LMFDB - Embedded newform 160.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(1,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, 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("160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.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:	$9.44030560092$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.89898 q^{3} +5.00000 q^{5} +20.4949 q^{7} +52.1918 q^{9} -61.3939 q^{11} +45.1918 q^{13} -44.4949 q^{15} -115.576 q^{17} -64.8082 q^{19} -182.384 q^{21} -6.11123 q^{23} +25.0000 q^{25} -224.182 q^{27} -224.384 q^{29} -58.6061 q^{31} +546.343 q^{33} +102.474 q^{35} -99.6163 q^{37} -402.161 q^{39} -145.959 q^{41} +6.73776 q^{43} +260.959 q^{45} +203.687 q^{47} +77.0408 q^{49} +1028.50 q^{51} +275.576 q^{53} -306.969 q^{55} +576.727 q^{57} -262.020 q^{59} -790.767 q^{61} +1069.67 q^{63} +225.959 q^{65} +141.748 q^{67} +54.3837 q^{69} -1043.98 q^{71} -1057.88 q^{73} -222.474 q^{75} -1258.26 q^{77} +826.624 q^{79} +585.808 q^{81} +1026.25 q^{83} -577.878 q^{85} +1996.79 q^{87} -154.849 q^{89} +926.202 q^{91} +521.535 q^{93} -324.041 q^{95} +414.041 q^{97} -3204.26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} + 10 q^{5} - 8 q^{7} + 26 q^{9} - 64 q^{11} + 12 q^{13} - 40 q^{15} + 4 q^{17} - 208 q^{19} - 208 q^{21} - 120 q^{23} + 50 q^{25} - 272 q^{27} - 292 q^{29} - 176 q^{31} + 544 q^{33} - 40 q^{35}+ \cdots - 3136 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 + 10 * q^5 - 8 * q^7 + 26 * q^9 - 64 * q^11 + 12 * q^13 - 40 * q^15 + 4 * q^17 - 208 * q^19 - 208 * q^21 - 120 * q^23 + 50 * q^25 - 272 * q^27 - 292 * q^29 - 176 * q^31 + 544 * q^33 - 40 * q^35 - 356 * q^37 - 432 * q^39 + 100 * q^41 + 376 * q^43 + 130 * q^45 + 280 * q^47 + 546 * q^49 + 1136 * q^51 + 316 * q^53 - 320 * q^55 + 448 * q^57 - 720 * q^59 - 1268 * q^61 + 1816 * q^63 + 60 * q^65 + 744 * q^67 - 48 * q^69 + 48 * q^71 - 940 * q^73 - 200 * q^75 - 1184 * q^77 - 32 * q^79 + 1250 * q^81 + 1592 * q^83 + 20 * q^85 + 1936 * q^87 - 780 * q^89 + 1872 * q^91 + 416 * q^93 - 1040 * q^95 + 1220 * q^97 - 3136 * 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
$3$	−8.89898	−1.71261	−0.856305	−	0.516471i	$-0.827245\pi$
$3$	−8.89898	−1.71261	−0.856305	+	0.516471i	$0.827245\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	20.4949	1.10662	0.553310	−	0.832975i	$-0.313364\pi$
$7$	20.4949	1.10662	0.553310	+	0.832975i	$0.313364\pi$
$8$	0	0
$9$	52.1918	1.93303
$10$	0	0
$11$	−61.3939	−1.68281	−0.841407	−	0.540402i	$-0.818272\pi$
$11$	−61.3939	−1.68281	−0.841407	+	0.540402i	$0.818272\pi$
$12$	0	0
$13$	45.1918	0.964151	0.482075	−	0.876130i	$-0.339883\pi$
$13$	45.1918	0.964151	0.482075	+	0.876130i	$0.339883\pi$
$14$	0	0
$15$	−44.4949	−0.765902
$16$	0	0
$17$	−115.576	−1.64889	−0.824446	−	0.565940i	$-0.808513\pi$
$17$	−115.576	−1.64889	−0.824446	+	0.565940i	$0.808513\pi$
$18$	0	0
$19$	−64.8082	−0.782527	−0.391263	−	0.920279i	$-0.627962\pi$
$19$	−64.8082	−0.782527	−0.391263	+	0.920279i	$0.627962\pi$
$20$	0	0
$21$	−182.384	−1.89521
$22$	0	0
$23$	−6.11123	−0.0554034	−0.0277017	−	0.999616i	$-0.508819\pi$
$23$	−6.11123	−0.0554034	−0.0277017	+	0.999616i	$0.508819\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−224.182	−1.59792
$28$	0	0
$29$	−224.384	−1.43679	−0.718397	−	0.695634i	$-0.755124\pi$
$29$	−224.384	−1.43679	−0.718397	+	0.695634i	$0.755124\pi$
$30$	0	0
$31$	−58.6061	−0.339547	−0.169774	−	0.985483i	$-0.554304\pi$
$31$	−58.6061	−0.339547	−0.169774	+	0.985483i	$0.554304\pi$
$32$	0	0
$33$	546.343	2.88200
$34$	0	0
$35$	102.474	0.494896
$36$	0	0
$37$	−99.6163	−0.442617	−0.221308	−	0.975204i	$-0.571033\pi$
$37$	−99.6163	−0.442617	−0.221308	+	0.975204i	$0.571033\pi$
$38$	0	0
$39$	−402.161	−1.65121
$40$	0	0
$41$	−145.959	−0.555975	−0.277988	−	0.960585i	$-0.589667\pi$
$41$	−145.959	−0.555975	−0.277988	+	0.960585i	$0.589667\pi$
$42$	0	0
$43$	6.73776	0.0238953	0.0119477	−	0.999929i	$-0.496197\pi$
$43$	6.73776	0.0238953	0.0119477	+	0.999929i	$0.496197\pi$
$44$	0	0
$45$	260.959	0.864478
$46$	0	0
$47$	203.687	0.632144	0.316072	−	0.948735i	$-0.397636\pi$
$47$	203.687	0.632144	0.316072	+	0.948735i	$0.397636\pi$
$48$	0	0
$49$	77.0408	0.224609
$50$	0	0
$51$	1028.50	2.82391
$52$	0	0
$53$	275.576	0.714211	0.357106	−	0.934064i	$-0.383764\pi$
$53$	275.576	0.714211	0.357106	+	0.934064i	$0.383764\pi$
$54$	0	0
$55$	−306.969	−0.752577
$56$	0	0
$57$	576.727	1.34016
$58$	0	0
$59$	−262.020	−0.578172	−0.289086	−	0.957303i	$-0.593351\pi$
$59$	−262.020	−0.578172	−0.289086	+	0.957303i	$0.593351\pi$
$60$	0	0
$61$	−790.767	−1.65979	−0.829897	−	0.557917i	$-0.811601\pi$
$61$	−790.767	−1.65979	−0.829897	+	0.557917i	$0.811601\pi$
$62$	0	0
$63$	1069.67	2.13913
$64$	0	0
$65$	225.959	0.431181
$66$	0	0
$67$	141.748	0.258467	0.129233	−	0.991614i	$-0.458748\pi$
