LMFDB - Embedded newform 441.4.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,4,Mod(1,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-2,0,34,0,0,0,-66,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.0198423125$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.94534$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.53113 q^{2} +4.46887 q^{4} -2.07730 q^{5} -12.4689 q^{8} -7.33521 q^{10} -49.1868 q^{11} +44.8559 q^{13} -79.7802 q^{16} +26.5179 q^{17} -77.7350 q^{19} -9.28317 q^{20} -173.685 q^{22} -55.7510 q^{23} -120.685 q^{25} +158.392 q^{26} -121.436 q^{29} -305.553 q^{31} -181.963 q^{32} +93.6380 q^{34} +77.1868 q^{37} -274.492 q^{38} +25.9016 q^{40} +248.720 q^{41} -147.179 q^{43} -219.809 q^{44} -196.864 q^{46} +269.851 q^{47} -426.154 q^{50} +200.455 q^{52} +141.121 q^{53} +102.176 q^{55} -428.805 q^{58} -424.834 q^{59} +587.996 q^{61} -1078.95 q^{62} -4.29373 q^{64} -93.1790 q^{65} -179.634 q^{67} +118.505 q^{68} -674.872 q^{71} -237.489 q^{73} +272.556 q^{74} -347.388 q^{76} +495.852 q^{79} +165.727 q^{80} +878.262 q^{82} -24.4406 q^{83} -55.0855 q^{85} -519.708 q^{86} +613.304 q^{88} +1072.29 q^{89} -249.144 q^{92} +952.877 q^{94} +161.479 q^{95} -1667.43 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 34 q^{4} - 66 q^{8} - 100 q^{11} - 174 q^{16} - 340 q^{22} - 352 q^{23} - 128 q^{25} - 260 q^{29} + 30 q^{32} + 212 q^{37} + 540 q^{43} - 460 q^{44} + 696 q^{46} - 1366 q^{50} - 16 q^{53}+ \cdots - 2192 q^{95}+O(q^{100}) $ 4 * q - 2 * q^2 + 34 * q^4 - 66 * q^8 - 100 * q^11 - 174 * q^16 - 340 * q^22 - 352 * q^23 - 128 * q^25 - 260 * q^29 + 30 * q^32 + 212 * q^37 + 540 * q^43 - 460 * q^44 + 696 * q^46 - 1366 * q^50 - 16 * q^53 - 780 * q^58 - 1678 * q^64 + 756 * q^65 - 1944 * q^67 - 2248 * q^71 + 284 * q^74 - 1048 * q^79 - 3284 * q^85 - 4820 * q^86 + 1260 * q^88 - 3512 * q^92 - 2192 * q^95

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 $

$2$	3.53113	1.24844	0.624221	−	0.781248i	$-0.285416\pi$
$2$	3.53113	1.24844	0.624221	+	0.781248i	$0.285416\pi$
$3$	0	0
$3$	0	0
$4$	4.46887	0.558609
$5$	−2.07730	−0.185799	−0.0928996	−	0.995675i	$-0.529614\pi$
$5$	−2.07730	−0.185799	−0.0928996	+	0.995675i	$0.529614\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−12.4689	−0.551051
$9$	0	0
$10$	−7.33521	−0.231960
$11$	−49.1868	−1.34822	−0.674108	−	0.738633i	$-0.735472\pi$
$11$	−49.1868	−1.34822	−0.674108	+	0.738633i	$0.735472\pi$
$12$	0	0
$13$	44.8559	0.956983	0.478492	−	0.878092i	$-0.341184\pi$
$13$	44.8559	0.956983	0.478492	+	0.878092i	$0.341184\pi$
$14$	0	0
$15$	0	0
$16$	−79.7802	−1.24656
$17$	26.5179	0.378325	0.189163	−	0.981946i	$-0.439423\pi$
$17$	26.5179	0.378325	0.189163	+	0.981946i	$0.439423\pi$
$18$	0	0
$19$	−77.7350	−0.938612	−0.469306	−	0.883036i	$-0.655496\pi$
$19$	−77.7350	−0.938612	−0.469306	+	0.883036i	$0.655496\pi$
$20$	−9.28317	−0.103789
$21$	0	0
$22$	−173.685	−1.68317
$23$	−55.7510	−0.505430	−0.252715	−	0.967541i	$-0.581323\pi$
$23$	−55.7510	−0.505430	−0.252715	+	0.967541i	$0.581323\pi$
$24$	0	0
$25$	−120.685	−0.965479
$26$	158.392	1.19474
$27$	0	0
$28$	0	0
$29$	−121.436	−0.777588	−0.388794	−	0.921325i	$-0.627108\pi$
$29$	−121.436	−0.777588	−0.388794	+	0.921325i	$0.627108\pi$
$30$	0	0
$31$	−305.553	−1.77029	−0.885143	−	0.465319i	$-0.845940\pi$
$31$	−305.553	−1.77029	−0.885143	+	0.465319i	$0.845940\pi$
$32$	−181.963	−1.00521
$33$	0	0
$34$	93.6380	0.472317
$35$	0	0
$36$	0	0
$37$	77.1868	0.342957	0.171479	−	0.985188i	$-0.445146\pi$
$37$	77.1868	0.342957	0.171479	+	0.985188i	$0.445146\pi$
$38$	−274.492	−1.17180
$39$	0	0
$40$	25.9016	0.102385
$41$	248.720	0.947403	0.473702	−	0.880685i	$-0.342917\pi$
$41$	248.720	0.947403	0.473702	+	0.880685i	$0.342917\pi$
$42$	0	0
$43$	−147.179	−0.521967	−0.260984	−	0.965343i	$-0.584047\pi$
$43$	−147.179	−0.521967	−0.260984	+	0.965343i	$0.584047\pi$
$44$	−219.809	−0.753125
$45$	0	0
$46$	−196.864	−0.631000
$47$	269.851	0.837484	0.418742	−	0.908105i	$-0.362471\pi$
$47$	269.851	0.837484	0.418742	+	0.908105i	$0.362471\pi$
$48$	0	0
$49$	0	0
$50$	−426.154	−1.20534
$51$	0	0
$52$	200.455	0.534579
$53$	141.121	0.365744	0.182872	−	0.983137i	$-0.441461\pi$
$53$	141.121	0.365744	0.182872	+	0.983137i	$0.441461\pi$
$54$	0	0
$55$	102.176	0.250497
$56$	0	0
$57$	0	0
$58$	−428.805	−0.970774
$59$	−424.834	−0.937434	−0.468717	−	0.883348i	$-0.655284\pi$
$59$	−424.834	−0.937434	−0.468717	+	0.883348i	$0.655284\pi$
$60$	0	0
$61$	587.996	1.23418	0.617092	−	0.786891i	$-0.288311\pi$
$61$	587.996	1.23418	0.617092	+	0.786891i	$0.288311\pi$
$62$	−1078.95	−2.21010
$63$	0	0
$64$	−4.29373	−0.00838618
$65$	−93.1790	−0.177807
$66$	0	0
$67$	−179.634	−0.327549	−0.163775	−	0.986498i	$-0.552367\pi$
$67$	−179.634	−0.327549	−0.163775	+	0.986498i	$0.552367\pi$
$68$	118.505	0.211336
$69$	0	0
$70$	0	0
$71$	−674.872	−1.12806	−0.564032	−	0.825753i	$-0.690750\pi$
