809828	−0.404914	+	0.914355i	$0.632699\pi$
$930$	0	0
$931$	6833.56	0.240559
$932$	0	0
$933$	16030.8	0.562513
$934$	0	0
$935$	−1124.44	−0.0393297
$936$	0	0
$937$	39354.7	1.37210	0.686052	−	0.727553i	$-0.259342\pi$
$937$	39354.7	1.37210	0.686052	+	0.727553i	$0.259342\pi$
$938$	0	0
$939$	−37616.2	−1.30730
$940$	0	0
$941$	34211.2	1.18518	0.592589	−	0.805505i	$-0.298106\pi$
$941$	34211.2	1.18518	0.592589	+	0.805505i	$0.298106\pi$
$942$	0	0
$943$	−2584.55	−0.0892519
$944$	0	0
$945$	4103.57	0.141259
$946$	0	0
$947$	−12021.8	−0.412521	−0.206260	−	0.978497i	$-0.566129\pi$
$947$	−12021.8	−0.412521	−0.206260	+	0.978497i	$0.566129\pi$
$948$	0	0
$949$	15756.2	0.538956
$950$	0	0
$951$	42802.4	1.45948
$952$	0	0
$953$	−25741.2	−0.874963	−0.437482	−	0.899227i	$-0.644129\pi$
$953$	−25741.2	−0.874963	−0.437482	+	0.899227i	$0.644129\pi$
$954$	0	0
$955$	−15421.6	−0.522544
$956$	0	0
$957$	17902.6	0.604711
$958$	0	0
$959$	3555.79	0.119731
$960$	0	0
$961$	−24573.1	−0.824851
$962$	0	0
$963$	−4889.57	−0.163618
$964$	0	0
$965$	−11659.9	−0.388958
$966$	0	0
$967$	11475.4	0.381619	0.190809	−	0.981627i	$-0.438889\pi$
$967$	11475.4	0.381619	0.190809	+	0.981627i	$0.438889\pi$
$968$	0	0
$969$	2797.18	0.0927331
$970$	0	0
$971$	−40315.6	−1.33243	−0.666215	−	0.745760i	$-0.732087\pi$
$971$	−40315.6	−1.33243	−0.666215	+	0.745760i	$0.732087\pi$
$972$	0	0
$973$	−11159.3	−0.367679
$974$	0	0
$975$	44917.9	1.47541
$976$	0	0
$977$	58589.5	1.91857	0.959286	−	0.282435i	$-0.0911422\pi$
$977$	58589.5	1.91857	0.959286	+	0.282435i	$0.0911422\pi$
$978$	0	0
$979$	10399.0	0.339484
$980$	0	0
$981$	13408.9	0.436404
$982$	0	0
$983$	−39871.4	−1.29369	−0.646846	−	0.762621i	$-0.723912\pi$
$983$	−39871.4	−1.29369	−0.646846	+	0.762621i	$0.723912\pi$
$984$	0	0
$985$	−4151.55	−0.134294
$986$	0	0
$987$	6997.40	0.225663
$988$	0	0
$989$	87005.5	2.79739
$990$	0	0
$991$	50100.6	1.60595	0.802976	−	0.596012i	$-0.203249\pi$
$991$	50100.6	1.60595	0.802976	+	0.596012i	$0.203249\pi$
$992$	0	0
$993$	53013.7	1.69420
$994$	0	0
$995$	−903.475	−0.0287860
$996$	0	0
$997$	55441.5	1.76113	0.880566	−	0.473924i	$-0.157163\pi$
$997$	55441.5	1.76113	0.880566	+	0.473924i	$0.157163\pi$
$998$	0	0
$999$	13063.1	0.413711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.r.1.2		3
4.3	odd	2		1088.4.a.ba.1.2		3
8.3	odd	2		544.4.a.f.1.2	✓	3
8.5	even	2		544.4.a.g.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.f.1.2	✓	3	8.3	odd	2
544.4.a.g.1.2	yes	3	8.5	even	2
1088.4.a.r.1.2		3	1.1	even	1	trivial
1088.4.a.ba.1.2		3	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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-5.76319 q^{3} -3.36478 q^{5} -10.1807 q^{7} +6.21436 q^{9} +O(q^{10})\)	\(q-5.76319 q^{3} -3.36478 q^{5} -10.1807 q^{7} +6.21436 q^{9} +19.6577 q^{11} +68.5613 q^{13} +19.3919 q^{15} +17.0000 q^{17} -28.5502 q^{19} +58.6735 q^{21} -201.822 q^{23} -113.678 q^{25} +119.792 q^{27} -158.023 q^{29} -72.2347 q^{31} -113.291 q^{33} +34.2559 q^{35} +109.048 q^{37} -395.132 q^{39} +12.8061 q^{41} -431.101 q^{43} -20.9100 q^{45} +119.260 q^{47} -239.353 q^{49} -97.9742 q^{51} +341.030 q^{53} -66.1438 q^{55} +164.540 q^{57} -390.813 q^{59} +700.829 q^{61} -63.2667 q^{63} -230.694 q^{65} +113.307 q^{67} +1163.14 q^{69} +273.472 q^{71} +229.812 q^{73} +655.149 q^{75} -200.129 q^{77} +628.935 q^{79} -858.169 q^{81} -853.016 q^{83} -57.2013 q^{85} +910.717 q^{87} +529.007 q^{89} -698.004 q^{91} +416.302 q^{93} +96.0651 q^{95} +1236.68 q^{97} +122.160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9	\( 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9} + 82 q^{11} + 6 q^{13} - 160 q^{15} + 51 q^{17} + 172 q^{19} + 72 q^{21} - 158 q^{23} + 69 q^{25} - 80 q^{27} + 330 q^{29} - 34 q^{31} - 84 q^{33} + 40 q^{35} + 378 q^{37} - 416 q^{39} - 282 q^{41} - 536 q^{43} + 954 q^{45} - 564 q^{47} - 321 q^{49} - 238 q^{51} + 894 q^{53} - 40 q^{55} - 984 q^{57} - 1216 q^{59} + 978 q^{61} - 742 q^{63} - 276 q^{65} + 272 q^{67} + 816 q^{69} - 158 q^{71} - 498 q^{73} - 2050 q^{75} + 1344 q^{77} + 1410 q^{79} + 111 q^{81} + 1624 q^{83} + 306 q^{85} - 1240 q^{87} + 42 q^{89} - 2224 q^{91} + 2088 q^{93} + 2960 q^{95} + 822 q^{97} - 1586 q^{99}+O(q^{100}) \) 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9 + 82 * q^11 + 6 * q^13 - 160 * q^15 + 51 * q^17 + 172 * q^19 + 72 * q^21 - 158 * q^23 + 69 * q^25 - 80 * q^27 + 330 * q^29 - 34 * q^31 - 84 * q^33 + 40 * q^35 + 378 * q^37 - 416 * q^39 - 282 * q^41 - 536 * q^43 + 954 * q^45 - 564 * q^47 - 321 * q^49 - 238 * q^51 + 894 * q^53 - 40 * q^55 - 984 * q^57 - 1216 * q^59 + 978 * q^61 - 742 * q^63 - 276 * q^65 + 272 * q^67 + 816 * q^69 - 158 * q^71 - 498 * q^73 - 2050 * q^75 + 1344 * q^77 + 1410 * q^79 + 111 * q^81 + 1624 * q^83 + 306 * q^85 - 1240 * q^87 + 42 * q^89 - 2224 * q^91 + 2088 * q^93 + 2960 * q^95 + 822 * q^97 - 1586 * q^99

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