$67$	141.748	0.258467	0.129233	+	0.991614i	$0.458748\pi$
$68$	0	0
$69$	54.3837	0.0948844
$70$	0	0
$71$	−1043.98	−1.74503	−0.872516	−	0.488585i	$-0.837513\pi$
$71$	−1043.98	−1.74503	−0.872516	+	0.488585i	$0.837513\pi$
$72$	0	0
$73$	−1057.88	−1.69610	−0.848049	−	0.529917i	$-0.822223\pi$
$73$	−1057.88	−1.69610	−0.848049	+	0.529917i	$0.822223\pi$
$74$	0	0
$75$	−222.474	−0.342522
$76$	0	0
$77$	−1258.26	−1.86224
$78$	0	0
$79$	826.624	1.17725	0.588624	−	0.808407i	$-0.299670\pi$
$79$	826.624	1.17725	0.588624	+	0.808407i	$0.299670\pi$
$80$	0	0
$81$	585.808	0.803578
$82$	0	0
$83$	1026.25	1.35718	0.678589	−	0.734518i	$-0.262592\pi$
$83$	1026.25	1.35718	0.678589	+	0.734518i	$0.262592\pi$
$84$	0	0
$85$	−577.878	−0.737407
$86$	0	0
$87$	1996.79	2.46067
$88$	0	0
$89$	−154.849	−0.184427	−0.0922133	−	0.995739i	$-0.529394\pi$
$89$	−154.849	−0.184427	−0.0922133	+	0.995739i	$0.529394\pi$
$90$	0	0
$91$	926.202	1.06695
$92$	0	0
$93$	521.535	0.581512
$94$	0	0
$95$	−324.041	−0.349957
$96$	0	0
$97$	414.041	0.433397	0.216698	−	0.976239i	$-0.430471\pi$
$97$	414.041	0.433397	0.216698	+	0.976239i	$0.430471\pi$
$98$	0	0
$99$	−3204.26	−3.25293
$100$	0	0
$101$	1776.99	1.75066	0.875331	−	0.483524i	$-0.160643\pi$
$101$	1776.99	1.75066	0.875331	+	0.483524i	$0.160643\pi$
$102$	0	0
$103$	767.991	0.734683	0.367342	−	0.930086i	$-0.380268\pi$
$103$	767.991	0.734683	0.367342	+	0.930086i	$0.380268\pi$
$104$	0	0
$105$	−911.918	−0.847563
$106$	0	0
$107$	−1649.00	−1.48986	−0.744929	−	0.667144i	$-0.767517\pi$
$107$	−1649.00	−1.48986	−0.744929	+	0.667144i	$0.767517\pi$
$108$	0	0
$109$	884.220	0.777000	0.388500	−	0.921449i	$-0.372993\pi$
$109$	884.220	0.777000	0.388500	+	0.921449i	$0.372993\pi$
$110$	0	0
$111$	886.484	0.758030
$112$	0	0
$113$	−1138.85	−0.948088	−0.474044	−	0.880501i	$-0.657206\pi$
$113$	−1138.85	−0.948088	−0.474044	+	0.880501i	$0.657206\pi$
$114$	0	0
$115$	−30.5561	−0.0247772
$116$	0	0
$117$	2358.64	1.86373
$118$	0	0
$119$	−2368.71	−1.82470
$120$	0	0
$121$	2438.21	1.83186
$122$	0	0
$123$	1298.89	0.952169
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1701.60	−1.18892	−0.594460	−	0.804125i	$-0.702634\pi$
$127$	−1701.60	−1.18892	−0.594460	+	0.804125i	$0.702634\pi$
$128$	0	0
$129$	−59.9592	−0.0409233
$130$	0	0
$131$	1540.99	1.02776	0.513880	−	0.857862i	$-0.328208\pi$
$131$	1540.99	1.02776	0.513880	+	0.857862i	$0.328208\pi$
$132$	0	0
$133$	−1328.24	−0.865960
$134$	0	0
$135$	−1120.91	−0.714610
$136$	0	0
$137$	−1391.53	−0.867787	−0.433894	−	0.900964i	$-0.642861\pi$
$137$	−1391.53	−0.867787	−0.433894	+	0.900964i	$0.642861\pi$
$138$	0	0
$139$	−156.649	−0.0955885	−0.0477942	−	0.998857i	$-0.515219\pi$
$139$	−156.649	−0.0955885	−0.0477942	+	0.998857i	$0.515219\pi$
$140$	0	0
$141$	−1812.60	−1.08262
$142$	0	0
$143$	−2774.50	−1.62249
$144$	0	0
$145$	−1121.92	−0.642553
$146$	0	0
$147$	−685.585	−0.384667
$148$	0	0
$149$	214.245	0.117796	0.0588981	−	0.998264i	$-0.481241\pi$
$149$	214.245	0.117796	0.0588981	+	0.998264i	$0.481241\pi$
$150$	0	0
$151$	35.0510	0.0188901	0.00944507	−	0.999955i	$-0.496993\pi$
$151$	35.0510	0.0188901	0.00944507	+	0.999955i	$0.496993\pi$
$152$	0	0
$153$	−6032.10	−3.18736
$154$	0	0
$155$	−293.031	−0.151850
$156$	0	0
$157$	349.592	0.177710	0.0888550	−	0.996045i	$-0.471679\pi$
$157$	349.592	0.177710	0.0888550	+	0.996045i	$0.471679\pi$
$158$	0	0
$159$	−2452.34	−1.22317
$160$	0	0
$161$	−125.249	−0.0613106
$162$	0	0
$163$	2029.04	0.975010	0.487505	−	0.873120i	$-0.337907\pi$
$163$	2029.04	0.975010	0.487505	+	0.873120i	$0.337907\pi$
$164$	0	0
$165$	2731.71	1.28887
$166$	0	0
$167$	−214.619	−0.0994476	−0.0497238	−	0.998763i	$-0.515834\pi$
$167$	−214.619	−0.0994476	−0.0497238	+	0.998763i	$0.515834\pi$
$168$	0	0
$169$	−154.698	−0.0704133
$170$	0	0
$171$	−3382.46	−1.51265
$172$	0	0
$173$	3424.99	1.50518	0.752592	−	0.658487i	$-0.228803\pi$
$173$	3424.99	1.50518	0.752592	+	0.658487i	$0.228803\pi$
$174$	0	0
$175$	512.372	0.221324
$176$	0	0
$177$	2331.71	0.990183
$178$	0	0
$179$	−2030.10	−0.847692	−0.423846	−	0.905734i	$-0.639320\pi$
$179$	−2030.10	−0.847692	−0.423846	+	0.905734i	$0.639320\pi$
$180$	0	0
$181$	204.139	0.0838316	0.0419158	−	0.999121i	$-0.486654\pi$
$181$	204.139	0.0838316	0.0419158	+	0.999121i	$0.486654\pi$
$182$	0	0
$183$	7037.02	2.84258
$184$	0	0
$185$	−498.082	−0.197944
$186$	0	0
$187$	7095.63	2.77478
$188$	0	0
$189$	−4594.58	−1.76829
$190$	0	0
$191$	−617.353	−0.233875	−0.116937	−	0.993139i	$-0.537308\pi$
$191$	−617.353	−0.233875	−0.116937	+	0.993139i	$0.537308\pi$
$192$	0	0
$193$	3071.33	1.14549	0.572744	−	0.819734i	$-0.305879\pi$
$193$	3071.33	1.14549	0.572744	+	0.819734i	$0.305879\pi$
$194$	0	0
$195$	−2010.81	−0.738445
$196$	0	0
$197$	−812.098	−0.293703	−0.146852	−	0.989159i	$-0.546914\pi$
$197$	−812.098	−0.293703	−0.146852	+	0.989159i	$0.546914\pi$
$198$	0	0