$71$	−674.872	−1.12806	−0.564032	+	0.825753i	$0.690750\pi$
$72$	0	0
$73$	−237.489	−0.380767	−0.190383	−	0.981710i	$-0.560973\pi$
$73$	−237.489	−0.380767	−0.190383	+	0.981710i	$0.560973\pi$
$74$	272.556	0.428163
$75$	0	0
$76$	−347.388	−0.524317
$77$	0	0
$78$	0	0
$79$	495.852	0.706174	0.353087	−	0.935591i	$-0.385132\pi$
$79$	495.852	0.706174	0.353087	+	0.935591i	$0.385132\pi$
$80$	165.727	0.231611
$81$	0	0
$82$	878.262	1.18278
$83$	−24.4406	−0.0323217	−0.0161609	−	0.999869i	$-0.505144\pi$
$83$	−24.4406	−0.0323217	−0.0161609	+	0.999869i	$0.505144\pi$
$84$	0	0
$85$	−55.0855	−0.0702925
$86$	−519.708	−0.651646
$87$	0	0
$88$	613.304	0.742936
$89$	1072.29	1.27710	0.638552	−	0.769579i	$-0.279534\pi$
$89$	1072.29	1.27710	0.638552	+	0.769579i	$0.279534\pi$
$90$	0	0
$91$	0	0
$92$	−249.144	−0.282337
$93$	0	0
$94$	952.877	1.04555
$95$	161.479	0.174393
$96$	0	0
$97$	−1667.43	−1.74538	−0.872690	−	0.488275i	$-0.837626\pi$
$97$	−1667.43	−1.74538	−0.872690	+	0.488275i	$0.837626\pi$
$98$	0	0
$99$	0	0
$100$	−539.325	−0.539325
$101$	−77.1187	−0.0759762	−0.0379881	−	0.999278i	$-0.512095\pi$
$101$	−77.1187	−0.0759762	−0.0379881	+	0.999278i	$0.512095\pi$
$102$	0	0
$103$	−164.693	−0.157550	−0.0787749	−	0.996892i	$-0.525101\pi$
$103$	−164.693	−0.157550	−0.0787749	+	0.996892i	$0.525101\pi$
$104$	−559.302	−0.527347
$105$	0	0
$106$	498.315	0.456610
$107$	−1022.62	−0.923931	−0.461966	−	0.886898i	$-0.652856\pi$
$107$	−1022.62	−0.923931	−0.461966	+	0.886898i	$0.652856\pi$
$108$	0	0
$109$	1362.52	1.19730	0.598649	−	0.801011i	$-0.295704\pi$
$109$	1362.52	1.19730	0.598649	+	0.801011i	$0.295704\pi$
$110$	360.795	0.312731
$111$	0	0
$112$	0	0
$113$	1538.41	1.28072	0.640360	−	0.768075i	$-0.278785\pi$
$113$	1538.41	1.28072	0.640360	+	0.768075i	$0.278785\pi$
$114$	0	0
$115$	115.811	0.0939084
$116$	−542.681	−0.434368
$117$	0	0
$118$	−1500.14	−1.17033
$119$	0	0
$120$	0	0
$121$	1088.34	0.817685
$122$	2076.29	1.54081
$123$	0	0
$124$	−1365.48	−0.988898
$125$	510.360	0.365184
$126$	0	0
$127$	−170.358	−0.119030	−0.0595151	−	0.998227i	$-0.518955\pi$
$127$	−170.358	−0.119030	−0.0595151	+	0.998227i	$0.518955\pi$
$128$	1440.54	0.994744
$129$	0	0
$130$	−329.027	−0.221981
$131$	−751.935	−0.501503	−0.250751	−	0.968051i	$-0.580678\pi$
$131$	−751.935	−0.501503	−0.250751	+	0.968051i	$0.580678\pi$
$132$	0	0
$133$	0	0
$134$	−634.312	−0.408927
$135$	0	0
$136$	−330.648	−0.208477
$137$	−518.623	−0.323423	−0.161711	−	0.986838i	$-0.551701\pi$
$137$	−518.623	−0.323423	−0.161711	+	0.986838i	$0.551701\pi$
$138$	0	0
$139$	2975.72	1.81581	0.907905	−	0.419177i	$-0.137681\pi$
$139$	2975.72	1.81581	0.907905	+	0.419177i	$0.137681\pi$
$140$	0	0
$141$	0	0
$142$	−2383.06	−1.40832
$143$	−2206.32	−1.29022
$144$	0	0
$145$	252.258	0.144475
$146$	−838.604	−0.475365
$147$	0	0
$148$	344.938	0.191579
$149$	2717.94	1.49438	0.747188	−	0.664612i	$-0.231403\pi$
$149$	2717.94	1.49438	0.747188	+	0.664612i	$0.231403\pi$
$150$	0	0
$151$	707.650	0.381376	0.190688	−	0.981651i	$-0.438928\pi$
$151$	707.650	0.381376	0.190688	+	0.981651i	$0.438928\pi$
$152$	969.267	0.517224
$153$	0	0
$154$	0	0
$155$	634.724	0.328918
$156$	0	0
$157$	−3117.91	−1.58495	−0.792473	−	0.609906i	$-0.791207\pi$
$157$	−3117.91	−1.58495	−0.792473	+	0.609906i	$0.791207\pi$
$158$	1750.92	0.881618
$159$	0	0
$160$	377.991	0.186768
$161$	0	0
$162$	0	0
$163$	1808.77	0.869167	0.434583	−	0.900632i	$-0.356896\pi$
$163$	1808.77	0.869167	0.434583	+	0.900632i	$0.356896\pi$
$164$	1111.50	0.529228
$165$	0	0
$166$	−86.3028	−0.0403518
$167$	3147.38	1.45839	0.729197	−	0.684303i	$-0.239894\pi$
$167$	3147.38	1.45839	0.729197	+	0.684303i	$0.239894\pi$
$168$	0	0
$169$	−184.949	−0.0841827
$170$	−194.514	−0.0877562
$171$	0	0
$172$	−657.724	−0.291576
$173$	−3284.36	−1.44338	−0.721691	−	0.692215i	$-0.756635\pi$
$173$	−3284.36	−1.44338	−0.721691	+	0.692215i	$0.756635\pi$
$174$	0	0
$175$	0	0
$176$	3924.13	1.68064
$177$	0	0
$178$	3786.39	1.59439
$179$	−2798.83	−1.16868	−0.584341	−	0.811508i	$-0.698647\pi$
$179$	−2798.83	−1.16868	−0.584341	+	0.811508i	$0.698647\pi$
$180$	0	0
$181$	3723.04	1.52890	0.764451	−	0.644682i	$-0.223010\pi$
$181$	3723.04	1.52890	0.764451	+	0.644682i	$0.223010\pi$
$182$	0	0
$183$	0	0
$184$	695.152	0.278518
$185$	−160.340	−0.0637212
$186$	0	0
$187$	−1304.33	−0.510064
$188$	1205.93	0.467826
$189$	0	0
$190$	570.202	0.217720
$191$	−959.650	−0.363549	−0.181774	−	0.983340i	$-0.558184\pi$
$191$	−959.650	−0.363549	−0.181774	+	0.983340i	$0.558184\pi$
$192$	0	0
$193$	−3790.25	−1.41362	−0.706808	−	0.707406i	$-0.749865\pi$
$193$	−3790.25	−1.41362	−0.706808	+	0.707406i	$0.749865\pi$
$194$	−5887.91	−2.17901
$195$	0	0
$196$	0	0
$197$	−5117.99	−1.85097	−0.925487	−	0.378779i	$-0.876344\pi$
$197$	−5117.99	−1.85097	−0.925487	+	0.378779i	$0.876344\pi$
$198$	0	0
$199$	−864.855	−0.308080	−0.154040	−	0.988065i	$-0.549228\pi$
$199$	−864.855	−0.308080	−0.154040	+	0.988065i	$0.549228\pi$
