LMFDB - Embedded newform 1600.4.a.cn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$0$
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:	no (minimal twist has level 160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.89898 q^{3} +20.4949 q^{7} +52.1918 q^{9} -61.3939 q^{11} +45.1918 q^{13} +115.576 q^{17} -64.8082 q^{19} +182.384 q^{21} -6.11123 q^{23} +224.182 q^{27} +224.384 q^{29} +58.6061 q^{31} -546.343 q^{33} -99.6163 q^{37} +402.161 q^{39} -145.959 q^{41} -6.73776 q^{43} +203.687 q^{47} +77.0408 q^{49} +1028.50 q^{51} +275.576 q^{53} -576.727 q^{57} -262.020 q^{59} +790.767 q^{61} +1069.67 q^{63} -141.748 q^{67} -54.3837 q^{69} +1043.98 q^{71} +1057.88 q^{73} -1258.26 q^{77} -826.624 q^{79} +585.808 q^{81} -1026.25 q^{83} +1996.79 q^{87} -154.849 q^{89} +926.202 q^{91} +521.535 q^{93} -414.041 q^{97} -3204.26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 8 q^{3} - 8 q^{7} + 26 q^{9} - 64 q^{11} + 12 q^{13} - 4 q^{17} - 208 q^{19} + 208 q^{21} - 120 q^{23} + 272 q^{27} + 292 q^{29} + 176 q^{31} - 544 q^{33} - 356 q^{37} + 432 q^{39} + 100 q^{41} - 376 q^{43}+ \cdots - 3136 q^{99}+O(q^{100})$ 2 * q + 8 * q^3 - 8 * q^7 + 26 * q^9 - 64 * q^11 + 12 * q^13 - 4 * q^17 - 208 * q^19 + 208 * q^21 - 120 * q^23 + 272 * q^27 + 292 * q^29 + 176 * q^31 - 544 * q^33 - 356 * q^37 + 432 * q^39 + 100 * q^41 - 376 * q^43 + 280 * q^47 + 546 * q^49 + 1136 * q^51 + 316 * q^53 - 448 * q^57 - 720 * q^59 + 1268 * q^61 + 1816 * q^63 - 744 * q^67 + 48 * q^69 - 48 * q^71 + 940 * q^73 - 1184 * q^77 + 32 * q^79 + 1250 * q^81 - 1592 * q^83 + 1936 * q^87 - 780 * q^89 + 1872 * q^91 + 416 * q^93 - 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	$-0.172755\pi$
$3$	8.89898	1.71261	0.856305	+	0.516471i	$0.172755\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	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
$15$	0	0
$16$	0	0
$17$	115.576	1.64889	0.824446	−	0.565940i	$-0.191487\pi$
$17$	115.576	1.64889	0.824446	+	0.565940i	$0.191487\pi$
$18$	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
$25$	0	0
$26$	0	0
$27$	224.182	1.59792
$28$	0	0
$29$	224.384	1.43679	0.718397	−	0.695634i	$-0.244876\pi$
$29$	224.384	1.43679	0.718397	+	0.695634i	$0.244876\pi$
$30$	0	0
$31$	58.6061	0.339547	0.169774	−	0.985483i	$-0.445696\pi$
$31$	58.6061	0.339547	0.169774	+	0.985483i	$0.445696\pi$
$32$	0	0
$33$	−546.343	−2.88200
$34$	0	0
$35$	0	0
$36$	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	$-0.503803\pi$
$43$	−6.73776	−0.0238953	−0.0119477	+	0.999929i	$0.503803\pi$
$44$	0	0
$45$	0	0
$46$	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
$55$	0	0
$56$	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	$-0.188399\pi$
$61$	790.767	1.65979	0.829897	+	0.557917i	$0.188399\pi$
$62$	0	0
$63$	1069.67	2.13913
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−141.748	−0.258467	−0.129233	−	0.991614i	$-0.541252\pi$
$67$	−141.748	−0.258467	−0.129233	+	0.991614i	$0.541252\pi$
$68$	0	0
$69$	−54.3837	−0.0948844
$70$	0	0
$71$	1043.98	1.74503	0.872516	−	0.488585i	$-0.162487\pi$
$71$	1043.98	1.74503	0.872516	+	0.488585i	$0.162487\pi$
$72$	0	0
$73$	1057.88	1.69610	0.848049	−	0.529917i	$-0.177777\pi$
$73$	1057.88	1.69610	0.848049	+	0.529917i	$0.177777\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1258.26	−1.86224
$78$	0	0
$79$	−826.624	−1.17725	−0.588624	−	0.808407i	$-0.700330\pi$
$79$	−826.624	−1.17725	−0.588624	+	0.808407i	$0.700330\pi$
$80$	0	0
$81$	585.808	0.803578