$199$	−1065.93	−0.379709	−0.189855	−	0.981812i	$-0.560802\pi$
$199$	−1065.93	−0.379709	−0.189855	+	0.981812i	$0.560802\pi$
$200$	0	0
$201$	−1261.41	−0.442653
$202$	0	0
$203$	−4598.72	−1.58998
$204$	0	0
$205$	−729.796	−0.248640
$206$	0	0
$207$	−318.956	−0.107097
$208$	0	0
$209$	3978.82	1.31685
$210$	0	0
$211$	−1222.24	−0.398779	−0.199390	−	0.979920i	$-0.563896\pi$
$211$	−1222.24	−0.398779	−0.199390	+	0.979920i	$0.563896\pi$
$212$	0	0
$213$	9290.33	2.98856
$214$	0	0
$215$	33.6888	0.0106863
$216$	0	0
$217$	−1201.13	−0.375750
$218$	0	0
$219$	9414.03	2.90475
$220$	0	0
$221$	−5223.07	−1.58978
$222$	0	0
$223$	414.374	0.124433	0.0622165	−	0.998063i	$-0.480183\pi$
$223$	414.374	0.124433	0.0622165	+	0.998063i	$0.480183\pi$
$224$	0	0
$225$	1304.80	0.386606
$226$	0	0
$227$	−2293.88	−0.670707	−0.335353	−	0.942092i	$-0.608856\pi$
$227$	−2293.88	−0.670707	−0.335353	+	0.942092i	$0.608856\pi$
$228$	0	0
$229$	3214.72	0.927662	0.463831	−	0.885924i	$-0.346475\pi$
$229$	3214.72	0.927662	0.463831	+	0.885924i	$0.346475\pi$
$230$	0	0
$231$	11197.2	3.18928
$232$	0	0
$233$	3598.60	1.01181	0.505905	−	0.862589i	$-0.331159\pi$
$233$	3598.60	1.01181	0.505905	+	0.862589i	$0.331159\pi$
$234$	0	0
$235$	1018.43	0.282703
$236$	0	0
$237$	−7356.11	−2.01616
$238$	0	0
$239$	2462.58	0.666491	0.333245	−	0.942840i	$-0.391856\pi$
$239$	2462.58	0.666491	0.333245	+	0.942840i	$0.391856\pi$
$240$	0	0
$241$	4153.27	1.11011	0.555054	−	0.831815i	$-0.312698\pi$
$241$	4153.27	1.11011	0.555054	+	0.831815i	$0.312698\pi$
$242$	0	0
$243$	839.809	0.221703
$244$	0	0
$245$	385.204	0.100448
$246$	0	0
$247$	−2928.80	−0.754474
$248$	0	0
$249$	−9132.60	−2.32432
$250$	0	0
$251$	−1335.26	−0.335779	−0.167890	−	0.985806i	$-0.553695\pi$
$251$	−1335.26	−0.335779	−0.167890	+	0.985806i	$0.553695\pi$
$252$	0	0
$253$	375.192	0.0932336
$254$	0	0
$255$	5142.52	1.26289
$256$	0	0
$257$	−530.359	−0.128727	−0.0643636	−	0.997927i	$-0.520502\pi$
$257$	−530.359	−0.128727	−0.0643636	+	0.997927i	$0.520502\pi$
$258$	0	0
$259$	−2041.63	−0.489809
$260$	0	0
$261$	−11711.0	−2.77737
$262$	0	0
$263$	−2234.23	−0.523836	−0.261918	−	0.965090i	$-0.584355\pi$
$263$	−2234.23	−0.523836	−0.261918	+	0.965090i	$0.584355\pi$
$264$	0	0
$265$	1377.88	0.319405
$266$	0	0
$267$	1378.00	0.315851
$268$	0	0
$269$	5096.09	1.15507	0.577535	−	0.816366i	$-0.304015\pi$
$269$	5096.09	1.15507	0.577535	+	0.816366i	$0.304015\pi$
$270$	0	0
$271$	1353.39	0.303367	0.151684	−	0.988429i	$-0.451530\pi$
$271$	1353.39	0.303367	0.151684	+	0.988429i	$0.451530\pi$
$272$	0	0
$273$	−8242.25	−1.82727
$274$	0	0
$275$	−1534.85	−0.336563
$276$	0	0
$277$	−8879.57	−1.92607	−0.963035	−	0.269376i	$-0.913183\pi$
$277$	−8879.57	−1.92607	−0.963035	+	0.269376i	$0.913183\pi$
$278$	0	0
$279$	−3058.76	−0.656356
$280$	0	0
$281$	−1145.55	−0.243195	−0.121598	−	0.992579i	$-0.538802\pi$
$281$	−1145.55	−0.243195	−0.121598	+	0.992579i	$0.538802\pi$
$282$	0	0
$283$	385.399	0.0809526	0.0404763	−	0.999180i	$-0.487112\pi$
$283$	385.399	0.0809526	0.0404763	+	0.999180i	$0.487112\pi$
$284$	0	0
$285$	2883.63	0.599339
$286$	0	0
$287$	−2991.42	−0.615254
$288$	0	0
$289$	8444.70	1.71885
$290$	0	0
$291$	−3684.54	−0.742239
$292$	0	0
$293$	3812.30	0.760126	0.380063	−	0.924961i	$-0.375902\pi$
$293$	3812.30	0.760126	0.380063	+	0.924961i	$0.375902\pi$
$294$	0	0
$295$	−1310.10	−0.258566
$296$	0	0
$297$	13763.4	2.68900
$298$	0	0
$299$	−276.178	−0.0534172
$300$	0	0
$301$	138.090	0.0264430
$302$	0	0
$303$	−15813.4	−2.99820
$304$	0	0
$305$	−3953.84	−0.742282
$306$	0	0
$307$	6897.45	1.28228	0.641138	−	0.767426i	$-0.278463\pi$
$307$	6897.45	1.28228	0.641138	+	0.767426i	$0.278463\pi$
$308$	0	0
$309$	−6834.33	−1.25823
$310$	0	0
$311$	−8125.39	−1.48151	−0.740753	−	0.671778i	$-0.765531\pi$
$311$	−8125.39	−1.48151	−0.740753	+	0.671778i	$0.765531\pi$
$312$	0	0
$313$	2052.66	0.370681	0.185341	−	0.982674i	$-0.440661\pi$
$313$	2052.66	0.370681	0.185341	+	0.982674i	$0.440661\pi$
$314$	0	0
$315$	5348.33	0.956649
$316$	0	0
$317$	4175.12	0.739741	0.369871	−	0.929083i	$-0.379402\pi$
$317$	4175.12	0.739741	0.369871	+	0.929083i	$0.379402\pi$
$318$	0	0
$319$	13775.8	2.41786
$320$	0	0
$321$	14674.4	2.55154
$322$	0	0
$323$	7490.24	1.29030
$324$	0	0
$325$	1129.80	0.192830
$326$	0	0
$327$	−7868.66	−1.33070
$328$	0	0
$329$	4174.54	0.699543
$330$	0	0
$331$	−11738.2	−1.94921	−0.974607	−	0.223923i	$-0.928114\pi$
$331$	−11738.2	−1.94921	−0.974607	+	0.223923i	$0.928114\pi$
$332$	0	0
$333$	−5199.16	−0.855592
$334$	0	0
$335$	708.740	0.115590
$336$	0	0
$337$	−1698.85	−0.274606	−0.137303	−	0.990529i	$-0.543843\pi$
$337$	−1698.85	−0.274606	−0.137303	+	0.990529i	$0.543843\pi$
$338$	0	0
$339$	10134.6	1.62370
$340$	0	0
$341$	3598.06	0.571395
$342$	0	0
$343$	−5450.81	−0.858064
$344$	0	0
$345$	271.918	0.0424336
$346$	0	0