$200$	1504.80	0.532028
$201$	0	0
$202$	−272.316	−0.0948519
$203$	0	0
$204$	0	0
$205$	−516.665	−0.176027
$206$	−581.550	−0.196692
$207$	0	0
$208$	−3578.61	−1.19294
$209$	3823.53	1.26545
$210$	0	0
$211$	−1344.61	−0.438707	−0.219353	−	0.975645i	$-0.570395\pi$
$211$	−1344.61	−0.438707	−0.219353	+	0.975645i	$0.570395\pi$
$212$	630.650	0.204308
$213$	0	0
$214$	−3611.01	−1.15348
$215$	305.735	0.0969811
$216$	0	0
$217$	0	0
$218$	4811.23	1.49476
$219$	0	0
$220$	456.609	0.139930
$221$	1189.48	0.362051
$222$	0	0
$223$	864.916	0.259727	0.129863	−	0.991532i	$-0.458546\pi$
$223$	864.916	0.259727	0.129863	+	0.991532i	$0.458546\pi$
$224$	0	0
$225$	0	0
$226$	5432.32	1.59890
$227$	1715.34	0.501548	0.250774	−	0.968046i	$-0.419315\pi$
$227$	1715.34	0.501548	0.250774	+	0.968046i	$0.419315\pi$
$228$	0	0
$229$	−1045.46	−0.301685	−0.150842	−	0.988558i	$-0.548199\pi$
$229$	−1045.46	−0.301685	−0.150842	+	0.988558i	$0.548199\pi$
$230$	408.945	0.117239
$231$	0	0
$232$	1514.17	0.428491
$233$	−1448.67	−0.407320	−0.203660	−	0.979042i	$-0.565284\pi$
$233$	−1448.67	−0.407320	−0.203660	+	0.979042i	$0.565284\pi$
$234$	0	0
$235$	−560.560	−0.155604
$236$	−1898.53	−0.523659
$237$	0	0
$238$	0	0
$239$	3153.12	0.853383	0.426691	−	0.904397i	$-0.359679\pi$
$239$	3153.12	0.853383	0.426691	+	0.904397i	$0.359679\pi$
$240$	0	0
$241$	−381.012	−0.101839	−0.0509194	−	0.998703i	$-0.516215\pi$
$241$	−381.012	−0.101839	−0.0509194	+	0.998703i	$0.516215\pi$
$242$	3843.06	1.02083
$243$	0	0
$244$	2627.68	0.689426
$245$	0	0
$246$	0	0
$247$	−3486.87	−0.898236
$248$	3809.90	0.975519
$249$	0	0
$250$	1802.15	0.455912
$251$	−3776.23	−0.949617	−0.474808	−	0.880089i	$-0.657483\pi$
$251$	−3776.23	−0.949617	−0.474808	+	0.880089i	$0.657483\pi$
$252$	0	0
$253$	2742.21	0.681428
$254$	−601.556	−0.148602
$255$	0	0
$256$	5121.09	1.25027
$257$	−4258.42	−1.03359	−0.516795	−	0.856109i	$-0.672875\pi$
$257$	−4258.42	−1.03359	−0.516795	+	0.856109i	$0.672875\pi$
$258$	0	0
$259$	0	0
$260$	−416.405	−0.0993244
$261$	0	0
$262$	−2655.18	−0.626098
$263$	−4198.83	−0.984451	−0.492226	−	0.870468i	$-0.663816\pi$
$263$	−4198.83	−0.984451	−0.492226	+	0.870468i	$0.663816\pi$
$264$	0	0
$265$	−293.150	−0.0679548
$266$	0	0
$267$	0	0
$268$	−802.762	−0.182972
$269$	−3740.59	−0.847835	−0.423917	−	0.905701i	$-0.639345\pi$
$269$	−3740.59	−0.847835	−0.423917	+	0.905701i	$0.639345\pi$
$270$	0	0
$271$	4356.30	0.976480	0.488240	−	0.872709i	$-0.337639\pi$
$271$	4356.30	0.976480	0.488240	+	0.872709i	$0.337639\pi$
$272$	−2115.60	−0.471607
$273$	0	0
$274$	−1831.32	−0.403775
$275$	5936.10	1.30167
$276$	0	0
$277$	−1344.30	−0.291593	−0.145797	−	0.989315i	$-0.546575\pi$
$277$	−1344.30	−0.291593	−0.145797	+	0.989315i	$0.546575\pi$
$278$	10507.7	2.26693
$279$	0	0
$280$	0	0
$281$	−4205.54	−0.892817	−0.446408	−	0.894829i	$-0.647297\pi$
$281$	−4205.54	−0.892817	−0.446408	+	0.894829i	$0.647297\pi$
$282$	0	0
$283$	4752.03	0.998159	0.499079	−	0.866556i	$-0.333672\pi$
$283$	4752.03	0.998159	0.499079	+	0.866556i	$0.333672\pi$
$284$	−3015.91	−0.630146
$285$	0	0
$286$	−7790.79	−1.61077
$287$	0	0
$288$	0	0
$289$	−4209.80	−0.856870
$290$	890.757	0.180369
$291$	0	0
$292$	−1061.31	−0.212700
$293$	4961.17	0.989196	0.494598	−	0.869122i	$-0.335315\pi$
$293$	4961.17	0.989196	0.494598	+	0.869122i	$0.335315\pi$
$294$	0	0
$295$	882.506	0.174174
$296$	−962.432	−0.188987
$297$	0	0
$298$	9597.39	1.86564
$299$	−2500.76	−0.483688
$300$	0	0
$301$	0	0
$302$	2498.80	0.476126
$303$	0	0
$304$	6201.71	1.17004
$305$	−1221.44	−0.229310
$306$	0	0
$307$	−4234.00	−0.787124	−0.393562	−	0.919298i	$-0.628757\pi$
$307$	−4234.00	−0.787124	−0.393562	+	0.919298i	$0.628757\pi$
$308$	0	0
$309$	0	0
$310$	2241.29	0.410635
$311$	684.700	0.124842	0.0624209	−	0.998050i	$-0.480118\pi$
$311$	684.700	0.124842	0.0624209	+	0.998050i	$0.480118\pi$
$312$	0	0
$313$	−5944.07	−1.07341	−0.536707	−	0.843768i	$-0.680332\pi$
$313$	−5944.07	−1.07341	−0.536707	+	0.843768i	$0.680332\pi$
$314$	−11009.8	−1.97872
$315$	0	0
$316$	2215.90	0.394475
$317$	2823.89	0.500333	0.250166	−	0.968203i	$-0.419515\pi$
$317$	2823.89	0.500333	0.250166	+	0.968203i	$0.419515\pi$
$318$	0	0
$319$	5973.04	1.04836
$320$	8.91935	0.00155815
$321$	0	0
$322$	0	0
$323$	−2061.37	−0.355101
$324$	0	0
$325$	−5413.43	−0.923947
$326$	6387.02	1.08510
$327$	0	0
$328$	−3101.26	−0.522068
$329$	0	0
$330$	0	0
$331$	−2812.97	−0.467114	−0.233557	−	0.972343i	$-0.575037\pi$
$331$	−2812.97	−0.467114	−0.233557	+	0.972343i	$0.575037\pi$
$332$	−109.222	−0.0180552
$333$	0	0
$334$	11113.8	1.82072
$335$	373.154	0.0608584
$336$	0	0
$337$	4260.10	0.688612	0.344306	−	0.938857i	$-0.388114\pi$
$337$	4260.10	0.688612	0.344306	+	0.938857i	$0.388114\pi$
$338$	−653.080	−0.105097
$339$	0	0
$340$	−246.170	−0.0392660
$341$	15029.2	2.38673
$342$	0	0
$343$	0	0
$344$	1835.16	0.287631
$345$	0	0
$346$	−11597.5	−1.80198
$347$	−36.0584	−0.00557843	−0.00278922	−	0.999996i	$-0.500888\pi$
