LMFDB - Embedded newform 1472.4.a.w.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,4,0,10,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 92)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.841083$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.13366 q^{3} -14.8525 q^{5} -19.8565 q^{7} +10.6218 q^{9} +55.2128 q^{11} +1.83551 q^{13} -91.1001 q^{15} -130.231 q^{17} +17.3027 q^{19} -121.793 q^{21} -23.0000 q^{23} +95.5963 q^{25} -100.458 q^{27} -77.6495 q^{29} +206.274 q^{31} +338.657 q^{33} +294.918 q^{35} +251.748 q^{37} +11.2584 q^{39} -157.141 q^{41} +336.389 q^{43} -157.760 q^{45} +488.790 q^{47} +51.2805 q^{49} -798.794 q^{51} -562.510 q^{53} -820.047 q^{55} +106.129 q^{57} +125.362 q^{59} -163.153 q^{61} -210.912 q^{63} -27.2618 q^{65} +769.127 q^{67} -141.074 q^{69} +113.905 q^{71} -747.272 q^{73} +586.356 q^{75} -1096.33 q^{77} +1112.03 q^{79} -902.966 q^{81} +988.932 q^{83} +1934.26 q^{85} -476.276 q^{87} +1381.04 q^{89} -36.4467 q^{91} +1265.22 q^{93} -256.988 q^{95} +1182.04 q^{97} +586.460 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 10 q^{5} - 46 q^{7} - 31 q^{9} + 64 q^{11} + 44 q^{13} - 134 q^{15} - 88 q^{17} + 94 q^{19} + 6 q^{21} - 69 q^{23} + 181 q^{25} - 20 q^{27} - 308 q^{29} - 140 q^{31} + 510 q^{33} - 192 q^{35}+ \cdots + 18 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 10 * q^5 - 46 * q^7 - 31 * q^9 + 64 * q^11 + 44 * q^13 - 134 * q^15 - 88 * q^17 + 94 * q^19 + 6 * q^21 - 69 * q^23 + 181 * q^25 - 20 * q^27 - 308 * q^29 - 140 * q^31 + 510 * q^33 - 192 * q^35 - 26 * q^37 + 180 * q^39 + 584 * q^41 + 478 * q^43 - 640 * q^45 - 28 * q^47 + 695 * q^49 - 334 * q^51 - 356 * q^53 - 1004 * q^55 + 520 * q^57 - 144 * q^59 + 1052 * q^61 + 120 * q^63 + 150 * q^65 + 2008 * q^67 - 92 * q^69 + 360 * q^71 - 252 * q^73 + 88 * q^75 + 576 * q^77 - 720 * q^79 - 325 * q^81 + 1404 * q^83 + 1632 * q^85 - 156 * q^87 + 534 * q^89 + 1526 * q^91 + 2342 * q^93 - 108 * q^95 + 736 * q^97 + 18 * 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
$3$	6.13366	1.18042	0.590212	−	0.807248i	$-0.299044\pi$
$3$	6.13366	1.18042	0.590212	+	0.807248i	$0.299044\pi$
$4$	0	0
$5$	−14.8525	−1.32845	−0.664223	−	0.747534i	$-0.731238\pi$
$5$	−14.8525	−1.32845	−0.664223	+	0.747534i	$0.731238\pi$
$6$	0	0
$7$	−19.8565	−1.07215	−0.536075	−	0.844170i	$-0.680094\pi$
$7$	−19.8565	−1.07215	−0.536075	+	0.844170i	$0.680094\pi$
$8$	0	0
$9$	10.6218	0.393401
$10$	0	0
$11$	55.2128	1.51339	0.756695	−	0.653768i	$-0.226813\pi$
$11$	55.2128	1.51339	0.756695	+	0.653768i	$0.226813\pi$
$12$	0	0
$13$	1.83551	0.0391598	0.0195799	−	0.999808i	$-0.493767\pi$
$13$	1.83551	0.0391598	0.0195799	+	0.999808i	$0.493767\pi$
$14$	0	0
$15$	−91.1001	−1.56813
$16$	0	0
$17$	−130.231	−1.85798	−0.928991	−	0.370103i	$-0.879322\pi$
$17$	−130.231	−1.85798	−0.928991	+	0.370103i	$0.879322\pi$
$18$	0	0
$19$	17.3027	0.208921	0.104461	−	0.994529i	$-0.466688\pi$
$19$	17.3027	0.208921	0.104461	+	0.994529i	$0.466688\pi$
$20$	0	0
$21$	−121.793	−1.26559
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	95.5963	0.764771
$26$	0	0
$27$	−100.458	−0.716044
$28$	0	0
$29$	−77.6495	−0.497212	−0.248606	−	0.968605i	$-0.579972\pi$
$29$	−77.6495	−0.497212	−0.248606	+	0.968605i	$0.579972\pi$
$30$	0	0
$31$	206.274	1.19509	0.597547	−	0.801834i	$-0.296142\pi$
$31$	206.274	1.19509	0.597547	+	0.801834i	$0.296142\pi$
$32$	0	0
$33$	338.657	1.78644
$34$	0	0
$35$	294.918	1.42429
$36$	0	0
$37$	251.748	1.11857	0.559286	−	0.828975i	$-0.311075\pi$
$37$	251.748	1.11857	0.559286	+	0.828975i	$0.311075\pi$
$38$	0	0
$39$	11.2584	0.0462252
$40$	0	0
$41$	−157.141	−0.598568	−0.299284	−	0.954164i	$-0.596748\pi$
$41$	−157.141	−0.598568	−0.299284	+	0.954164i	$0.596748\pi$
$42$	0	0
$43$	336.389	1.19300	0.596498	−	0.802615i	$-0.296558\pi$
$43$	336.389	1.19300	0.596498	+	0.802615i	$0.296558\pi$
$44$	0	0
$45$	−157.760	−0.522612