$347$	−9682.53	−1.49794	−0.748970	−	0.662604i	$-0.769451\pi$
$347$	−9682.53	−1.49794	−0.748970	+	0.662604i	$0.769451\pi$
$348$	0	0
$349$	−10755.1	−1.64959	−0.824796	−	0.565430i	$-0.808710\pi$
$349$	−10755.1	−1.64959	−0.824796	+	0.565430i	$0.808710\pi$
$350$	0	0
$351$	−10131.2	−1.54063
$352$	0	0
$353$	−7044.94	−1.06222	−0.531111	−	0.847302i	$-0.678225\pi$
$353$	−7044.94	−1.06222	−0.531111	+	0.847302i	$0.678225\pi$
$354$	0	0
$355$	−5219.89	−0.780402
$356$	0	0
$357$	21079.1	3.12500
$358$	0	0
$359$	−805.576	−0.118431	−0.0592154	−	0.998245i	$-0.518860\pi$
$359$	−805.576	−0.118431	−0.0592154	+	0.998245i	$0.518860\pi$
$360$	0	0
$361$	−2658.90	−0.387652
$362$	0	0
$363$	−21697.6	−3.13726
$364$	0	0
$365$	−5289.39	−0.758518
$366$	0	0
$367$	11157.2	1.58692	0.793460	−	0.608623i	$-0.208278\pi$
$367$	11157.2	1.58692	0.793460	+	0.608623i	$0.208278\pi$
$368$	0	0
$369$	−7617.88	−1.07472
$370$	0	0
$371$	5647.89	0.790361
$372$	0	0
$373$	−4634.24	−0.643303	−0.321652	−	0.946858i	$-0.604238\pi$
$373$	−4634.24	−0.643303	−0.321652	+	0.946858i	$0.604238\pi$
$374$	0	0
$375$	−1112.37	−0.153180
$376$	0	0
$377$	−10140.3	−1.38529
$378$	0	0
$379$	754.678	0.102283	0.0511414	−	0.998691i	$-0.483714\pi$
$379$	754.678	0.102283	0.0511414	+	0.998691i	$0.483714\pi$
$380$	0	0
$381$	15142.5	2.03616
$382$	0	0
$383$	984.709	0.131374	0.0656871	−	0.997840i	$-0.479076\pi$
$383$	984.709	0.131374	0.0656871	+	0.997840i	$0.479076\pi$
$384$	0	0
$385$	−6291.31	−0.832817
$386$	0	0
$387$	351.656	0.0461904
$388$	0	0
$389$	−3259.21	−0.424803	−0.212402	−	0.977182i	$-0.568128\pi$
$389$	−3259.21	−0.424803	−0.212402	+	0.977182i	$0.568128\pi$
$390$	0	0
$391$	706.308	0.0913543
$392$	0	0
$393$	−13713.2	−1.76015
$394$	0	0
$395$	4133.12	0.526481
$396$	0	0
$397$	−13713.2	−1.73361	−0.866807	−	0.498644i	$-0.833831\pi$
$397$	−13713.2	−1.73361	−0.866807	+	0.498644i	$0.833831\pi$
$398$	0	0
$399$	11820.0	1.48305
$400$	0	0
$401$	−671.306	−0.0835996	−0.0417998	−	0.999126i	$-0.513309\pi$
$401$	−671.306	−0.0835996	−0.0417998	+	0.999126i	$0.513309\pi$
$402$	0	0
$403$	−2648.52	−0.327375
$404$	0	0
$405$	2929.04	0.359371
$406$	0	0
$407$	6115.83	0.744842
$408$	0	0
$409$	12957.4	1.56651	0.783253	−	0.621703i	$-0.213559\pi$
$409$	12957.4	1.56651	0.783253	+	0.621703i	$0.213559\pi$
$410$	0	0
$411$	12383.2	1.48618
$412$	0	0
$413$	−5370.08	−0.639817
$414$	0	0
$415$	5131.26	0.606949
$416$	0	0
$417$	1394.02	0.163706
$418$	0	0
$419$	−14023.2	−1.63503	−0.817516	−	0.575906i	$-0.804649\pi$
$419$	−14023.2	−1.63503	−0.817516	+	0.575906i	$0.804649\pi$
$420$	0	0
$421$	−4825.76	−0.558653	−0.279326	−	0.960196i	$-0.590111\pi$
$421$	−4825.76	−0.558653	−0.279326	+	0.960196i	$0.590111\pi$
$422$	0	0
$423$	10630.8	1.22195
$424$	0	0
$425$	−2889.39	−0.329779
$426$	0	0
$427$	−16206.7	−1.83676
$428$	0	0
$429$	24690.2	2.77868
$430$	0	0
$431$	−3901.30	−0.436007	−0.218003	−	0.975948i	$-0.569954\pi$
$431$	−3901.30	−0.436007	−0.218003	+	0.975948i	$0.569954\pi$
$432$	0	0
$433$	1027.04	0.113987	0.0569933	−	0.998375i	$-0.481849\pi$
$433$	1027.04	0.113987	0.0569933	+	0.998375i	$0.481849\pi$
$434$	0	0
$435$	9983.93	1.10044
$436$	0	0
$437$	396.057	0.0433547
$438$	0	0
$439$	255.290	0.0277547	0.0138774	−	0.999904i	$-0.495583\pi$
$439$	255.290	0.0277547	0.0138774	+	0.999904i	$0.495583\pi$
$440$	0	0
$441$	4020.90	0.434176
$442$	0	0
$443$	−6551.19	−0.702611	−0.351305	−	0.936261i	$-0.614262\pi$
$443$	−6551.19	−0.702611	−0.351305	+	0.936261i	$0.614262\pi$
$444$	0	0
$445$	−774.245	−0.0824780
$446$	0	0
$447$	−1906.56	−0.201739
$448$	0	0
$449$	−2364.38	−0.248512	−0.124256	−	0.992250i	$-0.539654\pi$
$449$	−2364.38	−0.248512	−0.124256	+	0.992250i	$0.539654\pi$
$450$	0	0
$451$	8961.00	0.935603
$452$	0	0
$453$	−311.918	−0.0323514
$454$	0	0
$455$	4631.01	0.477154
$456$	0	0
$457$	4440.07	0.454481	0.227241	−	0.973839i	$-0.427030\pi$
$457$	4440.07	0.454481	0.227241	+	0.973839i	$0.427030\pi$
$458$	0	0
$459$	25909.9	2.63479
$460$	0	0
$461$	−891.518	−0.0900697	−0.0450349	−	0.998985i	$-0.514340\pi$
$461$	−891.518	−0.0900697	−0.0450349	+	0.998985i	$0.514340\pi$
$462$	0	0
$463$	3294.85	0.330723	0.165361	−	0.986233i	$-0.447121\pi$
$463$	3294.85	0.330723	0.165361	+	0.986233i	$0.447121\pi$
$464$	0	0
$465$	2607.67	0.260060
$466$	0	0
$467$	1488.22	0.147466	0.0737329	−	0.997278i	$-0.476509\pi$
$467$	1488.22	0.147466	0.0737329	+	0.997278i	$0.476509\pi$
$468$	0	0
$469$	2905.11	0.286025
$470$	0	0
$471$	−3111.01	−0.304348
$472$	0	0
$473$	−413.657	−0.0402114
$474$	0	0
$475$	−1620.20	−0.156505
$476$	0	0
$477$	14382.8	1.38059
$478$	0	0
$479$	17534.7	1.67261	0.836304	−	0.548266i	$-0.184712\pi$
$479$	17534.7	1.67261	0.836304	+	0.548266i	$0.184712\pi$
$480$	0	0
$481$	−4501.84	−0.426749
$482$	0	0
$483$	1114.59	0.105001
$484$	0	0
$485$	2070.20	0.193821
$486$	0	0
$487$	10113.8	0.941067	0.470533	−	0.882382i	$-0.344062\pi$