$347$	−36.0584	−0.00557843	−0.00278922	+	0.999996i	$0.500888\pi$
$348$	0	0
$349$	−242.692	−0.0372236	−0.0186118	−	0.999827i	$-0.505925\pi$
$349$	−242.692	−0.0372236	−0.0186118	+	0.999827i	$0.505925\pi$
$350$	0	0
$351$	0	0
$352$	8950.18	1.35524
$353$	109.990	0.0165840	0.00829201	−	0.999966i	$-0.497361\pi$
$353$	109.990	0.0165840	0.00829201	+	0.999966i	$0.497361\pi$
$354$	0	0
$355$	1401.91	0.209593
$356$	4791.92	0.713402
$357$	0	0
$358$	−9883.01	−1.45903
$359$	−12404.5	−1.82363	−0.911814	−	0.410604i	$-0.865318\pi$
$359$	−12404.5	−1.82363	−0.911814	+	0.410604i	$0.865318\pi$
$360$	0	0
$361$	−816.273	−0.119008
$362$	13146.5	1.90875
$363$	0	0
$364$	0	0
$365$	493.335	0.0707461
$366$	0	0
$367$	13859.6	1.97130	0.985649	−	0.168807i	$-0.0539913\pi$
$367$	13859.6	1.97130	0.985649	+	0.168807i	$0.0539913\pi$
$368$	4447.82	0.630051
$369$	0	0
$370$	−566.181	−0.0795523
$371$	0	0
$372$	0	0
$373$	4898.06	0.679925	0.339963	−	0.940439i	$-0.389586\pi$
$373$	4898.06	0.679925	0.339963	+	0.940439i	$0.389586\pi$
$374$	−4605.75	−0.636786
$375$	0	0
$376$	−3364.73	−0.461497
$377$	−5447.11	−0.744139
$378$	0	0
$379$	−9806.25	−1.32906	−0.664530	−	0.747262i	$-0.731368\pi$
$379$	−9806.25	−1.32906	−0.664530	+	0.747262i	$0.731368\pi$
$380$	721.627	0.0974176
$381$	0	0
$382$	−3388.65	−0.453870
$383$	−10729.7	−1.43149	−0.715746	−	0.698361i	$-0.753913\pi$
$383$	−10729.7	−1.43149	−0.715746	+	0.698361i	$0.753913\pi$
$384$	0	0
$385$	0	0
$386$	−13383.8	−1.76482
$387$	0	0
$388$	−7451.53	−0.974984
$389$	−5264.05	−0.686113	−0.343057	−	0.939315i	$-0.611462\pi$
$389$	−5264.05	−0.686113	−0.343057	+	0.939315i	$0.611462\pi$
$390$	0	0
$391$	−1478.40	−0.191217
$392$	0	0
$393$	0	0
$394$	−18072.3	−2.31083
$395$	−1030.03	−0.131206
$396$	0	0
$397$	1214.90	0.153587	0.0767935	−	0.997047i	$-0.475532\pi$
$397$	1214.90	0.153587	0.0767935	+	0.997047i	$0.475532\pi$
$398$	−3053.91	−0.384620
$399$	0	0
$400$	9628.26	1.20353
$401$	−2295.45	−0.285859	−0.142929	−	0.989733i	$-0.545652\pi$
$401$	−2295.45	−0.285859	−0.142929	+	0.989733i	$0.545652\pi$
$402$	0	0
$403$	−13705.8	−1.69413
$404$	−344.633	−0.0424410
$405$	0	0
$406$	0	0
$407$	−3796.57	−0.462381
$408$	0	0
$409$	4646.54	0.561753	0.280876	−	0.959744i	$-0.409375\pi$
$409$	4646.54	0.561753	0.280876	+	0.959744i	$0.409375\pi$
$410$	−1824.41	−0.219759
$411$	0	0
$412$	−735.990	−0.0880087
$413$	0	0
$414$	0	0
$415$	50.7703	0.00600535
$416$	−8162.11	−0.961973
$417$	0	0
$418$	13501.4	1.57984
$419$	−7541.24	−0.879269	−0.439634	−	0.898177i	$-0.644892\pi$
$419$	−7541.24	−0.879269	−0.439634	+	0.898177i	$0.644892\pi$
$420$	0	0
$421$	−6243.63	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$421$	−6243.63	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$422$	−4748.01	−0.547700
$423$	0	0
$424$	−1759.62	−0.201544
$425$	−3200.31	−0.365265
$426$	0	0
$427$	0	0
$428$	−4569.97	−0.516116
$429$	0	0
$430$	1079.59	0.121075
$431$	−11465.8	−1.28141	−0.640706	−	0.767786i	$-0.721358\pi$
$431$	−11465.8	−1.28141	−0.640706	+	0.767786i	$0.721358\pi$
$432$	0	0
$433$	5156.40	0.572289	0.286144	−	0.958187i	$-0.407626\pi$
$433$	5156.40	0.572289	0.286144	+	0.958187i	$0.407626\pi$
$434$	0	0
$435$	0	0
$436$	6088.92	0.668822
$437$	4333.80	0.474402
$438$	0	0
$439$	−5064.25	−0.550577	−0.275289	−	0.961362i	$-0.588773\pi$
$439$	−5064.25	−0.550577	−0.275289	+	0.961362i	$0.588773\pi$
$440$	−1274.01	−0.138037
$441$	0	0
$442$	4200.22	0.452000
$443$	−12703.6	−1.36246	−0.681228	−	0.732071i	$-0.738554\pi$
$443$	−12703.6	−1.36246	−0.681228	+	0.732071i	$0.738554\pi$
$444$	0	0
$445$	−2227.46	−0.237285
$446$	3054.13	0.324254
$447$	0	0
$448$	0	0
$449$	−13942.2	−1.46542	−0.732709	−	0.680542i	$-0.761744\pi$
$449$	−13942.2	−1.46542	−0.732709	+	0.680542i	$0.761744\pi$
$450$	0	0
$451$	−12233.7	−1.27730
$452$	6874.95	0.715421
$453$	0	0
$454$	6057.10	0.626154
$455$	0	0
$456$	0	0
$457$	−15214.0	−1.55729	−0.778646	−	0.627464i	$-0.784093\pi$
$457$	−15214.0	−1.55729	−0.778646	+	0.627464i	$0.784093\pi$
$458$	−3691.65	−0.376636
$459$	0	0
$460$	517.546	0.0524581
$461$	11430.2	1.15479	0.577394	−	0.816465i	$-0.304070\pi$
$461$	11430.2	1.15479	0.577394	+	0.816465i	$0.304070\pi$
$462$	0	0
$463$	−9347.88	−0.938300	−0.469150	−	0.883119i	$-0.655440\pi$
$463$	−9347.88	−0.938300	−0.469150	+	0.883119i	$0.655440\pi$
$464$	9688.17	0.969314
$465$	0	0
$466$	−5115.44	−0.508515
$467$	−3630.84	−0.359776	−0.179888	−	0.983687i	$-0.557573\pi$
$467$	−3630.84	−0.359776	−0.179888	+	0.983687i	$0.557573\pi$
$468$	0	0
$469$	0	0
$470$	−1979.41	−0.194262
$471$	0	0
$472$	5297.20	0.516575
$473$	7239.26	0.703724
$474$	0	0
$475$	9381.43	0.906210
$476$	0	0
$477$	0	0
$478$	11134.1	1.06540
$479$	−6521.25	−0.622053	−0.311027	−	0.950401i	$-0.600673\pi$
$479$	−6521.25	−0.622053	−0.311027	+	0.950401i	$0.600673\pi$
$480$	0	0
$481$	3462.28	0.328205
$482$	−1345.40	−0.127140
$483$	0	0
$484$	4863.65	0.456766