$46$	0	0
$47$	488.790	1.51696	0.758482	−	0.651694i	$-0.225941\pi$
$47$	488.790	1.51696	0.758482	+	0.651694i	$0.225941\pi$
$48$	0	0
$49$	51.2805	0.149506
$50$	0	0
$51$	−798.794	−2.19321
$52$	0	0
$53$	−562.510	−1.45786	−0.728931	−	0.684588i	$-0.759982\pi$
$53$	−562.510	−1.45786	−0.728931	+	0.684588i	$0.759982\pi$
$54$	0	0
$55$	−820.047	−2.01046
$56$	0	0
$57$	106.129	0.246616
$58$	0	0
$59$	125.362	0.276622	0.138311	−	0.990389i	$-0.455833\pi$
$59$	125.362	0.276622	0.138311	+	0.990389i	$0.455833\pi$
$60$	0	0
$61$	−163.153	−0.342452	−0.171226	−	0.985232i	$-0.554773\pi$
$61$	−163.153	−0.342452	−0.171226	+	0.985232i	$0.554773\pi$
$62$	0	0
$63$	−210.912	−0.421785
$64$	0	0
$65$	−27.2618	−0.0520217
$66$	0	0
$67$	769.127	1.40245	0.701223	−	0.712942i	$-0.252638\pi$
$67$	769.127	1.40245	0.701223	+	0.712942i	$0.252638\pi$
$68$	0	0
$69$	−141.074	−0.246135
$70$	0	0
$71$	113.905	0.190394	0.0951972	−	0.995458i	$-0.469652\pi$
$71$	113.905	0.190394	0.0951972	+	0.995458i	$0.469652\pi$
$72$	0	0
$73$	−747.272	−1.19810	−0.599052	−	0.800710i	$-0.704456\pi$
$73$	−747.272	−1.19810	−0.599052	+	0.800710i	$0.704456\pi$
$74$	0	0
$75$	586.356	0.902753
$76$	0	0
$77$	−1096.33	−1.62258
$78$	0	0
$79$	1112.03	1.58370	0.791852	−	0.610713i	$-0.209117\pi$
$79$	1112.03	1.58370	0.791852	+	0.610713i	$0.209117\pi$
$80$	0	0
$81$	−902.966	−1.23864
$82$	0	0
$83$	988.932	1.30782	0.653912	−	0.756570i	$-0.273127\pi$
$83$	988.932	1.30782	0.653912	+	0.756570i	$0.273127\pi$
$84$	0	0
$85$	1934.26	2.46823
$86$	0	0
$87$	−476.276	−0.586921
$88$	0	0
$89$	1381.04	1.64483	0.822413	−	0.568891i	$-0.192627\pi$
$89$	1381.04	1.64483	0.822413	+	0.568891i	$0.192627\pi$
$90$	0	0
$91$	−36.4467	−0.0419852
$92$	0	0
$93$	1265.22	1.41072
$94$	0	0
$95$	−256.988	−0.277541
$96$	0	0
$97$	1182.04	1.23730	0.618651	−	0.785666i	$-0.287680\pi$
$97$	1182.04	1.23730	0.618651	+	0.785666i	$0.287680\pi$
$98$	0	0
$99$	586.460	0.595368
$100$	0	0
$101$	−136.284	−0.134265	−0.0671324	−	0.997744i	$-0.521385\pi$
$101$	−136.284	−0.134265	−0.0671324	+	0.997744i	$0.521385\pi$
$102$	0	0
$103$	−776.639	−0.742956	−0.371478	−	0.928442i	$-0.621149\pi$
$103$	−776.639	−0.742956	−0.371478	+	0.928442i	$0.621149\pi$
$104$	0	0
$105$	1808.93	1.68127
$106$	0	0
$107$	73.6862	0.0665749	0.0332874	−	0.999446i	$-0.489402\pi$
$107$	73.6862	0.0665749	0.0332874	+	0.999446i	$0.489402\pi$
$108$	0	0
$109$	407.725	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$109$	407.725	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$110$	0	0
$111$	1544.14	1.32039
$112$	0	0
$113$	−1609.37	−1.33980	−0.669899	−	0.742452i	$-0.733663\pi$
$113$	−1609.37	−1.33980	−0.669899	+	0.742452i	$0.733663\pi$
$114$	0	0
$115$	341.607	0.277000
$116$	0	0
$117$	19.4964	0.0154055
$118$	0	0
$119$	2585.93	1.99203
$120$	0	0
$121$	1717.45	1.29035
$122$	0	0
$123$	−963.849	−0.706564
$124$	0	0
$125$	436.718	0.312490
$126$	0	0
$127$	−244.033	−0.170507	−0.0852536	−	0.996359i	$-0.527170\pi$
$127$	−244.033	−0.170507	−0.0852536	+	0.996359i	$0.527170\pi$
$128$	0	0
$129$	2063.30	1.40824
$130$	0	0
$131$	511.050	0.340845	0.170422	−	0.985371i	$-0.445487\pi$
$131$	511.050	0.340845	0.170422	+	0.985371i	$0.445487\pi$
$132$	0	0
$133$	−343.570	−0.223995
$134$	0	0
$135$	1492.05	0.951227
$136$	0	0
$137$	1468.41	0.915729	0.457865	−	0.889022i	$-0.348614\pi$
$137$	1468.41	0.915729	0.457865	+	0.889022i	$0.348614\pi$
$138$	0	0
$139$	212.472	0.129652	0.0648259	−	0.997897i	$-0.479351\pi$
$139$	212.472	0.129652	0.0648259	+	0.997897i	$0.479351\pi$
$140$	0	0
$141$	2998.07	1.79066
$142$	0	0
$143$	101.343	0.0592641
$144$	0	0
$145$	1153.29	0.660520
$146$	0	0
$147$	314.537	0.176480
$148$	0	0
$149$	1632.09	0.897357	0.448678	−	0.893693i	$-0.351895\pi$
$149$	1632.09	0.897357	0.448678	+	0.893693i	$0.351895\pi$
$150$	0	0
$151$	376.006	0.202642	0.101321	−	0.994854i	$-0.467693\pi$
$151$	376.006	0.202642	0.101321	+	0.994854i	$0.467693\pi$
$152$	0	0