$487$	10113.8	0.941067	0.470533	+	0.882382i	$0.344062\pi$
$488$	0	0
$489$	−18056.4	−1.66981
$490$	0	0
$491$	17855.8	1.64119	0.820594	−	0.571512i	$-0.193643\pi$
$491$	17855.8	1.64119	0.820594	+	0.571512i	$0.193643\pi$
$492$	0	0
$493$	25933.3	2.36912
$494$	0	0
$495$	−16021.3	−1.45475
$496$	0	0
$497$	−21396.2	−1.93109
$498$	0	0
$499$	2985.84	0.267865	0.133932	−	0.990990i	$-0.457240\pi$
$499$	2985.84	0.267865	0.133932	+	0.990990i	$0.457240\pi$
$500$	0	0
$501$	1909.89	0.170315
$502$	0	0
$503$	2159.75	0.191449	0.0957243	−	0.995408i	$-0.469483\pi$
$503$	2159.75	0.191449	0.0957243	+	0.995408i	$0.469483\pi$
$504$	0	0
$505$	8884.94	0.782920
$506$	0	0
$507$	1376.65	0.120590
$508$	0	0
$509$	16880.6	1.46998	0.734989	−	0.678079i	$-0.237187\pi$
$509$	16880.6	1.46998	0.734989	+	0.678079i	$0.237187\pi$
$510$	0	0
$511$	−21681.1	−1.87694
$512$	0	0
$513$	14528.8	1.25041
$514$	0	0
$515$	3839.95	0.328560
$516$	0	0
$517$	−12505.1	−1.06378
$518$	0	0
$519$	−30478.9	−2.57779
$520$	0	0
$521$	3915.76	0.329276	0.164638	−	0.986354i	$-0.447354\pi$
$521$	3915.76	0.329276	0.164638	+	0.986354i	$0.447354\pi$
$522$	0	0
$523$	14478.8	1.21054	0.605271	−	0.796019i	$-0.293065\pi$
$523$	14478.8	1.21054	0.605271	+	0.796019i	$0.293065\pi$
$524$	0	0
$525$	−4559.59	−0.379042
$526$	0	0
$527$	6773.43	0.559877
$528$	0	0
$529$	−12129.7	−0.996930
$530$	0	0
$531$	−13675.3	−1.11762
$532$	0	0
$533$	−6596.16	−0.536044
$534$	0	0
$535$	−8244.99	−0.666285
$536$	0	0
$537$	18065.8	1.45177
$538$	0	0
$539$	−4729.83	−0.377975
$540$	0	0
$541$	−11827.6	−0.939940	−0.469970	−	0.882682i	$-0.655735\pi$
$541$	−11827.6	−0.939940	−0.469970	+	0.882682i	$0.655735\pi$
$542$	0	0
$543$	−1816.63	−0.143571
$544$	0	0
$545$	4421.10	0.347485
$546$	0	0
$547$	6446.22	0.503877	0.251938	−	0.967743i	$-0.418932\pi$
$547$	6446.22	0.503877	0.251938	+	0.967743i	$0.418932\pi$
$548$	0	0
$549$	−41271.6	−3.20843
$550$	0	0
$551$	14541.9	1.12433
$552$	0	0
$553$	16941.6	1.30277
$554$	0	0
$555$	4432.42	0.339001
$556$	0	0
$557$	−14419.5	−1.09690	−0.548451	−	0.836183i	$-0.684782\pi$
$557$	−14419.5	−1.09690	−0.548451	+	0.836183i	$0.684782\pi$
$558$	0	0
$559$	304.492	0.0230387
$560$	0	0
$561$	−63143.9	−4.75211
$562$	0	0
$563$	−817.038	−0.0611617	−0.0305808	−	0.999532i	$-0.509736\pi$
$563$	−817.038	−0.0611617	−0.0305808	+	0.999532i	$0.509736\pi$
$564$	0	0
$565$	−5694.24	−0.423998
$566$	0	0
$567$	12006.1	0.889256
$568$	0	0
$569$	−16795.5	−1.23744	−0.618720	−	0.785612i	$-0.712348\pi$
$569$	−16795.5	−1.23744	−0.618720	+	0.785612i	$0.712348\pi$
$570$	0	0
$571$	−15342.6	−1.12447	−0.562233	−	0.826979i	$-0.690057\pi$
$571$	−15342.6	−1.12447	−0.562233	+	0.826979i	$0.690057\pi$
$572$	0	0
$573$	5493.81	0.400536
$574$	0	0
$575$	−152.781	−0.0110807
$576$	0	0
$577$	1287.65	0.0929039	0.0464519	−	0.998921i	$-0.485209\pi$
$577$	1287.65	0.0929039	0.0464519	+	0.998921i	$0.485209\pi$
$578$	0	0
$579$	−27331.7	−1.96177
$580$	0	0
$581$	21032.9	1.50188
$582$	0	0
$583$	−16918.6	−1.20188
$584$	0	0
$585$	11793.2	0.833487
$586$	0	0
$587$	2074.66	0.145878	0.0729391	−	0.997336i	$-0.476762\pi$
$587$	2074.66	0.145878	0.0729391	+	0.997336i	$0.476762\pi$
$588$	0	0
$589$	3798.16	0.265705
$590$	0	0
$591$	7226.84	0.502999
$592$	0	0
$593$	16263.0	1.12621	0.563104	−	0.826386i	$-0.309607\pi$
$593$	16263.0	1.12621	0.563104	+	0.826386i	$0.309607\pi$
$594$	0	0
$595$	−11843.5	−0.816030
$596$	0	0
$597$	9485.73	0.650293
$598$	0	0
$599$	22423.7	1.52956	0.764780	−	0.644292i	$-0.222848\pi$
$599$	22423.7	1.52956	0.764780	+	0.644292i	$0.222848\pi$
$600$	0	0
$601$	−2915.58	−0.197885	−0.0989424	−	0.995093i	$-0.531546\pi$
$601$	−2915.58	−0.197885	−0.0989424	+	0.995093i	$0.531546\pi$
$602$	0	0
$603$	7398.09	0.499624
$604$	0	0
$605$	12191.0	0.819234
$606$	0	0
$607$	−11897.3	−0.795546	−0.397773	−	0.917484i	$-0.630217\pi$
$607$	−11897.3	−0.795546	−0.397773	+	0.917484i	$0.630217\pi$
$608$	0	0
$609$	40923.9	2.72302
$610$	0	0
$611$	9204.98	0.609482
$612$	0	0
$613$	−22791.7	−1.50171	−0.750856	−	0.660466i	$-0.770359\pi$
$613$	−22791.7	−1.50171	−0.750856	+	0.660466i	$0.770359\pi$
$614$	0	0
$615$	6494.44	0.425823
$616$	0	0
$617$	11894.0	0.776071	0.388036	−	0.921644i	$-0.373154\pi$
$617$	11894.0	0.776071	0.388036	+	0.921644i	$0.373154\pi$
$618$	0	0
$619$	−15567.8	−1.01086	−0.505431	−	0.862867i	$-0.668667\pi$
$619$	−15567.8	−1.01086	−0.505431	+	0.862867i	$0.668667\pi$
$620$	0	0
$621$	1370.02	0.0885301
$622$	0	0
$623$	−3173.61	−0.204090
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−35407.5	−2.25524
$628$	0	0
$629$	11513.2	0.729828
$630$	0	0
$631$	−23681.8	−1.49407	−0.747034	−	0.664786i	$-0.768523\pi$
$631$	−23681.8	−1.49407	−0.747034	+	0.664786i	$0.768523\pi$
$632$	0	0
$633$	10876.7	0.682953
$634$	0	0
$635$	−8508.02	−0.531701
$636$	0	0
$637$	3481.62	0.216557
$638$	0	0
$639$	−54487.1	−3.37320
$640$	0	0