$485$	3463.75	0.324290
$486$	0	0
$487$	−3666.29	−0.341140	−0.170570	−	0.985346i	$-0.554561\pi$
$487$	−3666.29	−0.341140	−0.170570	+	0.985346i	$0.554561\pi$
$488$	−7331.65	−0.680099
$489$	0	0
$490$	0	0
$491$	12470.7	1.14623	0.573113	−	0.819476i	$-0.305736\pi$
$491$	12470.7	1.14623	0.573113	+	0.819476i	$0.305736\pi$
$492$	0	0
$493$	−3220.22	−0.294181
$494$	−12312.6	−1.12140
$495$	0	0
$496$	24377.0	2.20678
$497$	0	0
$498$	0	0
$499$	−2303.93	−0.206690	−0.103345	−	0.994646i	$-0.532955\pi$
$499$	−2303.93	−0.206690	−0.103345	+	0.994646i	$0.532955\pi$
$500$	2280.74	0.203995
$501$	0	0
$502$	−13334.4	−1.18554
$503$	10520.4	0.932570	0.466285	−	0.884635i	$-0.345592\pi$
$503$	10520.4	0.932570	0.466285	+	0.884635i	$0.345592\pi$
$504$	0	0
$505$	160.198	0.0141163
$506$	9683.10	0.850724
$507$	0	0
$508$	−761.308	−0.0664913
$509$	9662.22	0.841395	0.420698	−	0.907201i	$-0.361785\pi$
$509$	9662.22	0.841395	0.420698	+	0.907201i	$0.361785\pi$
$510$	0	0
$511$	0	0
$512$	6558.89	0.566142
$513$	0	0
$514$	−15037.0	−1.29038
$515$	342.115	0.0292726
$516$	0	0
$517$	−13273.1	−1.12911
$518$	0	0
$519$	0	0
$520$	1161.84	0.0979806
$521$	−8607.81	−0.723829	−0.361914	−	0.932211i	$-0.617877\pi$
$521$	−8607.81	−0.723829	−0.361914	+	0.932211i	$0.617877\pi$
$522$	0	0
$523$	−10482.7	−0.876439	−0.438219	−	0.898868i	$-0.644391\pi$
$523$	−10482.7	−0.876439	−0.438219	+	0.898868i	$0.644391\pi$
$524$	−3360.30	−0.280144
$525$	0	0
$526$	−14826.6	−1.22903
$527$	−8102.61	−0.669744
$528$	0	0
$529$	−9058.83	−0.744541
$530$	−1035.15	−0.0848377
$531$	0	0
$532$	0	0
$533$	11156.6	0.906649
$534$	0	0
$535$	2124.29	0.171666
$536$	2239.84	0.180497
$537$	0	0
$538$	−13208.5	−1.05847
$539$	0	0
$540$	0	0
$541$	20722.6	1.64683	0.823416	−	0.567438i	$-0.192066\pi$
$541$	20722.6	1.64683	0.823416	+	0.567438i	$0.192066\pi$
$542$	15382.6	1.21908
$543$	0	0
$544$	−4825.27	−0.380298
$545$	−2830.35	−0.222457
$546$	0	0
$547$	−4175.09	−0.326351	−0.163176	−	0.986597i	$-0.552174\pi$
$547$	−4175.09	−0.326351	−0.163176	+	0.986597i	$0.552174\pi$
$548$	−2317.66	−0.180667
$549$	0	0
$550$	20961.1	1.62506
$551$	9439.81	0.729854
$552$	0	0
$553$	0	0
$554$	−4746.91	−0.364038
$555$	0	0
$556$	13298.1	1.01433
$557$	10161.7	0.773011	0.386505	−	0.922287i	$-0.373682\pi$
$557$	10161.7	0.773011	0.386505	+	0.922287i	$0.373682\pi$
$558$	0	0
$559$	−6601.85	−0.499514
$560$	0	0
$561$	0	0
$562$	−14850.3	−1.11463
$563$	17104.4	1.28040	0.640201	−	0.768208i	$-0.278851\pi$
$563$	17104.4	1.28040	0.640201	+	0.768208i	$0.278851\pi$
$564$	0	0
$565$	−3195.73	−0.237957
$566$	16780.0	1.24614
$567$	0	0
$568$	8414.89	0.621621
$569$	18257.6	1.34516	0.672581	−	0.740023i	$-0.265185\pi$
$569$	18257.6	1.34516	0.672581	+	0.740023i	$0.265185\pi$
$570$	0	0
$571$	13630.5	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$571$	13630.5	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$572$	−9859.74	−0.720728
$573$	0	0
$574$	0	0
$575$	6728.30	0.487981
$576$	0	0
$577$	4442.08	0.320496	0.160248	−	0.987077i	$-0.448771\pi$
$577$	4442.08	0.320496	0.160248	+	0.987077i	$0.448771\pi$
$578$	−14865.4	−1.06975
$579$	0	0
$580$	1127.31	0.0807052
$581$	0	0
$582$	0	0
$583$	−6941.27	−0.493101
$584$	2961.22	0.209822
$585$	0	0
$586$	17518.5	1.23495
$587$	−3103.38	−0.218211	−0.109106	−	0.994030i	$-0.534799\pi$
$587$	−3103.38	−0.218211	−0.109106	+	0.994030i	$0.534799\pi$
$588$	0	0
$589$	23752.1	1.66161
$590$	3116.24	0.217447
$591$	0	0
$592$	−6157.97	−0.427519
$593$	−5937.71	−0.411185	−0.205592	−	0.978638i	$-0.565912\pi$
$593$	−5937.71	−0.411185	−0.205592	+	0.978638i	$0.565912\pi$
$594$	0	0
$595$	0	0
$596$	12146.1	0.834772
$597$	0	0
$598$	−8830.50	−0.603856
$599$	2600.33	0.177373	0.0886866	−	0.996060i	$-0.471733\pi$
$599$	2600.33	0.177373	0.0886866	+	0.996060i	$0.471733\pi$
$600$	0	0
$601$	−13881.4	−0.942156	−0.471078	−	0.882092i	$-0.656135\pi$
$601$	−13881.4	−0.942156	−0.471078	+	0.882092i	$0.656135\pi$
$602$	0	0
$603$	0	0
$604$	3162.40	0.213040
$605$	−2260.80	−0.151925
$606$	0	0
$607$	−12284.6	−0.821442	−0.410721	−	0.911761i	$-0.634723\pi$
$607$	−12284.6	−0.821442	−0.410721	+	0.911761i	$0.634723\pi$
$608$	14144.9	0.943505
$609$	0	0
$610$	−4313.07	−0.286281
$611$	12104.4	0.801459
$612$	0	0
$613$	22062.0	1.45363	0.726815	−	0.686833i	$-0.241000\pi$
$613$	22062.0	1.45363	0.726815	+	0.686833i	$0.241000\pi$
$614$	−14950.8	−0.982679
$615$	0	0
$616$	0	0
$617$	12182.2	0.794871	0.397436	−	0.917630i	$-0.369900\pi$
$617$	12182.2	0.794871	0.397436	+	0.917630i	$0.369900\pi$
$618$	0	0
$619$	23248.6	1.50960	0.754799	−	0.655956i	$-0.227734\pi$
$619$	23248.6	1.50960	0.754799	+	0.655956i	$0.227734\pi$
$620$	2836.50	0.183736
$621$	0	0
$622$	2417.76	0.155858
$623$	0	0
$624$	0	0
$625$	14025.4	0.897628
$626$	−20989.3	−1.34010
$627$	0	0
$628$	−13933.6	−0.885365
$629$	2046.83	0.129749
$630$	0	0
$631$	19184.4	1.21033	0.605165	−	0.796100i	$-0.293107\pi$