$153$	−1383.29	−0.730931
$154$	0	0
$155$	−3063.68	−1.58762
$156$	0	0
$157$	3693.99	1.87779	0.938894	−	0.344207i	$-0.111852\pi$
$157$	3693.99	1.87779	0.938894	+	0.344207i	$0.111852\pi$
$158$	0	0
$159$	−3450.24	−1.72089
$160$	0	0
$161$	456.699	0.223559
$162$	0	0
$163$	421.000	0.202302	0.101151	−	0.994871i	$-0.467747\pi$
$163$	421.000	0.202302	0.101151	+	0.994871i	$0.467747\pi$
$164$	0	0
$165$	−5029.89	−2.37319
$166$	0	0
$167$	1922.49	0.890820	0.445410	−	0.895327i	$-0.353058\pi$
$167$	1922.49	0.890820	0.445410	+	0.895327i	$0.353058\pi$
$168$	0	0
$169$	−2193.63	−0.998467
$170$	0	0
$171$	183.786	0.0821898
$172$	0	0
$173$	4106.18	1.80455	0.902274	−	0.431163i	$-0.141897\pi$
$173$	4106.18	1.80455	0.902274	+	0.431163i	$0.141897\pi$
$174$	0	0
$175$	−1898.21	−0.819949
$176$	0	0
$177$	768.926	0.326531
$178$	0	0
$179$	−3547.01	−1.48110	−0.740548	−	0.672004i	$-0.765434\pi$
$179$	−3547.01	−1.48110	−0.740548	+	0.672004i	$0.765434\pi$
$180$	0	0
$181$	2538.65	1.04252	0.521260	−	0.853398i	$-0.325462\pi$
$181$	2538.65	1.04252	0.521260	+	0.853398i	$0.325462\pi$
$182$	0	0
$183$	−1000.72	−0.404238
$184$	0	0
$185$	−3739.09	−1.48596
$186$	0	0
$187$	−7190.42	−2.81185
$188$	0	0
$189$	1994.75	0.767707
$190$	0	0
$191$	2916.89	1.10502	0.552509	−	0.833507i	$-0.313670\pi$
$191$	2916.89	1.10502	0.552509	+	0.833507i	$0.313670\pi$
$192$	0	0
$193$	−897.820	−0.334852	−0.167426	−	0.985885i	$-0.553546\pi$
$193$	−897.820	−0.334852	−0.167426	+	0.985885i	$0.553546\pi$
$194$	0	0
$195$	−167.215	−0.0614077
$196$	0	0
$197$	−1708.26	−0.617811	−0.308906	−	0.951093i	$-0.599963\pi$
$197$	−1708.26	−0.617811	−0.308906	+	0.951093i	$0.599963\pi$
$198$	0	0
$199$	−5485.06	−1.95390	−0.976949	−	0.213472i	$-0.931523\pi$
$199$	−5485.06	−1.95390	−0.976949	+	0.213472i	$0.931523\pi$
$200$	0	0
$201$	4717.57	1.65548
$202$	0	0
$203$	1541.85	0.533086
$204$	0	0
$205$	2333.93	0.795165
$206$	0	0
$207$	−244.302	−0.0820297
$208$	0	0
$209$	955.329	0.316179
$210$	0	0
$211$	2018.90	0.658707	0.329353	−	0.944207i	$-0.393169\pi$
$211$	2018.90	0.658707	0.329353	+	0.944207i	$0.393169\pi$
$212$	0	0
$213$	698.653	0.224746
$214$	0	0
$215$	−4996.21	−1.58483
$216$	0	0
$217$	−4095.88	−1.28132
$218$	0	0
$219$	−4583.52	−1.41427
$220$	0	0
$221$	−239.040	−0.0727582
$222$	0	0
$223$	−1408.84	−0.423061	−0.211531	−	0.977371i	$-0.567845\pi$
$223$	−1408.84	−0.423061	−0.211531	+	0.977371i	$0.567845\pi$
$224$	0	0
$225$	1015.41	0.300861
$226$	0	0
$227$	4218.53	1.23345	0.616725	−	0.787178i	$-0.288459\pi$
$227$	4218.53	1.23345	0.616725	+	0.787178i	$0.288459\pi$
$228$	0	0
$229$	−3145.64	−0.907727	−0.453864	−	0.891071i	$-0.649955\pi$
$229$	−3145.64	−0.907727	−0.453864	+	0.891071i	$0.649955\pi$
$230$	0	0
$231$	−6724.54	−1.91533
$232$	0	0
$233$	−2725.56	−0.766340	−0.383170	−	0.923678i	$-0.625168\pi$
$233$	−2725.56	−0.766340	−0.383170	+	0.923678i	$0.625168\pi$
$234$	0	0
$235$	−7259.75	−2.01521
$236$	0	0
$237$	6820.79	1.86944
$238$	0	0
$239$	253.806	0.0686917	0.0343459	−	0.999410i	$-0.489065\pi$
$239$	253.806	0.0686917	0.0343459	+	0.999410i	$0.489065\pi$
$240$	0	0
$241$	546.857	0.146167	0.0730833	−	0.997326i	$-0.476716\pi$
$241$	546.857	0.146167	0.0730833	+	0.997326i	$0.476716\pi$
$242$	0	0
$243$	−2826.12	−0.746072
$244$	0	0
$245$	−761.643	−0.198611
$246$	0	0
$247$	31.7591	0.00818132
$248$	0	0
$249$	6065.78	1.54379
$250$	0	0
$251$	1293.39	0.325250	0.162625	−	0.986688i	$-0.448004\pi$
$251$	1293.39	0.325250	0.162625	+	0.986688i	$0.448004\pi$
$252$	0	0
$253$	−1269.89	−0.315564
$254$	0	0
$255$	11864.1	2.91356
$256$	0	0
$257$	4457.95	1.08202	0.541010	−	0.841016i	$-0.318042\pi$
$257$	4457.95	1.08202	0.541010	+	0.841016i	$0.318042\pi$
$258$	0	0
$259$	−4998.84	−1.19928
$260$	0	0
$261$	−824.779	−0.195604
$262$	0	0
$263$	7213.46	1.69126	0.845629	−	0.533771i	$-0.179225\pi$
$263$	7213.46	1.69126	0.845629	+	0.533771i	$0.179225\pi$
$264$	0	0
$265$	8354.67	1.93669