$641$	21351.6	1.31566	0.657831	−	0.753166i	$-0.271474\pi$
$641$	21351.6	1.31566	0.657831	+	0.753166i	$0.271474\pi$
$642$	0	0
$643$	7196.04	0.441344	0.220672	−	0.975348i	$-0.429175\pi$
$643$	7196.04	0.441344	0.220672	+	0.975348i	$0.429175\pi$
$644$	0	0
$645$	−299.796	−0.0183015
$646$	0	0
$647$	−20897.9	−1.26983	−0.634915	−	0.772582i	$-0.718965\pi$
$647$	−20897.9	−1.26983	−0.634915	+	0.772582i	$0.718965\pi$
$648$	0	0
$649$	16086.4	0.972956
$650$	0	0
$651$	10688.8	0.643513
$652$	0	0
$653$	8655.43	0.518703	0.259352	−	0.965783i	$-0.416491\pi$
$653$	8655.43	0.518703	0.259352	+	0.965783i	$0.416491\pi$
$654$	0	0
$655$	7704.93	0.459628
$656$	0	0
$657$	−55212.6	−3.27861
$658$	0	0
$659$	−13004.3	−0.768705	−0.384352	−	0.923186i	$-0.625575\pi$
$659$	−13004.3	−0.768705	−0.384352	+	0.923186i	$0.625575\pi$
$660$	0	0
$661$	−19135.2	−1.12598	−0.562991	−	0.826463i	$-0.690349\pi$
$661$	−19135.2	−1.12598	−0.562991	+	0.826463i	$0.690349\pi$
$662$	0	0
$663$	46480.0	2.72267
$664$	0	0
$665$	−6641.18	−0.387269
$666$	0	0
$667$	1371.26	0.0796033
$668$	0	0
$669$	−3687.51	−0.213105
$670$	0	0
$671$	48548.3	2.79312
$672$	0	0
$673$	24101.3	1.38044	0.690221	−	0.723599i	$-0.257513\pi$
$673$	24101.3	1.38044	0.690221	+	0.723599i	$0.257513\pi$
$674$	0	0
$675$	−5604.54	−0.319584
$676$	0	0
$677$	17892.9	1.01577	0.507887	−	0.861424i	$-0.330427\pi$
$677$	17892.9	1.01577	0.507887	+	0.861424i	$0.330427\pi$
$678$	0	0
$679$	8485.72	0.479606
$680$	0	0
$681$	20413.2	1.14866
$682$	0	0
$683$	27545.4	1.54319	0.771593	−	0.636117i	$-0.219460\pi$
$683$	27545.4	1.54319	0.771593	+	0.636117i	$0.219460\pi$
$684$	0	0
$685$	−6957.67	−0.388086
$686$	0	0
$687$	−28607.7	−1.58872
$688$	0	0
$689$	12453.8	0.688608
$690$	0	0
$691$	−11360.1	−0.625408	−0.312704	−	0.949851i	$-0.601235\pi$
$691$	−11360.1	−0.625408	−0.312704	+	0.949851i	$0.601235\pi$
$692$	0	0
$693$	−65671.0	−3.59976
$694$	0	0
$695$	−783.245	−0.0427485
$696$	0	0
$697$	16869.3	0.916744
$698$	0	0
$699$	−32023.8	−1.73284
$700$	0	0
$701$	−148.008	−0.00797459	−0.00398729	−	0.999992i	$-0.501269\pi$
$701$	−148.008	−0.00797459	−0.00398729	+	0.999992i	$0.501269\pi$
$702$	0	0
$703$	6455.95	0.346360
$704$	0	0
$705$	−9063.02	−0.484160
$706$	0	0
$707$	36419.2	1.93732
$708$	0	0
$709$	2401.61	0.127213	0.0636067	−	0.997975i	$-0.479740\pi$
$709$	2401.61	0.127213	0.0636067	+	0.997975i	$0.479740\pi$
$710$	0	0
$711$	43143.0	2.27566
$712$	0	0
$713$	358.155	0.0188121
$714$	0	0
$715$	−13872.5	−0.725598
$716$	0	0
$717$	−21914.5	−1.14144
$718$	0	0
$719$	−13003.4	−0.674472	−0.337236	−	0.941420i	$-0.609492\pi$
$719$	−13003.4	−0.674472	−0.337236	+	0.941420i	$0.609492\pi$
$720$	0	0
$721$	15739.9	0.813016
$722$	0	0
$723$	−36959.9	−1.90118
$724$	0	0
$725$	−5609.59	−0.287359
$726$	0	0
$727$	36506.5	1.86238	0.931190	−	0.364533i	$-0.118771\pi$
$727$	36506.5	1.86238	0.931190	+	0.364533i	$0.118771\pi$
$728$	0	0
$729$	−23290.3	−1.18327
$730$	0	0
$731$	−778.720	−0.0394008
$732$	0	0
$733$	−18661.5	−0.940350	−0.470175	−	0.882573i	$-0.655809\pi$
$733$	−18661.5	−0.940350	−0.470175	+	0.882573i	$0.655809\pi$
$734$	0	0
$735$	−3427.92	−0.172028
$736$	0	0
$737$	−8702.46	−0.434951
$738$	0	0
$739$	27675.0	1.37759	0.688797	−	0.724954i	$-0.258139\pi$
$739$	27675.0	1.37759	0.688797	+	0.724954i	$0.258139\pi$
$740$	0	0
$741$	26063.3	1.29212
$742$	0	0
$743$	−17622.0	−0.870103	−0.435052	−	0.900406i	$-0.643270\pi$
$743$	−17622.0	−0.870103	−0.435052	+	0.900406i	$0.643270\pi$
$744$	0	0
$745$	1071.22	0.0526800
$746$	0	0
$747$	53562.0	2.62347
$748$	0	0
$749$	−33796.1	−1.64871
$750$	0	0
$751$	−34581.0	−1.68027	−0.840133	−	0.542381i	$-0.817523\pi$
$751$	−34581.0	−1.68027	−0.840133	+	0.542381i	$0.817523\pi$
$752$	0	0
$753$	11882.4	0.575058
$754$	0	0
$755$	175.255	0.00844793
$756$	0	0
$757$	33994.4	1.63216	0.816082	−	0.577936i	$-0.196142\pi$
$757$	33994.4	1.63216	0.816082	+	0.577936i	$0.196142\pi$
$758$	0	0
$759$	−3338.82	−0.159673
$760$	0	0
$761$	−34917.8	−1.66330	−0.831649	−	0.555302i	$-0.812603\pi$
$761$	−34917.8	−1.66330	−0.831649	+	0.555302i	$0.812603\pi$
$762$	0	0
$763$	18122.0	0.859844
$764$	0	0
$765$	−30160.5	−1.42543
$766$	0	0
$767$	−11841.2	−0.557445
$768$	0	0
$769$	−20714.9	−0.971388	−0.485694	−	0.874129i	$-0.661433\pi$
$769$	−20714.9	−0.971388	−0.485694	+	0.874129i	$0.661433\pi$
$770$	0	0
$771$	4719.66	0.220459
$772$	0	0
$773$	−12461.4	−0.579826	−0.289913	−	0.957053i	$-0.593626\pi$
$773$	−12461.4	−0.579826	−0.289913	+	0.957053i	$0.593626\pi$
$774$	0	0
$775$	−1465.15	−0.0679095
$776$	0	0
$777$	18168.4	0.838851
$778$	0	0
$779$	9459.35	0.435066
$780$	0	0
$781$	64093.8	2.93657
$782$	0	0
$783$	50302.7	2.29588
$784$	0	0
$785$	1747.96	0.0794743
$786$	0	0
$787$	32821.0	1.48658	0.743292	−	0.668967i	$-0.233263\pi$
$787$	32821.0	1.48658	0.743292	+	0.668967i	$0.233263\pi$
$788$	0	0
$789$	19882.4	0.897126
$790$	0	0