$631$	19184.4	1.21033	0.605165	+	0.796100i	$0.293107\pi$
$632$	−6182.72	−0.389138
$633$	0	0
$634$	9971.52	0.624637
$635$	353.884	0.0221157
$636$	0	0
$637$	0	0
$638$	21091.6	1.30881
$639$	0	0
$640$	−2992.44	−0.184823
$641$	19433.4	1.19746	0.598730	−	0.800951i	$-0.295672\pi$
$641$	19433.4	1.19746	0.598730	+	0.800951i	$0.295672\pi$
$642$	0	0
$643$	5777.47	0.354341	0.177170	−	0.984180i	$-0.443306\pi$
$643$	5777.47	0.354341	0.177170	+	0.984180i	$0.443306\pi$
$644$	0	0
$645$	0	0
$646$	−7278.95	−0.443323
$647$	29231.5	1.77621	0.888106	−	0.459640i	$-0.152021\pi$
$647$	29231.5	1.77621	0.888106	+	0.459640i	$0.152021\pi$
$648$	0	0
$649$	20896.2	1.26386
$650$	−19115.5	−1.15349
$651$	0	0
$652$	8083.18	0.485524
$653$	7093.27	0.425086	0.212543	−	0.977152i	$-0.431826\pi$
$653$	7093.27	0.425086	0.212543	+	0.977152i	$0.431826\pi$
$654$	0	0
$655$	1561.99	0.0931788
$656$	−19842.9	−1.18100
$657$	0	0
$658$	0	0
$659$	−19014.2	−1.12396	−0.561980	−	0.827151i	$-0.689960\pi$
$659$	−19014.2	−1.12396	−0.561980	+	0.827151i	$0.689960\pi$
$660$	0	0
$661$	−21058.4	−1.23915	−0.619573	−	0.784939i	$-0.712694\pi$
$661$	−21058.4	−1.23915	−0.619573	+	0.784939i	$0.712694\pi$
$662$	−9932.96	−0.583165
$663$	0	0
$664$	304.746	0.0178109
$665$	0	0
$666$	0	0
$667$	6770.16	0.393016
$668$	14065.3	0.814672
$669$	0	0
$670$	1317.65	0.0759782
$671$	−28921.6	−1.66395
$672$	0	0
$673$	9634.87	0.551853	0.275926	−	0.961179i	$-0.411015\pi$
$673$	9634.87	0.551853	0.275926	+	0.961179i	$0.411015\pi$
$674$	15043.0	0.859693
$675$	0	0
$676$	−826.515	−0.0470252
$677$	−8371.31	−0.475237	−0.237619	−	0.971359i	$-0.576367\pi$
$677$	−8371.31	−0.475237	−0.237619	+	0.971359i	$0.576367\pi$
$678$	0	0
$679$	0	0
$680$	686.854	0.0387348
$681$	0	0
$682$	53069.9	2.97969
$683$	12068.8	0.676137	0.338069	−	0.941121i	$-0.390226\pi$
$683$	12068.8	0.676137	0.338069	+	0.941121i	$0.390226\pi$
$684$	0	0
$685$	1077.33	0.0600917
$686$	0	0
$687$	0	0
$688$	11742.0	0.650666
$689$	6330.09	0.350011
$690$	0	0
$691$	2981.29	0.164130	0.0820648	−	0.996627i	$-0.473849\pi$
$691$	2981.29	0.164130	0.0820648	+	0.996627i	$0.473849\pi$
$692$	−14677.4	−0.806286
$693$	0	0
$694$	−127.327	−0.00696435
$695$	−6181.46	−0.337376
$696$	0	0
$697$	6595.53	0.358427
$698$	−856.978	−0.0464715
$699$	0	0
$700$	0	0
$701$	28978.0	1.56132	0.780660	−	0.624956i	$-0.214883\pi$
$701$	28978.0	1.56132	0.780660	+	0.624956i	$0.214883\pi$
$702$	0	0
$703$	−6000.11	−0.321904
$704$	211.195	0.0113064
$705$	0	0
$706$	388.388	0.0207042
$707$	0	0
$708$	0	0
$709$	−16372.4	−0.867249	−0.433625	−	0.901094i	$-0.642766\pi$
$709$	−16372.4	−0.867249	−0.433625	+	0.901094i	$0.642766\pi$
$710$	4950.32	0.261665
$711$	0	0
$712$	−13370.2	−0.703750
$713$	17034.9	0.894755
$714$	0	0
$715$	4583.18	0.239722
$716$	−12507.6	−0.652836
$717$	0	0
$718$	−43801.7	−2.27669
$719$	−23010.5	−1.19353	−0.596765	−	0.802416i	$-0.703547\pi$
$719$	−23010.5	−1.19353	−0.596765	+	0.802416i	$0.703547\pi$
$720$	0	0
$721$	0	0
$722$	−2882.36	−0.148574
$723$	0	0
$724$	16637.8	0.854058
$725$	14655.5	0.750745
$726$	0	0
$727$	−24636.8	−1.25685	−0.628423	−	0.777872i	$-0.716299\pi$
$727$	−24636.8	−1.25685	−0.628423	+	0.777872i	$0.716299\pi$
$728$	0	0
$729$	0	0
$730$	1742.03	0.0883225
$731$	−3902.87	−0.197473
$732$	0	0
$733$	6904.76	0.347931	0.173965	−	0.984752i	$-0.444342\pi$
$733$	6904.76	0.347931	0.173965	+	0.984752i	$0.444342\pi$
$734$	48940.1	2.46105
$735$	0	0
$736$	10144.6	0.508065
$737$	8835.63	0.441607
$738$	0	0
$739$	−9234.89	−0.459690	−0.229845	−	0.973227i	$-0.573822\pi$
$739$	−9234.89	−0.459690	−0.229845	+	0.973227i	$0.573822\pi$
$740$	−716.538	−0.0355952
$741$	0	0
$742$	0	0
$743$	−20216.9	−0.998232	−0.499116	−	0.866535i	$-0.666342\pi$
$743$	−20216.9	−0.998232	−0.499116	+	0.866535i	$0.666342\pi$
$744$	0	0
$745$	−5645.97	−0.277654
$746$	17295.7	0.848847
$747$	0	0
$748$	−5828.88	−0.284926
$749$	0	0
$750$	0	0
$751$	24054.9	1.16881	0.584405	−	0.811462i	$-0.301328\pi$
$751$	24054.9	1.16881	0.584405	+	0.811462i	$0.301328\pi$
$752$	−21528.7	−1.04398
$753$	0	0
$754$	−19234.5	−0.929015
$755$	−1470.00	−0.0708593
$756$	0	0
$757$	−30328.2	−1.45614	−0.728069	−	0.685504i	$-0.759582\pi$
$757$	−30328.2	−1.45614	−0.728069	+	0.685504i	$0.759582\pi$
$758$	−34627.1	−1.65925
$759$	0	0
$760$	−2013.46	−0.0960997
$761$	33834.1	1.61168	0.805839	−	0.592135i	$-0.201715\pi$
$761$	33834.1	1.61168	0.805839	+	0.592135i	$0.201715\pi$
$762$	0	0
$763$	0	0
$764$	−4288.55	−0.203082
$765$	0	0
$766$	−37887.9	−1.78713
$767$	−19056.3	−0.897109
$768$	0	0
$769$	31738.1	1.48830	0.744151	−	0.668011i	$-0.232854\pi$
$769$	31738.1	1.48830	0.744151	+	0.668011i	$0.232854\pi$
$770$	0	0
$771$	0	0
$772$	−16938.1	−0.789658
$773$	−27494.2	−1.27930	−0.639650	−	0.768667i	$-0.720920\pi$
$773$	−27494.2	−1.27930	−0.639650	+	0.768667i	$0.720920\pi$
$774$	0	0
$775$	36875.6	1.70917
$776$	20791.0	0.961794
$777$	0	0
$778$	−18588.0	−0.856573
$779$	−19334.2	−0.889244
$780$	0	0