$266$	0	0
$267$	8470.81	1.94159
$268$	0	0
$269$	−6970.90	−1.58001	−0.790006	−	0.613099i	$-0.789923\pi$
$269$	−6970.90	−1.58001	−0.790006	+	0.613099i	$0.789923\pi$
$270$	0	0
$271$	−5950.25	−1.33377	−0.666886	−	0.745160i	$-0.732373\pi$
$271$	−5950.25	−1.33377	−0.666886	+	0.745160i	$0.732373\pi$
$272$	0	0
$273$	−223.552	−0.0495603
$274$	0	0
$275$	5278.14	1.15740
$276$	0	0
$277$	318.533	0.0690932	0.0345466	−	0.999403i	$-0.489001\pi$
$277$	318.533	0.0690932	0.0345466	+	0.999403i	$0.489001\pi$
$278$	0	0
$279$	2191.01	0.470151
$280$	0	0
$281$	−5155.42	−1.09447	−0.547236	−	0.836978i	$-0.684320\pi$
$281$	−5155.42	−1.09447	−0.547236	+	0.836978i	$0.684320\pi$
$282$	0	0
$283$	7996.15	1.67958	0.839791	−	0.542910i	$-0.182678\pi$
$283$	7996.15	1.67958	0.839791	+	0.542910i	$0.182678\pi$
$284$	0	0
$285$	−1576.28	−0.327616
$286$	0	0
$287$	3120.27	0.641755
$288$	0	0
$289$	12047.1	2.45209
$290$	0	0
$291$	7250.25	1.46054
$292$	0	0
$293$	−3089.70	−0.616048	−0.308024	−	0.951379i	$-0.599668\pi$
$293$	−3089.70	−0.616048	−0.308024	+	0.951379i	$0.599668\pi$
$294$	0	0
$295$	−1861.93	−0.367477
$296$	0	0
$297$	−5546.58	−1.08365
$298$	0	0
$299$	−42.2166	−0.00816539
$300$	0	0
$301$	−6679.51	−1.27907
$302$	0	0
$303$	−835.919	−0.158489
$304$	0	0
$305$	2423.22	0.454929
$306$	0	0
$307$	9221.38	1.71431	0.857153	−	0.515062i	$-0.172231\pi$
$307$	9221.38	1.71431	0.857153	+	0.515062i	$0.172231\pi$
$308$	0	0
$309$	−4763.64	−0.877003
$310$	0	0
$311$	−325.204	−0.0592946	−0.0296473	−	0.999560i	$-0.509438\pi$
$311$	−325.204	−0.0592946	−0.0296473	+	0.999560i	$0.509438\pi$
$312$	0	0
$313$	−7771.70	−1.40346	−0.701729	−	0.712444i	$-0.747588\pi$
$313$	−7771.70	−1.40346	−0.701729	+	0.712444i	$0.747588\pi$
$314$	0	0
$315$	3132.57	0.560318
$316$	0	0
$317$	3888.41	0.688943	0.344472	−	0.938797i	$-0.388058\pi$
$317$	3888.41	0.688943	0.344472	+	0.938797i	$0.388058\pi$
$318$	0	0
$319$	−4287.25	−0.752476
$320$	0	0
$321$	451.966	0.0785865
$322$	0	0
$323$	−2253.35	−0.388172
$324$	0	0
$325$	175.468	0.0299483
$326$	0	0
$327$	2500.85	0.422927
$328$	0	0
$329$	−9705.66	−1.62641
$330$	0	0
$331$	−7587.98	−1.26004	−0.630020	−	0.776579i	$-0.716953\pi$
$331$	−7587.98	−1.26004	−0.630020	+	0.776579i	$0.716953\pi$
$332$	0	0
$333$	2674.02	0.440047
$334$	0	0
$335$	−11423.4	−1.86307
$336$	0	0
$337$	10567.0	1.70807	0.854036	−	0.520215i	$-0.174148\pi$
$337$	10567.0	1.70807	0.854036	+	0.520215i	$0.174148\pi$
$338$	0	0
$339$	−9871.36	−1.58153
$340$	0	0
$341$	11389.0	1.80864
$342$	0	0
$343$	5792.53	0.911857
$344$	0	0
$345$	2095.30	0.326978
$346$	0	0
$347$	−7634.31	−1.18107	−0.590535	−	0.807012i	$-0.701083\pi$
$347$	−7634.31	−1.18107	−0.590535	+	0.807012i	$0.701083\pi$
$348$	0	0
$349$	1494.92	0.229287	0.114644	−	0.993407i	$-0.463427\pi$
$349$	1494.92	0.229287	0.114644	+	0.993407i	$0.463427\pi$
$350$	0	0
$351$	−184.392	−0.0280402
$352$	0	0
$353$	1352.12	0.203870	0.101935	−	0.994791i	$-0.467497\pi$
$353$	1352.12	0.203870	0.101935	+	0.994791i	$0.467497\pi$
$354$	0	0
$355$	−1691.77	−0.252929
$356$	0	0
$357$	15861.2	2.35145
$358$	0	0
$359$	−3598.20	−0.528985	−0.264492	−	0.964388i	$-0.585204\pi$
$359$	−3598.20	−0.528985	−0.264492	+	0.964388i	$0.585204\pi$
$360$	0	0
$361$	−6559.62	−0.956352
$362$	0	0
$363$	10534.3	1.52316
$364$	0	0
$365$	11098.9	1.59162
$366$	0	0
$367$	5389.41	0.766553	0.383277	−	0.923634i	$-0.374796\pi$
$367$	5389.41	0.766553	0.383277	+	0.923634i	$0.374796\pi$
$368$	0	0
$369$	−1669.12	−0.235477
$370$	0	0
$371$	11169.5	1.56305
$372$	0	0
$373$	−11554.0	−1.60387	−0.801934	−	0.597412i	$-0.796196\pi$
$373$	−11554.0	−1.60387	−0.801934	+	0.597412i	$0.796196\pi$
$374$	0	0
$375$	2678.68	0.368870
$376$	0	0
$377$	−142.526	−0.0194707
$378$	0	0
$379$	−899.535	−0.121916	−0.0609578	−	0.998140i	$-0.519416\pi$
$379$	−899.535	−0.121916	−0.0609578	+	0.998140i	$0.519416\pi$
$380$	0	0
$381$	−1496.81	−0.201271
$382$	0	0
$383$	−7099.99	−0.947239	−0.473619	−	0.880730i	$-0.657053\pi$