$791$	−23340.6	−1.04917
$792$	0	0
$793$	−35736.2	−1.60029
$794$	0	0
$795$	−12261.7	−0.547016
$796$	0	0
$797$	7777.06	0.345643	0.172822	−	0.984953i	$-0.444712\pi$
$797$	7777.06	0.345643	0.172822	+	0.984953i	$0.444712\pi$
$798$	0	0
$799$	−23541.2	−1.04234
$800$	0	0
$801$	−8081.85	−0.356502
$802$	0	0
$803$	64947.2	2.85422
$804$	0	0
$805$	−626.245	−0.0274189
$806$	0	0
$807$	−45350.0	−1.97819
$808$	0	0
$809$	20407.4	0.886880	0.443440	−	0.896304i	$-0.353758\pi$
$809$	20407.4	0.886880	0.443440	+	0.896304i	$0.353758\pi$
$810$	0	0
$811$	−25876.6	−1.12041	−0.560205	−	0.828354i	$-0.689277\pi$
$811$	−25876.6	−1.12041	−0.560205	+	0.828354i	$0.689277\pi$
$812$	0	0
$813$	−12043.8	−0.519550
$814$	0	0
$815$	10145.2	0.436038
$816$	0	0
$817$	−436.662	−0.0186987
$818$	0	0
$819$	48340.2	2.06245
$820$	0	0
$821$	1100.58	0.0467850	0.0233925	−	0.999726i	$-0.492553\pi$
$821$	1100.58	0.0467850	0.0233925	+	0.999726i	$0.492553\pi$
$822$	0	0
$823$	−36245.7	−1.53517	−0.767585	−	0.640948i	$-0.778542\pi$
$823$	−36245.7	−1.53517	−0.767585	+	0.640948i	$0.778542\pi$
$824$	0	0
$825$	13658.6	0.576401
$826$	0	0
$827$	1996.72	0.0839572	0.0419786	−	0.999119i	$-0.486634\pi$
$827$	1996.72	0.0839572	0.0419786	+	0.999119i	$0.486634\pi$
$828$	0	0
$829$	43393.8	1.81801	0.909004	−	0.416788i	$-0.136844\pi$
$829$	43393.8	1.81801	0.909004	+	0.416788i	$0.136844\pi$
$830$	0	0
$831$	79019.1	3.29861
$832$	0	0
$833$	−8904.03	−0.370356
$834$	0	0
$835$	−1073.10	−0.0444743
$836$	0	0
$837$	13138.4	0.542569
$838$	0	0
$839$	−3424.53	−0.140915	−0.0704576	−	0.997515i	$-0.522446\pi$
$839$	−3424.53	−0.140915	−0.0704576	+	0.997515i	$0.522446\pi$
$840$	0	0
$841$	25959.0	1.06437
$842$	0	0
$843$	10194.2	0.416498
$844$	0	0
$845$	−773.490	−0.0314898
$846$	0	0
$847$	49970.8	2.02718
$848$	0	0
$849$	−3429.66	−0.138640
$850$	0	0
$851$	608.778	0.0245225
$852$	0	0
$853$	−5173.63	−0.207669	−0.103835	−	0.994595i	$-0.533111\pi$
$853$	−5173.63	−0.207669	−0.103835	+	0.994595i	$0.533111\pi$
$854$	0	0
$855$	−16912.3	−0.676477
$856$	0	0
$857$	28323.8	1.12896	0.564482	−	0.825445i	$-0.309076\pi$
$857$	28323.8	1.12896	0.564482	+	0.825445i	$0.309076\pi$
$858$	0	0
$859$	−42970.7	−1.70680	−0.853401	−	0.521255i	$-0.825464\pi$
$859$	−42970.7	−1.70680	−0.853401	+	0.521255i	$0.825464\pi$
$860$	0	0
$861$	26620.6	1.05369
$862$	0	0
$863$	12664.1	0.499527	0.249764	−	0.968307i	$-0.419647\pi$
$863$	12664.1	0.499527	0.249764	+	0.968307i	$0.419647\pi$
$864$	0	0
$865$	17124.9	0.673139
$866$	0	0
$867$	−75149.2	−2.94371
$868$	0	0
$869$	−50749.7	−1.98109
$870$	0	0
$871$	6405.85	0.249201
$872$	0	0
$873$	21609.6	0.837769
$874$	0	0
$875$	2561.86	0.0989791
$876$	0	0
$877$	−7403.14	−0.285047	−0.142524	−	0.989791i	$-0.545522\pi$
$877$	−7403.14	−0.285047	−0.142524	+	0.989791i	$0.545522\pi$
$878$	0	0
$879$	−33925.6	−1.30180
$880$	0	0
$881$	−5691.74	−0.217661	−0.108831	−	0.994060i	$-0.534711\pi$
$881$	−5691.74	−0.217661	−0.108831	+	0.994060i	$0.534711\pi$
$882$	0	0
$883$	−13915.1	−0.530329	−0.265164	−	0.964203i	$-0.585426\pi$
$883$	−13915.1	−0.530329	−0.265164	+	0.964203i	$0.585426\pi$
$884$	0	0
$885$	11658.6	0.442823
$886$	0	0
$887$	−35631.2	−1.34879	−0.674396	−	0.738370i	$-0.735596\pi$
$887$	−35631.2	−1.34879	−0.674396	+	0.738370i	$0.735596\pi$
$888$	0	0
$889$	−34874.2	−1.31568
$890$	0	0
$891$	−35965.0	−1.35227
$892$	0	0
$893$	−13200.6	−0.494670
$894$	0	0
$895$	−10150.5	−0.379100
$896$	0	0
$897$	2457.70	0.0914829
$898$	0	0
$899$	13150.3	0.487859
$900$	0	0
$901$	−31849.8	−1.17766
$902$	0	0
$903$	−1228.86	−0.0452866
$904$	0	0
$905$	1020.69	0.0374906
$906$	0	0
$907$	22734.4	0.832285	0.416143	−	0.909299i	$-0.363382\pi$
$907$	22734.4	0.832285	0.416143	+	0.909299i	$0.363382\pi$
$908$	0	0
$909$	92744.3	3.38408
$910$	0	0
$911$	32714.8	1.18978	0.594890	−	0.803807i	$-0.297195\pi$
$911$	32714.8	1.18978	0.594890	+	0.803807i	$0.297195\pi$
$912$	0	0
$913$	−63005.6	−2.28388
$914$	0	0
$915$	35185.1	1.27124
$916$	0	0
$917$	31582.3	1.13734
$918$	0	0
$919$	41077.0	1.47443	0.737217	−	0.675656i	$-0.236139\pi$
$919$	41077.0	1.47443	0.737217	+	0.675656i	$0.236139\pi$
$920$	0	0
$921$	−61380.3	−2.19604
$922$	0	0
$923$	−47179.3	−1.68247
$924$	0	0
$925$	−2490.41	−0.0885234
$926$	0	0
$927$	40082.9	1.42017
$928$	0	0
$929$	27678.8	0.977515	0.488758	−	0.872420i	$-0.337450\pi$
$929$	27678.8	0.977515	0.488758	+	0.872420i	$0.337450\pi$
$930$	0	0
$931$	−4992.87	−0.175762
$932$	0	0
$933$	72307.6	2.53724
$934$	0	0
$935$	35478.1	1.24092
$936$	0	0
$937$	−17120.7	−0.596915	−0.298457	−	0.954423i	$-0.596472\pi$
$937$	−17120.7	−0.596915	−0.298457	+	0.954423i	$0.596472\pi$
$938$	0	0
$939$	−18266.6	−0.634832
$940$	0	0
$941$	−30687.2	−1.06310	−0.531549	−	0.847028i	$-0.678390\pi$
$941$	−30687.2	−1.06310	−0.531549	+	0.847028i	$0.678390\pi$
$942$	0	0
$943$	891.989	0.0308029
$944$	0	0
$945$	−22972.9	−0.790803
$946$	0	0