$781$	33194.8	1.52087
$782$	−5220.41	−0.238723
$783$	0	0
$784$	0	0
$785$	6476.84	0.294482
$786$	0	0
$787$	−468.356	−0.0212136	−0.0106068	−	0.999944i	$-0.503376\pi$
$787$	−468.356	−0.0212136	−0.0106068	+	0.999944i	$0.503376\pi$
$788$	−22871.6	−1.03397
$789$	0	0
$790$	−3637.18	−0.163804
$791$	0	0
$792$	0	0
$793$	26375.1	1.18109
$794$	4289.97	0.191745
$795$	0	0
$796$	−3864.93	−0.172096
$797$	−37723.8	−1.67659	−0.838297	−	0.545214i	$-0.816449\pi$
$797$	−37723.8	−1.67659	−0.838297	+	0.545214i	$0.816449\pi$
$798$	0	0
$799$	7155.87	0.316841
$800$	21960.2	0.970512
$801$	0	0
$802$	−8105.53	−0.356878
$803$	11681.3	0.513356
$804$	0	0
$805$	0	0
$806$	−48397.1	−2.11503
$807$	0	0
$808$	961.583	0.0418668
$809$	7797.13	0.338854	0.169427	−	0.985543i	$-0.445808\pi$
$809$	7797.13	0.338854	0.169427	+	0.985543i	$0.445808\pi$
$810$	0	0
$811$	16925.9	0.732860	0.366430	−	0.930446i	$-0.380580\pi$
$811$	16925.9	0.732860	0.366430	+	0.930446i	$0.380580\pi$
$812$	0	0
$813$	0	0
$814$	−13406.2	−0.577256
$815$	−3757.36	−0.161490
$816$	0	0
$817$	11441.0	0.489925
$818$	16407.5	0.701316
$819$	0	0
$820$	−2308.91	−0.0983301
$821$	30009.3	1.27568	0.637840	−	0.770169i	$-0.279828\pi$
$821$	30009.3	1.27568	0.637840	+	0.770169i	$0.279828\pi$
$822$	0	0
$823$	−23385.6	−0.990486	−0.495243	−	0.868754i	$-0.664921\pi$
$823$	−23385.6	−0.990486	−0.495243	+	0.868754i	$0.664921\pi$
$824$	2053.53	0.0868181
$825$	0	0
$826$	0	0
$827$	37325.9	1.56947	0.784734	−	0.619833i	$-0.212800\pi$
$827$	37325.9	1.56947	0.784734	+	0.619833i	$0.212800\pi$
$828$	0	0
$829$	−24671.3	−1.03362	−0.516809	−	0.856100i	$-0.672880\pi$
$829$	−24671.3	−1.03362	−0.516809	+	0.856100i	$0.672880\pi$
$830$	179.277	0.00749733
$831$	0	0
$832$	−192.599	−0.00802544
$833$	0	0
$834$	0	0
$835$	−6538.05	−0.270969
$836$	17086.9	0.706892
$837$	0	0
$838$	−26629.1	−1.09772
$839$	−14147.4	−0.582147	−0.291074	−	0.956701i	$-0.594013\pi$
$839$	−14147.4	−0.582147	−0.291074	+	0.956701i	$0.594013\pi$
$840$	0	0
$841$	−9642.35	−0.395356
$842$	−22047.1	−0.902366
$843$	0	0
$844$	−6008.91	−0.245065
$845$	384.195	0.0156411
$846$	0	0
$847$	0	0
$848$	−11258.6	−0.455923
$849$	0	0
$850$	−11300.7	−0.456012
$851$	−4303.24	−0.173341
$852$	0	0
$853$	27963.6	1.12246	0.561229	−	0.827661i	$-0.310329\pi$
$853$	27963.6	1.12246	0.561229	+	0.827661i	$0.310329\pi$
$854$	0	0
$855$	0	0
$856$	12750.9	0.509134
$857$	−33855.8	−1.34947	−0.674734	−	0.738061i	$-0.735741\pi$
$857$	−33855.8	−1.34947	−0.674734	+	0.738061i	$0.735741\pi$
$858$	0	0
$859$	−24282.7	−0.964511	−0.482255	−	0.876031i	$-0.660182\pi$
$859$	−24282.7	−0.964511	−0.482255	+	0.876031i	$0.660182\pi$
$860$	1366.29	0.0541745
$861$	0	0
$862$	−40487.2	−1.59977
$863$	19667.0	0.775750	0.387875	−	0.921712i	$-0.373209\pi$
$863$	19667.0	0.775750	0.387875	+	0.921712i	$0.373209\pi$
$864$	0	0
$865$	6822.59	0.268179
$866$	18207.9	0.714469
$867$	0	0
$868$	0	0
$869$	−24389.4	−0.952075
$870$	0	0
$871$	−8057.65	−0.313459
$872$	−16989.1	−0.659773
$873$	0	0
$874$	15303.2	0.592264
$875$	0	0
$876$	0	0
$877$	−36061.0	−1.38848	−0.694238	−	0.719745i	$-0.744259\pi$
$877$	−36061.0	−1.38848	−0.694238	+	0.719745i	$0.744259\pi$
$878$	−17882.5	−0.687364
$879$	0	0
$880$	−8151.58	−0.312261
$881$	15889.7	0.607646	0.303823	−	0.952728i	$-0.401737\pi$
$881$	15889.7	0.607646	0.303823	+	0.952728i	$0.401737\pi$
$882$	0	0
$883$	14861.3	0.566390	0.283195	−	0.959062i	$-0.408606\pi$
$883$	14861.3	0.566390	0.283195	+	0.959062i	$0.408606\pi$
$884$	5315.65	0.202245
$885$	0	0
$886$	−44858.2	−1.70095
$887$	38189.9	1.44565	0.722824	−	0.691032i	$-0.242844\pi$
$887$	38189.9	1.44565	0.722824	+	0.691032i	$0.242844\pi$
$888$	0	0
$889$	0	0
$890$	−7865.45	−0.296237
$891$	0	0
$892$	3865.20	0.145086
$893$	−20976.8	−0.786073
$894$	0	0
$895$	5813.99	0.217140
$896$	0	0
$897$	0	0
$898$	−49231.7	−1.82949
$899$	37105.0	1.37655
$900$	0	0
$901$	3742.22	0.138370
$902$	−43198.9	−1.59464
$903$	0	0
$904$	−19182.2	−0.705742
$905$	−7733.86	−0.284069
$906$	0	0
$907$	16865.5	0.617431	0.308715	−	0.951154i	$-0.400101\pi$
$907$	16865.5	0.617431	0.308715	+	0.951154i	$0.400101\pi$
$908$	7665.65	0.280169
$909$	0	0
$910$	0	0
$911$	−26754.1	−0.973000	−0.486500	−	0.873681i	$-0.661727\pi$
$911$	−26754.1	−0.973000	−0.486500	+	0.873681i	$0.661727\pi$
$912$	0	0
$913$	1202.15	0.0435766
$914$	−53722.7	−1.94419
$915$	0	0
$916$	−4672.02	−0.168524
$917$	0	0
$918$	0	0
$919$	−41527.5	−1.49061	−0.745303	−	0.666726i	$-0.767695\pi$
$919$	−41527.5	−1.49061	−0.745303	+	0.666726i	$0.767695\pi$
$920$	−1444.04	−0.0517484
$921$	0	0
$922$	40361.5	1.44169
$923$	−30272.0	−1.07954
$924$	0	0
$925$	−9315.27	−0.331118
$926$	−33008.6	−1.17141
$927$	0	0
$928$	22096.8	0.781642
$929$	−4584.68	−0.161914	−0.0809572	−	0.996718i	$-0.525798\pi$
$929$	−4584.68	−0.161914	−0.0809572	+	0.996718i	$0.525798\pi$
$930$	0	0
$931$	0	0
$932$	−6473.92	−0.227532
$933$	0	0
$934$	−12821.0	−0.449159
$935$	2709.48	0.0947694