$383$	−7099.99	−0.947239	−0.473619	+	0.880730i	$0.657053\pi$
$384$	0	0
$385$	16283.3	2.15551
$386$	0	0
$387$	3573.06	0.469325
$388$	0	0
$389$	−7471.47	−0.973826	−0.486913	−	0.873450i	$-0.661877\pi$
$389$	−7471.47	−0.973826	−0.486913	+	0.873450i	$0.661877\pi$
$390$	0	0
$391$	2995.32	0.387416
$392$	0	0
$393$	3134.61	0.402341
$394$	0	0
$395$	−16516.3	−2.10387
$396$	0	0
$397$	2610.90	0.330069	0.165034	−	0.986288i	$-0.447226\pi$
$397$	2610.90	0.330069	0.165034	+	0.986288i	$0.447226\pi$
$398$	0	0
$399$	−2107.34	−0.264409
$400$	0	0
$401$	−3698.53	−0.460588	−0.230294	−	0.973121i	$-0.573969\pi$
$401$	−3698.53	−0.460588	−0.230294	+	0.973121i	$0.573969\pi$
$402$	0	0
$403$	378.617	0.0467997
$404$	0	0
$405$	13411.3	1.64546
$406$	0	0
$407$	13899.7	1.69284
$408$	0	0
$409$	12905.9	1.56028	0.780140	−	0.625605i	$-0.215148\pi$
$409$	12905.9	1.56028	0.780140	+	0.625605i	$0.215148\pi$
$410$	0	0
$411$	9006.74	1.08095
$412$	0	0
$413$	−2489.24	−0.296580
$414$	0	0
$415$	−14688.1	−1.73737
$416$	0	0
$417$	1303.23	0.153044
$418$	0	0
$419$	−12776.2	−1.48963	−0.744817	−	0.667268i	$-0.767463\pi$
$419$	−12776.2	−1.48963	−0.744817	+	0.667268i	$0.767463\pi$
$420$	0	0
$421$	1425.67	0.165043	0.0825213	−	0.996589i	$-0.473703\pi$
$421$	1425.67	0.165043	0.0825213	+	0.996589i	$0.473703\pi$
$422$	0	0
$423$	5191.84	0.596775
$424$	0	0
$425$	−12449.6	−1.42093
$426$	0	0
$427$	3239.64	0.367160
$428$	0	0
$429$	621.606	0.0699567
$430$	0	0
$431$	115.735	0.0129344	0.00646721	−	0.999979i	$-0.497941\pi$
$431$	115.735	0.0129344	0.00646721	+	0.999979i	$0.497941\pi$
$432$	0	0
$433$	−1530.96	−0.169915	−0.0849575	−	0.996385i	$-0.527075\pi$
$433$	−1530.96	−0.169915	−0.0849575	+	0.996385i	$0.527075\pi$
$434$	0	0
$435$	7073.88	0.779693
$436$	0	0
$437$	−397.961	−0.0435631
$438$	0	0
$439$	−11952.2	−1.29942	−0.649712	−	0.760181i	$-0.725110\pi$
$439$	−11952.2	−1.29942	−0.649712	+	0.760181i	$0.725110\pi$
$440$	0	0
$441$	544.692	0.0588157
$442$	0	0
$443$	−2081.30	−0.223218	−0.111609	−	0.993752i	$-0.535600\pi$
$443$	−2081.30	−0.223218	−0.111609	+	0.993752i	$0.535600\pi$
$444$	0	0
$445$	−20511.8	−2.18506
$446$	0	0
$447$	10010.7	1.05926
$448$	0	0
$449$	−3242.73	−0.340832	−0.170416	−	0.985372i	$-0.554511\pi$
$449$	−3242.73	−0.340832	−0.170416	+	0.985372i	$0.554511\pi$
$450$	0	0
$451$	−8676.19	−0.905866
$452$	0	0
$453$	2306.30	0.239204
$454$	0	0
$455$	541.324	0.0557751
$456$	0	0
$457$	16412.2	1.67994	0.839970	−	0.542633i	$-0.182573\pi$
$457$	16412.2	1.67994	0.839970	+	0.542633i	$0.182573\pi$
$458$	0	0
$459$	13082.8	1.33040
$460$	0	0
$461$	2985.88	0.301662	0.150831	−	0.988560i	$-0.451805\pi$
$461$	2985.88	0.301662	0.150831	+	0.988560i	$0.451805\pi$
$462$	0	0
$463$	−7994.42	−0.802445	−0.401223	−	0.915981i	$-0.631415\pi$
$463$	−7994.42	−0.802445	−0.401223	+	0.915981i	$0.631415\pi$
$464$	0	0
$465$	−18791.6	−1.87406
$466$	0	0
$467$	−621.345	−0.0615684	−0.0307842	−	0.999526i	$-0.509800\pi$
$467$	−621.345	−0.0615684	−0.0307842	+	0.999526i	$0.509800\pi$
$468$	0	0
$469$	−15272.2	−1.50363
$470$	0	0
$471$	22657.7	2.21659
$472$	0	0
$473$	18573.0	1.80547
$474$	0	0
$475$	1654.07	0.159777
$476$	0	0
$477$	−5974.87	−0.573524
$478$	0	0
$479$	5650.45	0.538989	0.269494	−	0.963002i	$-0.413143\pi$
$479$	5650.45	0.538989	0.269494	+	0.963002i	$0.413143\pi$
$480$	0	0
$481$	462.085	0.0438031
$482$	0	0
$483$	2801.24	0.263894
$484$	0	0
$485$	−17556.3	−1.64369
$486$	0	0
$487$	7687.95	0.715347	0.357674	−	0.933847i	$-0.383570\pi$
$487$	7687.95	0.715347	0.357674	+	0.933847i	$0.383570\pi$
$488$	0	0
$489$	2582.27	0.238802
$490$	0	0
$491$	−5517.82	−0.507160	−0.253580	−	0.967314i	$-0.581608\pi$
$491$	−5517.82	−0.507160	−0.253580	+	0.967314i	$0.581608\pi$
$492$	0	0
$493$	10112.4	0.923811
$494$	0	0
$495$	−8710.39	−0.790915
$496$	0	0
$497$	−2261.75	−0.204131
$498$	0	0
$499$	14399.6	1.29181	0.645907	−	0.763416i	$-0.276479\pi$