$82$	0	0
$83$	−1026.25	−1.35718	−0.678589	−	0.734518i	$-0.737408\pi$
$83$	−1026.25	−1.35718	−0.678589	+	0.734518i	$0.737408\pi$
$84$	0	0
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−414.041	−0.433397	−0.216698	−	0.976239i	$-0.569529\pi$
$97$	−414.041	−0.433397	−0.216698	+	0.976239i	$0.569529\pi$
$98$	0	0
$99$	−3204.26	−3.25293
$100$	0	0
$101$	−1776.99	−1.75066	−0.875331	−	0.483524i	$-0.839357\pi$
$101$	−1776.99	−1.75066	−0.875331	+	0.483524i	$0.839357\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	1649.00	1.48986	0.744929	−	0.667144i	$-0.232483\pi$
$107$	1649.00	1.48986	0.744929	+	0.667144i	$0.232483\pi$
$108$	0	0
$109$	−884.220	−0.777000	−0.388500	−	0.921449i	$-0.627007\pi$
$109$	−884.220	−0.777000	−0.388500	+	0.921449i	$0.627007\pi$
$110$	0	0
$111$	−886.484	−0.758030
$112$	0	0
$113$	1138.85	0.948088	0.474044	−	0.880501i	$-0.342794\pi$
$113$	1138.85	0.948088	0.474044	+	0.880501i	$0.342794\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	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
$135$	0	0
$136$	0	0
$137$	1391.53	0.867787	0.433894	−	0.900964i	$-0.357139\pi$
$137$	1391.53	0.867787	0.433894	+	0.900964i	$0.357139\pi$
$138$	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
$145$	0	0
$146$	0	0
$147$	685.585	0.384667
$148$	0	0
$149$	−214.245	−0.117796	−0.0588981	−	0.998264i	$-0.518759\pi$
$149$	−214.245	−0.117796	−0.0588981	+	0.998264i	$0.518759\pi$
$150$	0	0
$151$	−35.0510	−0.0188901	−0.00944507	−	0.999955i	$-0.503007\pi$
$151$	−35.0510	−0.0188901	−0.00944507	+	0.999955i	$0.503007\pi$
$152$	0	0
$153$	6032.10	3.18736
$154$	0	0
$155$	0	0
$156$	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	$-0.662093\pi$
$163$	−2029.04	−0.975010	−0.487505	+	0.873120i	$0.662093\pi$
$164$	0	0
$165$	0	0
$166$	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
$175$	0	0
$176$	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	$-0.513346\pi$
$181$	−204.139	−0.0838316	−0.0419158	+	0.999121i	$0.513346\pi$
$182$	0	0
$183$	7037.02	2.84258
$184$	0	0
$185$	0	0
$186$	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	$-0.462692\pi$
$191$	617.353	0.233875	0.116937	+	0.993139i	$0.462692\pi$
$192$	0	0
$193$	−3071.33	−1.14549	−0.572744	−	0.819734i	$-0.694121\pi$
$193$	−3071.33	−1.14549	−0.572744	+	0.819734i	$0.694121\pi$
$194$	0	0
$195$	0	0
$196$	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	$-0.439198\pi$
$199$	1065.93	0.379709	0.189855	+	0.981812i	$0.439198\pi$
$200$	0	0
$201$	−1261.41	−0.442653
$202$	0	0
$203$	4598.72	1.58998
$204$	0	0
$205$	0	0
$206$	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
$215$	0	0
$216$	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
$225$	0	0
$226$	0	0
$227$	2293.88	0.670707	0.335353	−	0.942092i	$-0.391144\pi$
$227$	2293.88	0.670707	0.335353	+	0.942092i	$0.391144\pi$
$228$	0	0
$229$	−3214.72	−0.927662	−0.463831	−	0.885924i	$-0.653525\pi$
$229$	−3214.72	−0.927662	−0.463831	+	0.885924i	$0.653525\pi$
$230$	0	0
$231$	−11197.2	−3.18928
$232$	0	0
$233$	−3598.60	−1.01181	−0.505905	−	0.862589i	$-0.668841\pi$
$233$	−3598.60	−1.01181	−0.505905	+	0.862589i	$0.668841\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7356.11	−2.01616
$238$	0	0
$239$	−2462.58	−0.666491	−0.333245	−	0.942840i	$-0.608144\pi$
$239$	−2462.58	−0.666491	−0.333245	+	0.942840i	$0.608144\pi$
$240$	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
$245$	0	0
$246$	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
$255$	0	0
$256$	0	0