$936$	0	0
$937$	−6928.18	−0.241552	−0.120776	−	0.992680i	$-0.538538\pi$
$937$	−6928.18	−0.241552	−0.120776	+	0.992680i	$0.538538\pi$
$938$	0	0
$939$	0	0
$940$	−2505.07	−0.0869217
$941$	−20944.9	−0.725596	−0.362798	−	0.931868i	$-0.618178\pi$
$941$	−20944.9	−0.725596	−0.362798	+	0.931868i	$0.618178\pi$
$942$	0	0
$943$	−13866.4	−0.478846
$944$	33893.3	1.16857
$945$	0	0
$946$	25562.8	0.878559
$947$	−29278.9	−1.00468	−0.502342	−	0.864669i	$-0.667528\pi$
$947$	−29278.9	−1.00468	−0.502342	+	0.864669i	$0.667528\pi$
$948$	0	0
$949$	−10652.8	−0.364387
$950$	33127.1	1.13135
$951$	0	0
$952$	0	0
$953$	−2136.81	−0.0726316	−0.0363158	−	0.999340i	$-0.511562\pi$
$953$	−2136.81	−0.0726316	−0.0363158	+	0.999340i	$0.511562\pi$
$954$	0	0
$955$	1993.48	0.0675470
$956$	14090.9	0.476707
$957$	0	0
$958$	−23027.4	−0.776597
$959$	0	0
$960$	0	0
$961$	63571.4	2.13391
$962$	12225.8	0.409745
$963$	0	0
$964$	−1702.70	−0.0568881
$965$	7873.47	0.262649
$966$	0	0
$967$	−3921.32	−0.130405	−0.0652023	−	0.997872i	$-0.520769\pi$
$967$	−3921.32	−0.130405	−0.0652023	+	0.997872i	$0.520769\pi$
$968$	−13570.4	−0.450586
$969$	0	0
$970$	12230.9	0.404857
$971$	−47809.1	−1.58009	−0.790045	−	0.613049i	$-0.789943\pi$
$971$	−47809.1	−1.58009	−0.790045	+	0.613049i	$0.789943\pi$
$972$	0	0
$973$	0	0
$974$	−12946.1	−0.425894
$975$	0	0
$976$	−46910.5	−1.53849
$977$	51329.1	1.68082	0.840411	−	0.541949i	$-0.182314\pi$
$977$	51329.1	1.68082	0.840411	+	0.541949i	$0.182314\pi$
$978$	0	0
$979$	−52742.4	−1.72181
$980$	0	0
$981$	0	0
$982$	44035.8	1.43100
$983$	16326.3	0.529734	0.264867	−	0.964285i	$-0.414672\pi$
$983$	16326.3	0.529734	0.264867	+	0.964285i	$0.414672\pi$
$984$	0	0
$985$	10631.6	0.343909
$986$	−11371.0	−0.367268
$987$	0	0
$988$	−15582.4	−0.501763
$989$	8205.37	0.263818
$990$	0	0
$991$	33770.0	1.08248	0.541242	−	0.840867i	$-0.317954\pi$
$991$	33770.0	1.08248	0.541242	+	0.840867i	$0.317954\pi$
$992$	55599.3	1.77952
$993$	0	0
$994$	0	0
$995$	1796.56	0.0572410
$996$	0	0
$997$	50695.4	1.61037	0.805185	−	0.593023i	$-0.202066\pi$
$997$	50695.4	1.61037	0.805185	+	0.593023i	$0.202066\pi$
$998$	−8135.48	−0.258040
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.u.1.3		4
3.2	odd	2		49.4.a.e.1.2	yes	4
7.2	even	3		441.4.e.y.361.2		8
7.3	odd	6		441.4.e.y.226.1		8
7.4	even	3		441.4.e.y.226.2		8
7.5	odd	6		441.4.e.y.361.1		8
7.6	odd	2	inner	441.4.a.u.1.4		4
12.11	even	2		784.4.a.bf.1.1		4
15.14	odd	2		1225.4.a.bb.1.3		4
21.2	odd	6		49.4.c.e.18.3		8
21.5	even	6		49.4.c.e.18.4		8
21.11	odd	6		49.4.c.e.30.3		8
21.17	even	6		49.4.c.e.30.4		8
21.20	even	2		49.4.a.e.1.1	✓	4
84.83	odd	2		784.4.a.bf.1.4		4
105.104	even	2		1225.4.a.bb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.e.1.1	✓	4	21.20	even	2
49.4.a.e.1.2	yes	4	3.2	odd	2
49.4.c.e.18.3		8	21.2	odd	6
49.4.c.e.18.4		8	21.5	even	6
49.4.c.e.30.3		8	21.11	odd	6
49.4.c.e.30.4		8	21.17	even	6
441.4.a.u.1.3		4	1.1	even	1	trivial
441.4.a.u.1.4		4	7.6	odd	2	inner
441.4.e.y.226.1		8	7.3	odd	6
441.4.e.y.226.2		8	7.4	even	3
441.4.e.y.361.1		8	7.5	odd	6
441.4.e.y.361.2		8	7.2	even	3
784.4.a.bf.1.1		4	12.11	even	2
784.4.a.bf.1.4		4	84.83	odd	2
1225.4.a.bb.1.3		4	15.14	odd	2
1225.4.a.bb.1.4		4	105.104	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+3.53113 q^{2} +4.46887 q^{4} -2.07730 q^{5} -12.4689 q^{8} -7.33521 q^{10} -49.1868 q^{11} +44.8559 q^{13} -79.7802 q^{16} +26.5179 q^{17} -77.7350 q^{19} -9.28317 q^{20} -173.685 q^{22} -55.7510 q^{23} -120.685 q^{25} +158.392 q^{26} -121.436 q^{29} -305.553 q^{31} -181.963 q^{32} +93.6380 q^{34} +77.1868 q^{37} -274.492 q^{38} +25.9016 q^{40} +248.720 q^{41} -147.179 q^{43} -219.809 q^{44} -196.864 q^{46} +269.851 q^{47} -426.154 q^{50} +200.455 q^{52} +141.121 q^{53} +102.176 q^{55} -428.805 q^{58} -424.834 q^{59} +587.996 q^{61} -1078.95 q^{62} -4.29373 q^{64} -93.1790 q^{65} -179.634 q^{67} +118.505 q^{68} -674.872 q^{71} -237.489 q^{73} +272.556 q^{74} -347.388 q^{76} +495.852 q^{79} +165.727 q^{80} +878.262 q^{82} -24.4406 q^{83} -55.0855 q^{85} -519.708 q^{86} +613.304 q^{88} +1072.29 q^{89} -249.144 q^{92} +952.877 q^{94} +161.479 q^{95} -1667.43 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 34 q^{4} - 66 q^{8} - 100 q^{11} - 174 q^{16} - 340 q^{22} - 352 q^{23} - 128 q^{25} - 260 q^{29} + 30 q^{32} + 212 q^{37} + 540 q^{43} - 460 q^{44} + 696 q^{46} - 1366 q^{50} - 16 q^{53}+ \cdots - 2192 q^{95}+O(q^{100}) \) 4 * q - 2 * q^2 + 34 * q^4 - 66 * q^8 - 100 * q^11 - 174 * q^16 - 340 * q^22 - 352 * q^23 - 128 * q^25 - 260 * q^29 + 30 * q^32 + 212 * q^37 + 540 * q^43 - 460 * q^44 + 696 * q^46 - 1366 * q^50 - 16 * q^53 - 780 * q^58 - 1678 * q^64 + 756 * q^65 - 1944 * q^67 - 2248 * q^71 + 284 * q^74 - 1048 * q^79 - 3284 * q^85 - 4820 * q^86 + 1260 * q^88 - 3512 * q^92 - 2192 * q^95

Introduction

L-functions

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