$499$	14399.6	1.29181	0.645907	+	0.763416i	$0.276479\pi$
$500$	0	0
$501$	11791.9	1.05155
$502$	0	0
$503$	1689.24	0.149741	0.0748703	−	0.997193i	$-0.476146\pi$
$503$	1689.24	0.149741	0.0748703	+	0.997193i	$0.476146\pi$
$504$	0	0
$505$	2024.15	0.178364
$506$	0	0
$507$	−13455.0	−1.17861
$508$	0	0
$509$	4595.43	0.400175	0.200087	−	0.979778i	$-0.435877\pi$
$509$	4595.43	0.400175	0.200087	+	0.979778i	$0.435877\pi$
$510$	0	0
$511$	14838.2	1.28455
$512$	0	0
$513$	−1738.20	−0.149597
$514$	0	0
$515$	11535.0	0.986978
$516$	0	0
$517$	26987.5	2.29576
$518$	0	0
$519$	25185.9	2.13013
$520$	0	0
$521$	−15259.0	−1.28313	−0.641564	−	0.767069i	$-0.721714\pi$
$521$	−15259.0	−1.28313	−0.641564	+	0.767069i	$0.721714\pi$
$522$	0	0
$523$	13742.9	1.14901	0.574507	−	0.818500i	$-0.305194\pi$
$523$	13742.9	1.14901	0.574507	+	0.818500i	$0.305194\pi$
$524$	0	0
$525$	−11643.0	−0.967887
$526$	0	0
$527$	−26863.3	−2.22046
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	1331.57	0.108823
$532$	0	0
$533$	−288.433	−0.0234398
$534$	0	0
$535$	−1094.42	−0.0884411
$536$	0	0
$537$	−21756.2	−1.74832
$538$	0	0
$539$	2831.34	0.226261
$540$	0	0
$541$	−14206.6	−1.12900	−0.564501	−	0.825432i	$-0.690931\pi$
$541$	−14206.6	−1.12900	−0.564501	+	0.825432i	$0.690931\pi$
$542$	0	0
$543$	15571.2	1.23061
$544$	0	0
$545$	−6055.72	−0.475961
$546$	0	0
$547$	4893.57	0.382512	0.191256	−	0.981540i	$-0.438744\pi$
$547$	4893.57	0.382512	0.191256	+	0.981540i	$0.438744\pi$
$548$	0	0
$549$	−1732.98	−0.134721
$550$	0	0
$551$	−1343.54	−0.103878
$552$	0	0
$553$	−22080.9	−1.69797
$554$	0	0
$555$	−22934.3	−1.75407
$556$	0	0
$557$	6114.45	0.465130	0.232565	−	0.972581i	$-0.425288\pi$
$557$	6114.45	0.465130	0.232565	+	0.972581i	$0.425288\pi$
$558$	0	0
$559$	617.444	0.0467175
$560$	0	0
$561$	−44103.6	−3.31917
$562$	0	0
$563$	−283.621	−0.0212312	−0.0106156	−	0.999944i	$-0.503379\pi$
$563$	−283.621	−0.0212312	−0.0106156	+	0.999944i	$0.503379\pi$
$564$	0	0
$565$	23903.2	1.77985
$566$	0	0
$567$	17929.7	1.32800
$568$	0	0
$569$	−22593.3	−1.66461	−0.832304	−	0.554319i	$-0.812979\pi$
$569$	−22593.3	−1.66461	−0.832304	+	0.554319i	$0.812979\pi$
$570$	0	0
$571$	−4659.78	−0.341516	−0.170758	−	0.985313i	$-0.554622\pi$
$571$	−4659.78	−0.341516	−0.170758	+	0.985313i	$0.554622\pi$
$572$	0	0
$573$	17891.2	1.30439
$574$	0	0
$575$	−2198.72	−0.159466
$576$	0	0
$577$	−11799.6	−0.851343	−0.425672	−	0.904878i	$-0.639962\pi$
$577$	−11799.6	−0.851343	−0.425672	+	0.904878i	$0.639962\pi$
$578$	0	0
$579$	−5506.93	−0.395268
$580$	0	0
$581$	−19636.7	−1.40218
$582$	0	0
$583$	−31057.7	−2.20631
$584$	0	0
$585$	−289.570	−0.0204654
$586$	0	0
$587$	−6391.44	−0.449409	−0.224704	−	0.974427i	$-0.572142\pi$
$587$	−6391.44	−0.449409	−0.224704	+	0.974427i	$0.572142\pi$
$588$	0	0
$589$	3569.09	0.249681
$590$	0	0
$591$	−10477.9	−0.729279
$592$	0	0
$593$	25789.3	1.78590	0.892950	−	0.450156i	$-0.148631\pi$
$593$	25789.3	1.78590	0.892950	+	0.450156i	$0.148631\pi$
$594$	0	0
$595$	−38407.5	−2.64631
$596$	0	0
$597$	−33643.5	−2.30643
$598$	0	0
$599$	−1019.78	−0.0695612	−0.0347806	−	0.999395i	$-0.511073\pi$
$599$	−1019.78	−0.0695612	−0.0347806	+	0.999395i	$0.511073\pi$
$600$	0	0
$601$	18088.1	1.22767	0.613835	−	0.789435i	$-0.289626\pi$
$601$	18088.1	1.22767	0.613835	+	0.789435i	$0.289626\pi$
$602$	0	0
$603$	8169.53	0.551723
$604$	0	0
$605$	−25508.5	−1.71416
$606$	0	0
$607$	8849.82	0.591768	0.295884	−	0.955224i	$-0.404386\pi$
$607$	8849.82	0.591768	0.295884	+	0.955224i	$0.404386\pi$
$608$	0	0
$609$	9457.17	0.629268
$610$	0	0
$611$	897.177	0.0594041
$612$	0	0
$613$	9371.04	0.617443	0.308722	−	0.951152i	$-0.400099\pi$
$613$	9371.04	0.617443	0.308722	+	0.951152i	$0.400099\pi$
$614$	0	0
$615$	14315.6	0.938632
$616$	0	0
$617$	−12177.9	−0.794593	−0.397296	−	0.917690i	$-0.630051\pi$
$617$	−12177.9	−0.794593	−0.397296	+	0.917690i	$0.630051\pi$
$618$	0	0
$619$	10466.5	0.679620	0.339810	−	0.940494i	$-0.389637\pi$