$257$	530.359	0.128727	0.0643636	−	0.997927i	$-0.479498\pi$
$257$	530.359	0.128727	0.0643636	+	0.997927i	$0.479498\pi$
$258$	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
$265$	0	0
$266$	0	0
$267$	−1378.00	−0.315851
$268$	0	0
$269$	−5096.09	−1.15507	−0.577535	−	0.816366i	$-0.695985\pi$
$269$	−5096.09	−1.15507	−0.577535	+	0.816366i	$0.695985\pi$
$270$	0	0
$271$	−1353.39	−0.303367	−0.151684	−	0.988429i	$-0.548470\pi$
$271$	−1353.39	−0.303367	−0.151684	+	0.988429i	$0.548470\pi$
$272$	0	0
$273$	8242.25	1.82727
$274$	0	0
$275$	0	0
$276$	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	$-0.512888\pi$
$283$	−385.399	−0.0809526	−0.0404763	+	0.999180i	$0.512888\pi$
$284$	0	0
$285$	0	0
$286$	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
$295$	0	0
$296$	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
$305$	0	0
$306$	0	0
$307$	−6897.45	−1.28228	−0.641138	−	0.767426i	$-0.721537\pi$
$307$	−6897.45	−1.28228	−0.641138	+	0.767426i	$0.721537\pi$
$308$	0	0
$309$	6834.33	1.25823
$310$	0	0
$311$	8125.39	1.48151	0.740753	−	0.671778i	$-0.234469\pi$
$311$	8125.39	1.48151	0.740753	+	0.671778i	$0.234469\pi$
$312$	0	0
$313$	−2052.66	−0.370681	−0.185341	−	0.982674i	$-0.559339\pi$
$313$	−2052.66	−0.370681	−0.185341	+	0.982674i	$0.559339\pi$
$314$	0	0
$315$	0	0
$316$	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
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	1698.85	0.274606	0.137303	−	0.990529i	$-0.456157\pi$
$337$	1698.85	0.274606	0.137303	+	0.990529i	$0.456157\pi$
$338$	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
$345$	0	0
$346$	0	0
$347$	9682.53	1.49794	0.748970	−	0.662604i	$-0.230549\pi$
$347$	9682.53	1.49794	0.748970	+	0.662604i	$0.230549\pi$
$348$	0	0
$349$	10755.1	1.64959	0.824796	−	0.565430i	$-0.191290\pi$
$349$	10755.1	1.64959	0.824796	+	0.565430i	$0.191290\pi$
$350$	0	0
$351$	10131.2	1.54063
$352$	0	0
$353$	7044.94	1.06222	0.531111	−	0.847302i	$-0.321775\pi$
$353$	7044.94	1.06222	0.531111	+	0.847302i	$0.321775\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21079.1	3.12500
$358$	0	0
$359$	805.576	0.118431	0.0592154	−	0.998245i	$-0.481140\pi$
$359$	805.576	0.118431	0.0592154	+	0.998245i	$0.481140\pi$
$360$	0	0
$361$	−2658.90	−0.387652
$362$	0	0
$363$	21697.6	3.13726
$364$	0	0
$365$	0	0
$366$	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
$375$	0	0
$376$	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
$385$	0	0
$386$	0	0
$387$	−351.656	−0.0461904
$388$	0	0
$389$	3259.21	0.424803	0.212402	−	0.977182i	$-0.431872\pi$
$389$	3259.21	0.424803	0.212402	+	0.977182i	$0.431872\pi$
$390$	0	0
$391$	−706.308	−0.0913543
$392$	0	0
$393$	13713.2	1.76015
$394$	0	0
$395$	0	0
$396$	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
$405$	0	0
$406$	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
$415$	0	0
$416$	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	$-0.409889\pi$
$421$	4825.76	0.558653	0.279326	+	0.960196i	$0.409889\pi$
$422$	0	0
$423$	10630.8	1.22195
$424$	0	0
$425$	0	0
$426$	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	$-0.430046\pi$
$431$	3901.30	0.436007	0.218003	+	0.975948i	$0.430046\pi$
$432$	0	0
$433$	−1027.04	−0.113987	−0.0569933	−	0.998375i	$-0.518151\pi$
$433$	−1027.04	−0.113987	−0.0569933	+	0.998375i	$0.518151\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	396.057	0.0433547
$438$	0	0
$439$	−255.290	−0.0277547	−0.0138774	−	0.999904i	$-0.504417\pi$
$439$	−255.290	−0.0277547	−0.0138774	+	0.999904i	$0.504417\pi$