$947$	−36649.9	−1.25761	−0.628807	−	0.777561i	$-0.716456\pi$
$947$	−36649.9	−1.25761	−0.628807	+	0.777561i	$0.716456\pi$
$948$	0	0
$949$	−47807.4	−1.63529
$950$	0	0
$951$	−37154.3	−1.26689
$952$	0	0
$953$	23768.8	0.807919	0.403959	−	0.914777i	$-0.367634\pi$
$953$	23768.8	0.807919	0.403959	+	0.914777i	$0.367634\pi$
$954$	0	0
$955$	−3086.77	−0.104592
$956$	0	0
$957$	−122590.	−4.14084
$958$	0	0
$959$	−28519.4	−0.960311
$960$	0	0
$961$	−26356.3	−0.884708
$962$	0	0
$963$	−86064.3	−2.87994
$964$	0	0
$965$	15356.7	0.512278
$966$	0	0
$967$	−36241.8	−1.20523	−0.602615	−	0.798032i	$-0.705875\pi$
$967$	−36241.8	−1.20523	−0.602615	+	0.798032i	$0.705875\pi$
$968$	0	0
$969$	−66655.5	−2.20979
$970$	0	0
$971$	16837.4	0.556476	0.278238	−	0.960512i	$-0.410250\pi$
$971$	16837.4	0.556476	0.278238	+	0.960512i	$0.410250\pi$
$972$	0	0
$973$	−3210.51	−0.105780
$974$	0	0
$975$	−10054.0	−0.330243
$976$	0	0
$977$	−26322.4	−0.861954	−0.430977	−	0.902363i	$-0.641831\pi$
$977$	−26322.4	−0.861954	−0.430977	+	0.902363i	$0.641831\pi$
$978$	0	0
$979$	9506.78	0.310355
$980$	0	0
$981$	46149.1	1.50196
$982$	0	0
$983$	−42438.7	−1.37699	−0.688497	−	0.725240i	$-0.741729\pi$
$983$	−42438.7	−1.37699	−0.688497	+	0.725240i	$0.741729\pi$
$984$	0	0
$985$	−4060.49	−0.131348
$986$	0	0
$987$	−37149.1	−1.19804
$988$	0	0
$989$	−41.1760	−0.00132388
$990$	0	0
$991$	−17890.3	−0.573464	−0.286732	−	0.958011i	$-0.592569\pi$
$991$	−17890.3	−0.573464	−0.286732	+	0.958011i	$0.592569\pi$
$992$	0	0
$993$	104458.	3.33824
$994$	0	0
$995$	−5329.67	−0.169811
$996$	0	0
$997$	16189.9	0.514283	0.257142	−	0.966374i	$-0.417219\pi$
$997$	16189.9	0.514283	0.257142	+	0.966374i	$0.417219\pi$
$998$	0	0
$999$	22332.2	0.707265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.a.c.1.1	✓	2
3.2	odd	2		1440.4.a.t.1.2		2
4.3	odd	2		160.4.a.g.1.2	yes	2
5.2	odd	4		800.4.c.i.449.4		4
5.3	odd	4		800.4.c.i.449.1		4
5.4	even	2		800.4.a.s.1.2		2
8.3	odd	2		320.4.a.o.1.1		2
8.5	even	2		320.4.a.s.1.2		2
12.11	even	2		1440.4.a.x.1.1		2
16.3	odd	4		1280.4.d.q.641.4		4
16.5	even	4		1280.4.d.x.641.4		4
16.11	odd	4		1280.4.d.q.641.1		4
16.13	even	4		1280.4.d.x.641.1		4
20.3	even	4		800.4.c.k.449.4		4
20.7	even	4		800.4.c.k.449.1		4
20.19	odd	2		800.4.a.m.1.1		2
40.19	odd	2		1600.4.a.cn.1.2		2
40.29	even	2		1600.4.a.cd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.c.1.1	✓	2	1.1	even	1	trivial
160.4.a.g.1.2	yes	2	4.3	odd	2
320.4.a.o.1.1		2	8.3	odd	2
320.4.a.s.1.2		2	8.5	even	2
800.4.a.m.1.1		2	20.19	odd	2
800.4.a.s.1.2		2	5.4	even	2
800.4.c.i.449.1		4	5.3	odd	4
800.4.c.i.449.4		4	5.2	odd	4
800.4.c.k.449.1		4	20.7	even	4
800.4.c.k.449.4		4	20.3	even	4
1280.4.d.q.641.1		4	16.11	odd	4
1280.4.d.q.641.4		4	16.3	odd	4
1280.4.d.x.641.1		4	16.13	even	4
1280.4.d.x.641.4		4	16.5	even	4
1440.4.a.t.1.2		2	3.2	odd	2
1440.4.a.x.1.1		2	12.11	even	2
1600.4.a.cd.1.1		2	40.29	even	2
1600.4.a.cn.1.2		2	40.19	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

\(f(q)\)	\(=\)	\(q-8.89898 q^{3} +5.00000 q^{5} +20.4949 q^{7} +52.1918 q^{9} -61.3939 q^{11} +45.1918 q^{13} -44.4949 q^{15} -115.576 q^{17} -64.8082 q^{19} -182.384 q^{21} -6.11123 q^{23} +25.0000 q^{25} -224.182 q^{27} -224.384 q^{29} -58.6061 q^{31} +546.343 q^{33} +102.474 q^{35} -99.6163 q^{37} -402.161 q^{39} -145.959 q^{41} +6.73776 q^{43} +260.959 q^{45} +203.687 q^{47} +77.0408 q^{49} +1028.50 q^{51} +275.576 q^{53} -306.969 q^{55} +576.727 q^{57} -262.020 q^{59} -790.767 q^{61} +1069.67 q^{63} +225.959 q^{65} +141.748 q^{67} +54.3837 q^{69} -1043.98 q^{71} -1057.88 q^{73} -222.474 q^{75} -1258.26 q^{77} +826.624 q^{79} +585.808 q^{81} +1026.25 q^{83} -577.878 q^{85} +1996.79 q^{87} -154.849 q^{89} +926.202 q^{91} +521.535 q^{93} -324.041 q^{95} +414.041 q^{97} -3204.26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} + 10 q^{5} - 8 q^{7} + 26 q^{9} - 64 q^{11} + 12 q^{13} - 40 q^{15} + 4 q^{17} - 208 q^{19} - 208 q^{21} - 120 q^{23} + 50 q^{25} - 272 q^{27} - 292 q^{29} - 176 q^{31} + 544 q^{33} - 40 q^{35}+ \cdots - 3136 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 + 10 * q^5 - 8 * q^7 + 26 * q^9 - 64 * q^11 + 12 * q^13 - 40 * q^15 + 4 * q^17 - 208 * q^19 - 208 * q^21 - 120 * q^23 + 50 * q^25 - 272 * q^27 - 292 * q^29 - 176 * q^31 + 544 * q^33 - 40 * q^35 - 356 * q^37 - 432 * q^39 + 100 * q^41 + 376 * q^43 + 130 * q^45 + 280 * q^47 + 546 * q^49 + 1136 * q^51 + 316 * q^53 - 320 * q^55 + 448 * q^57 - 720 * q^59 - 1268 * q^61 + 1816 * q^63 + 60 * q^65 + 744 * q^67 - 48 * q^69 + 48 * q^71 - 940 * q^73 - 200 * q^75 - 1184 * q^77 - 32 * q^79 + 1250 * q^81 + 1592 * q^83 + 20 * q^85 + 1936 * q^87 - 780 * q^89 + 1872 * q^91 + 416 * q^93 - 1040 * q^95 + 1220 * q^97 - 3136 * q^99

Introduction

L-functions

Modular forms

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