$619$	10466.5	0.679620	0.339810	+	0.940494i	$0.389637\pi$
$620$	0	0
$621$	2310.54	0.149306
$622$	0	0
$623$	−27422.5	−1.76350
$624$	0	0
$625$	−18435.9	−1.17990
$626$	0	0
$627$	5859.66	0.373226
$628$	0	0
$629$	−32785.5	−2.07829
$630$	0	0
$631$	12761.0	0.805083	0.402542	−	0.915402i	$-0.368127\pi$
$631$	12761.0	0.805083	0.402542	+	0.915402i	$0.368127\pi$
$632$	0	0
$633$	12383.3	0.777553
$634$	0	0
$635$	3624.49	0.226510
$636$	0	0
$637$	94.1257	0.00585463
$638$	0	0
$639$	1209.87	0.0749013
$640$	0	0
$641$	−16863.9	−1.03913	−0.519567	−	0.854429i	$-0.673907\pi$
$641$	−16863.9	−1.03913	−0.519567	+	0.854429i	$0.673907\pi$
$642$	0	0
$643$	7124.46	0.436954	0.218477	−	0.975842i	$-0.429891\pi$
$643$	7124.46	0.436954	0.218477	+	0.975842i	$0.429891\pi$
$644$	0	0
$645$	−30645.1	−1.87077
$646$	0	0
$647$	4611.30	0.280199	0.140100	−	0.990137i	$-0.455258\pi$
$647$	4611.30	0.280199	0.140100	+	0.990137i	$0.455258\pi$
$648$	0	0
$649$	6921.56	0.418637
$650$	0	0
$651$	−25122.8	−1.51250
$652$	0	0
$653$	24684.4	1.47929	0.739645	−	0.672997i	$-0.234993\pi$
$653$	24684.4	1.47929	0.739645	+	0.672997i	$0.234993\pi$
$654$	0	0
$655$	−7590.37	−0.452794
$656$	0	0
$657$	−7937.39	−0.471335
$658$	0	0
$659$	−7975.64	−0.471452	−0.235726	−	0.971820i	$-0.575747\pi$
$659$	−7975.64	−0.471452	−0.235726	+	0.971820i	$0.575747\pi$
$660$	0	0
$661$	18826.4	1.10781	0.553903	−	0.832581i	$-0.313138\pi$
$661$	18826.4	1.10781	0.553903	+	0.832581i	$0.313138\pi$
$662$	0	0
$663$	−1466.19	−0.0858855
$664$	0	0
$665$	5102.87	0.297565
$666$	0	0
$667$	1785.94	0.103676
$668$	0	0
$669$	−8641.32	−0.499391
$670$	0	0
$671$	−9008.11	−0.518263
$672$	0	0
$673$	2048.53	0.117333	0.0586665	−	0.998278i	$-0.481315\pi$
$673$	2048.53	0.117333	0.0586665	+	0.998278i	$0.481315\pi$
$674$	0	0
$675$	−9603.44	−0.547610
$676$	0	0
$677$	2817.95	0.159975	0.0799873	−	0.996796i	$-0.474512\pi$
$677$	2817.95	0.159975	0.0799873	+	0.996796i	$0.474512\pi$
$678$	0	0
$679$	−23471.2	−1.32657
$680$	0	0
$681$	25875.0	1.45600
$682$	0	0
$683$	−2315.98	−0.129749	−0.0648745	−	0.997893i	$-0.520665\pi$
$683$	−2315.98	−0.129749	−0.0648745	+	0.997893i	$0.520665\pi$
$684$	0	0
$685$	−21809.6	−1.21650
$686$	0	0
$687$	−19294.3	−1.07150
$688$	0	0
$689$	−1032.49	−0.0570896
$690$	0	0
$691$	32776.9	1.80447	0.902236	−	0.431243i	$-0.141925\pi$
$691$	32776.9	1.80447	0.902236	+	0.431243i	$0.141925\pi$
$692$	0	0
$693$	−11645.0	−0.638324
$694$	0	0
$695$	−3155.73	−0.172235
$696$	0	0
$697$	20464.6	1.11213
$698$	0	0
$699$	−16717.7	−0.904606
$700$	0	0
$701$	−19670.2	−1.05982	−0.529909	−	0.848054i	$-0.677774\pi$
$701$	−19670.2	−1.05982	−0.529909	+	0.848054i	$0.677774\pi$
$702$	0	0
$703$	4355.92	0.233694
$704$	0	0
$705$	−44528.8	−2.37880
$706$	0	0
$707$	2706.12	0.143952
$708$	0	0
$709$	−11162.8	−0.591297	−0.295648	−	0.955297i	$-0.595536\pi$
$709$	−11162.8	−0.591297	−0.295648	+	0.955297i	$0.595536\pi$
$710$	0	0
$711$	11811.7	0.623030
$712$	0	0
$713$	−4744.30	−0.249194
$714$	0	0
$715$	−1505.20	−0.0787292
$716$	0	0
$717$	1556.76	0.0810854
$718$	0	0
$719$	−9864.65	−0.511668	−0.255834	−	0.966721i	$-0.582350\pi$
$719$	−9864.65	−0.511668	−0.255834	+	0.966721i	$0.582350\pi$
$720$	0	0
$721$	15421.3	0.796561
$722$	0	0
$723$	3354.23	0.172538
$724$	0	0
$725$	−7423.01	−0.380253
$726$	0	0
$727$	14571.5	0.743367	0.371683	−	0.928360i	$-0.378781\pi$
$727$	14571.5	0.743367	0.371683	+	0.928360i	$0.378781\pi$
$728$	0	0
$729$	7045.64	0.357956
$730$	0	0
$731$	−43808.3	−2.21656
$732$	0	0
$733$	−4903.77	−0.247101	−0.123550	−	0.992338i	$-0.539428\pi$
$733$	−4903.77	−0.247101	−0.123550	+	0.992338i	$0.539428\pi$
$734$	0	0
$735$	−4671.66	−0.234445
$736$	0	0
$737$	42465.7	2.12245
$738$	0	0
$739$	21471.7	1.06881	0.534403	−	0.845230i	$-0.320536\pi$
$739$	21471.7	1.06881	0.534403	+	0.845230i	$0.320536\pi$
$740$	0	0
$741$	194.800	0.00965743
$742$	0	0
$743$	−15170.0	−0.749034	−0.374517	−	0.927220i	$-0.622191\pi$