$440$	0	0
$441$	4020.90	0.434176
$442$	0	0
$443$	6551.19	0.702611	0.351305	−	0.936261i	$-0.385738\pi$
$443$	6551.19	0.702611	0.351305	+	0.936261i	$0.385738\pi$
$444$	0	0
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	−4440.07	−0.454481	−0.227241	−	0.973839i	$-0.572970\pi$
$457$	−4440.07	−0.454481	−0.227241	+	0.973839i	$0.572970\pi$
$458$	0	0
$459$	25909.9	2.63479
$460$	0	0
$461$	891.518	0.0900697	0.0450349	−	0.998985i	$-0.485660\pi$
$461$	891.518	0.0900697	0.0450349	+	0.998985i	$0.485660\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	−1488.22	−0.147466	−0.0737329	−	0.997278i	$-0.523491\pi$
$467$	−1488.22	−0.147466	−0.0737329	+	0.997278i	$0.523491\pi$
$468$	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
$475$	0	0
$476$	0	0
$477$	14382.8	1.38059
$478$	0	0
$479$	−17534.7	−1.67261	−0.836304	−	0.548266i	$-0.815288\pi$
$479$	−17534.7	−1.67261	−0.836304	+	0.548266i	$0.815288\pi$
$480$	0	0
$481$	−4501.84	−0.426749
$482$	0	0
$483$	−1114.59	−0.105001
$484$	0	0
$485$	0	0
$486$	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
$495$	0	0
$496$	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
$505$	0	0
$506$	0	0
$507$	−1376.65	−0.120590
$508$	0	0
$509$	−16880.6	−1.46998	−0.734989	−	0.678079i	$-0.762813\pi$
$509$	−16880.6	−1.46998	−0.734989	+	0.678079i	$0.762813\pi$
$510$	0	0
$511$	21681.1	1.87694
$512$	0	0
$513$	−14528.8	−1.25041
$514$	0	0
$515$	0	0
$516$	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	$-0.706935\pi$
$523$	−14478.8	−1.21054	−0.605271	+	0.796019i	$0.706935\pi$
$524$	0	0
$525$	0	0
$526$	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
$535$	0	0
$536$	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	$-0.344265\pi$
$541$	11827.6	0.939940	0.469970	+	0.882682i	$0.344265\pi$
$542$	0	0
$543$	−1816.63	−0.143571
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6446.22	−0.503877	−0.251938	−	0.967743i	$-0.581068\pi$
$547$	−6446.22	−0.503877	−0.251938	+	0.967743i	$0.581068\pi$
$548$	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
$555$	0	0
$556$	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	$-0.490264\pi$
$563$	817.038	0.0611617	0.0305808	+	0.999532i	$0.490264\pi$
$564$	0	0
$565$	0	0
$566$	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
$575$	0	0
$576$	0	0
$577$	−1287.65	−0.0929039	−0.0464519	−	0.998921i	$-0.514791\pi$
$577$	−1287.65	−0.0929039	−0.0464519	+	0.998921i	$0.514791\pi$
$578$	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
$585$	0	0
$586$	0	0
$587$	−2074.66	−0.145878	−0.0729391	−	0.997336i	$-0.523238\pi$
$587$	−2074.66	−0.145878	−0.0729391	+	0.997336i	$0.523238\pi$
$588$	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	$-0.690393\pi$
$593$	−16263.0	−1.12621	−0.563104	+	0.826386i	$0.690393\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9485.73	0.650293
$598$	0	0
$599$	−22423.7	−1.52956	−0.764780	−	0.644292i	$-0.777152\pi$
$599$	−22423.7	−1.52956	−0.764780	+	0.644292i	$0.777152\pi$
$600$	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
$605$	0	0
$606$	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
$615$	0	0
$616$	0	0
$617$	−11894.0	−0.776071	−0.388036	−	0.921644i	$-0.626846\pi$
$617$	−11894.0	−0.776071	−0.388036	+	0.921644i	$0.626846\pi$
$618$	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
$625$	0	0
$626$	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	$-0.231477\pi$
$631$	23681.8	1.49407	0.747034	+	0.664786i	$0.231477\pi$
$632$	0	0
$633$	−10876.7	−0.682953
$634$	0	0
$635$	0	0
$636$	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	$-0.570825\pi$
$643$	−7196.04	−0.441344	−0.220672	+	0.975348i	$0.570825\pi$
$644$	0	0
$645$	0	0
$646$	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
$655$	0	0
$656$	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	$-0.309651\pi$
$661$	19135.2	1.12598	0.562991	+	0.826463i	$0.309651\pi$
$662$	0	0
$663$	46480.0	2.72267
$664$	0	0
$665$	0	0
$666$	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	$-0.742487\pi$
$673$	−24101.3	−1.38044	−0.690221	+	0.723599i	$0.742487\pi$
$674$	0	0
$675$	0	0
$676$	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	$-0.780540\pi$
$683$	−27545.4	−1.54319	−0.771593	+	0.636117i	$0.780540\pi$
$684$	0	0
$685$	0	0
$686$	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
$695$	0	0
$696$	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	$-0.498731\pi$
$701$	148.008	0.00797459	0.00398729	+	0.999992i	$0.498731\pi$
$702$	0	0
$703$	6455.95	0.346360
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36419.2	−1.93732
$708$	0	0
$709$	−2401.61	−0.127213	−0.0636067	−	0.997975i	$-0.520260\pi$
$709$	−2401.61	−0.127213	−0.0636067	+	0.997975i	$0.520260\pi$
$710$	0	0
$711$	−43143.0	−2.27566
$712$	0	0
$713$	−358.155	−0.0188121
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21914.5	−1.14144
$718$	0	0
$719$	13003.4	0.674472	0.337236	−	0.941420i	$-0.390508\pi$
$719$	13003.4	0.674472	0.337236	+	0.941420i	$0.390508\pi$
$720$	0	0
$721$	15739.9	0.813016
$722$	0	0
$723$	36959.9	1.90118
$724$	0	0
$725$	0	0
$726$	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
$735$	0	0
$736$	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
$745$	0	0
$746$	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	$-0.182477\pi$
$751$	34581.0	1.68027	0.840133	+	0.542381i	$0.182477\pi$
$752$	0	0
$753$	−11882.4	−0.575058
$754$	0	0
$755$	0	0
$756$	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
$765$	0	0
$766$	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
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	−32821.0	−1.48658	−0.743292	−	0.668967i	$-0.766737\pi$
$787$	−32821.0	−1.48658	−0.743292	+	0.668967i	$0.766737\pi$
$788$	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
$795$	0	0
$796$	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
$805$	0	0
$806$	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
$815$	0	0
$816$	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	$-0.507447\pi$
$821$	−1100.58	−0.0467850	−0.0233925	+	0.999726i	$0.507447\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	−1996.72	−0.0839572	−0.0419786	−	0.999119i	$-0.513366\pi$
$827$	−1996.72	−0.0839572	−0.0419786	+	0.999119i	$0.513366\pi$
$828$	0	0
$829$	−43393.8	−1.81801	−0.909004	−	0.416788i	$-0.863156\pi$
$829$	−43393.8	−1.81801	−0.909004	+	0.416788i	$0.863156\pi$
$830$	0	0
$831$	−79019.1	−3.29861
$832$	0	0
$833$	8904.03	0.370356
$834$	0	0
$835$	0	0
$836$	0	0
$837$	13138.4	0.542569
$838$	0	0
$839$	3424.53	0.140915	0.0704576	−	0.997515i	$-0.477554\pi$
$839$	3424.53	0.140915	0.0704576	+	0.997515i	$0.477554\pi$
$840$	0	0
$841$	25959.0	1.06437
$842$	0	0
$843$	−10194.2	−0.416498
$844$	0	0
$845$	0	0
$846$	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
$855$	0	0
$856$	0	0
$857$	−28323.8	−1.12896	−0.564482	−	0.825445i	$-0.690924\pi$
$857$	−28323.8	−1.12896	−0.564482	+	0.825445i	$0.690924\pi$
$858$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	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	$-0.414574\pi$
$883$	13915.1	0.530329	0.265164	+	0.964203i	$0.414574\pi$
$884$	0	0
$885$	0	0
$886$	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
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	−22734.4	−0.832285	−0.416143	−	0.909299i	$-0.636618\pi$
$907$	−22734.4	−0.832285	−0.416143	+	0.909299i	$0.636618\pi$
$908$	0	0
$909$	−92744.3	−3.38408
$910$	0	0
$911$	−32714.8	−1.18978	−0.594890	−	0.803807i	$-0.702805\pi$
$911$	−32714.8	−1.18978	−0.594890	+	0.803807i	$0.702805\pi$
$912$	0	0
$913$	63005.6	2.28388
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31582.3	1.13734
$918$	0	0
$919$	−41077.0	−1.47443	−0.737217	−	0.675656i	$-0.763861\pi$
$919$	−41077.0	−1.47443	−0.737217	+	0.675656i	$0.763861\pi$
$920$	0	0
$921$	−61380.3	−2.19604
$922$	0	0
$923$	47179.3	1.68247
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	17120.7	0.596915	0.298457	−	0.954423i	$-0.403528\pi$
$937$	17120.7	0.596915	0.298457	+	0.954423i	$0.403528\pi$
$938$	0	0
$939$	−18266.6	−0.634832
$940$	0	0
$941$	30687.2	1.06310	0.531549	−	0.847028i	$-0.321610\pi$
$941$	30687.2	1.06310	0.531549	+	0.847028i	$0.321610\pi$
$942$	0	0
$943$	891.989	0.0308029
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36649.9	1.25761	0.628807	−	0.777561i	$-0.283544\pi$
$947$	36649.9	1.25761	0.628807	+	0.777561i	$0.283544\pi$
$948$	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	$-0.632366\pi$
$953$	−23768.8	−0.807919	−0.403959	+	0.914777i	$0.632366\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	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
$975$	0	0
$976$	0	0
$977$	26322.4	0.861954	0.430977	−	0.902363i	$-0.358169\pi$
$977$	26322.4	0.861954	0.430977	+	0.902363i	$0.358169\pi$
$978$	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
$985$	0	0
$986$	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	$-0.407431\pi$
$991$	17890.3	0.573464	0.286732	+	0.958011i	$0.407431\pi$
$992$	0	0
$993$	−104458.	−3.33824
$994$	0	0
$995$	0	0
$996$	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

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.cn.1.2		2
4.3	odd	2		1600.4.a.cd.1.1		2
5.4	even	2		320.4.a.o.1.1		2
8.3	odd	2		800.4.a.s.1.2		2
8.5	even	2		800.4.a.m.1.1		2
20.19	odd	2		320.4.a.s.1.2		2
40.3	even	4		800.4.c.i.449.4		4
40.13	odd	4		800.4.c.k.449.1		4
40.19	odd	2		160.4.a.c.1.1	✓	2
40.27	even	4		800.4.c.i.449.1		4
40.29	even	2		160.4.a.g.1.2	yes	2
40.37	odd	4		800.4.c.k.449.4		4
80.19	odd	4		1280.4.d.x.641.4		4
80.29	even	4		1280.4.d.q.641.1		4
80.59	odd	4		1280.4.d.x.641.1		4
80.69	even	4		1280.4.d.q.641.4		4
120.29	odd	2		1440.4.a.x.1.1		2
120.59	even	2		1440.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.c.1.1	✓	2	40.19	odd	2
160.4.a.g.1.2	yes	2	40.29	even	2
320.4.a.o.1.1		2	5.4	even	2
320.4.a.s.1.2		2	20.19	odd	2
800.4.a.m.1.1		2	8.5	even	2
800.4.a.s.1.2		2	8.3	odd	2
800.4.c.i.449.1		4	40.27	even	4
800.4.c.i.449.4		4	40.3	even	4
800.4.c.k.449.1		4	40.13	odd	4
800.4.c.k.449.4		4	40.37	odd	4
1280.4.d.q.641.1		4	80.29	even	4
1280.4.d.q.641.4		4	80.69	even	4
1280.4.d.x.641.1		4	80.59	odd	4
1280.4.d.x.641.4		4	80.19	odd	4
1440.4.a.t.1.2		2	120.59	even	2
1440.4.a.x.1.1		2	120.29	odd	2
1600.4.a.cd.1.1		2	4.3	odd	2
1600.4.a.cn.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1600.4.a.cn.1.2

Level	$1600$
Weight	$4$
Character	1600.1
Self dual	yes
Analytic conductor	$94.403$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$