$743$	−15170.0	−0.749034	−0.374517	+	0.927220i	$0.622191\pi$
$744$	0	0
$745$	−24240.6	−1.19209
$746$	0	0
$747$	10504.3	0.514499
$748$	0	0
$749$	−1463.15	−0.0713782
$750$	0	0
$751$	8003.54	0.388886	0.194443	−	0.980914i	$-0.437710\pi$
$751$	8003.54	0.388886	0.194443	+	0.980914i	$0.437710\pi$
$752$	0	0
$753$	7933.19	0.383933
$754$	0	0
$755$	−5584.63	−0.269199
$756$	0	0
$757$	−1899.42	−0.0911963	−0.0455981	−	0.998960i	$-0.514519\pi$
$757$	−1899.42	−0.0911963	−0.0455981	+	0.998960i	$0.514519\pi$
$758$	0	0
$759$	−7789.10	−0.372499
$760$	0	0
$761$	−13058.9	−0.622058	−0.311029	−	0.950400i	$-0.600674\pi$
$761$	−13058.9	−0.622058	−0.311029	+	0.950400i	$0.600674\pi$
$762$	0	0
$763$	−8095.98	−0.384134
$764$	0	0
$765$	20545.3	0.971003
$766$	0	0
$767$	230.102	0.0108325
$768$	0	0
$769$	1522.42	0.0713913	0.0356956	−	0.999363i	$-0.488635\pi$
$769$	1522.42	0.0713913	0.0356956	+	0.999363i	$0.488635\pi$
$770$	0	0
$771$	27343.5	1.27724
$772$	0	0
$773$	−19506.8	−0.907646	−0.453823	−	0.891092i	$-0.649940\pi$
$773$	−19506.8	−0.907646	−0.453823	+	0.891092i	$0.649940\pi$
$774$	0	0
$775$	19719.0	0.913973
$776$	0	0
$777$	−30661.2	−1.41566
$778$	0	0
$779$	−2718.96	−0.125054
$780$	0	0
$781$	6289.00	0.288141
$782$	0	0
$783$	7800.53	0.356026
$784$	0	0
$785$	−54865.0	−2.49454
$786$	0	0
$787$	7210.38	0.326585	0.163292	−	0.986578i	$-0.447789\pi$
$787$	7210.38	0.326585	0.163292	+	0.986578i	$0.447789\pi$
$788$	0	0
$789$	44244.9	1.99640
$790$	0	0
$791$	31956.6	1.43647
$792$	0	0
$793$	−299.467	−0.0134103
$794$	0	0
$795$	51244.7	2.28612
$796$	0	0
$797$	−38586.8	−1.71495	−0.857474	−	0.514528i	$-0.827967\pi$
$797$	−38586.8	−1.71495	−0.857474	+	0.514528i	$0.827967\pi$
$798$	0	0
$799$	−63655.7	−2.81849
$800$	0	0
$801$	14669.1	0.647075
$802$	0	0
$803$	−41259.0	−1.81320
$804$	0	0
$805$	−6783.12	−0.296986
$806$	0	0
$807$	−42757.2	−1.86508
$808$	0	0
$809$	−9927.23	−0.431425	−0.215712	−	0.976457i	$-0.569207\pi$
$809$	−9927.23	−0.431425	−0.215712	+	0.976457i	$0.569207\pi$
$810$	0	0
$811$	26808.0	1.16074	0.580368	−	0.814354i	$-0.302909\pi$
$811$	26808.0	1.16074	0.580368	+	0.814354i	$0.302909\pi$
$812$	0	0
$813$	−36496.8	−1.57442
$814$	0	0
$815$	−6252.89	−0.268748
$816$	0	0
$817$	5820.42	0.249242
$818$	0	0
$819$	−387.130	−0.0165170
$820$	0	0
$821$	−18620.9	−0.791565	−0.395783	−	0.918344i	$-0.629527\pi$
$821$	−18620.9	−0.791565	−0.395783	+	0.918344i	$0.629527\pi$
$822$	0	0
$823$	−29918.1	−1.26717	−0.633585	−	0.773673i	$-0.718417\pi$
$823$	−29918.1	−1.26717	−0.633585	+	0.773673i	$0.718417\pi$
$824$	0	0
$825$	32374.3	1.36622
$826$	0	0
$827$	26360.3	1.10839	0.554195	−	0.832387i	$-0.313026\pi$
$827$	26360.3	1.10839	0.554195	+	0.832387i	$0.313026\pi$
$828$	0	0
$829$	−6260.01	−0.262267	−0.131133	−	0.991365i	$-0.541862\pi$
$829$	−6260.01	−0.262267	−0.131133	+	0.991365i	$0.541862\pi$
$830$	0	0
$831$	1953.78	0.0815593
$832$	0	0
$833$	−6678.32	−0.277779
$834$	0	0
$835$	−28553.8	−1.18341
$836$	0	0
$837$	−20721.9	−0.855741
$838$	0	0
$839$	−9837.11	−0.404785	−0.202393	−	0.979304i	$-0.564872\pi$
$839$	−9837.11	−0.404785	−0.202393	+	0.979304i	$0.564872\pi$
$840$	0	0
$841$	−18359.6	−0.752780
$842$	0	0
$843$	−31621.6	−1.29194
$844$	0	0
$845$	32580.9	1.32641
$846$	0	0
$847$	−34102.6	−1.38345
$848$	0	0
$849$	49045.7	1.98262
$850$	0	0
$851$	−5790.21	−0.233238
$852$	0	0
$853$	−2590.82	−0.103995	−0.0519976	−	0.998647i	$-0.516559\pi$
$853$	−2590.82	−0.103995	−0.0519976	+	0.998647i	$0.516559\pi$
$854$	0	0
$855$	−2729.68	−0.109185
$856$	0	0
$857$	42982.0	1.71323	0.856614	−	0.515958i	$-0.172564\pi$
$857$	42982.0	1.71323	0.856614	+	0.515958i	$0.172564\pi$
$858$	0	0
$859$	11858.2	0.471008	0.235504	−	0.971873i	$-0.424326\pi$
$859$	11858.2	0.471008	0.235504	+	0.971873i	$0.424326\pi$
$860$	0	0
$861$	19138.7	0.757543
$862$	0	0
$863$	9020.00	0.355787	0.177894	−	0.984050i	$-0.443072\pi$
$863$	9020.00	0.355787	0.177894	+	0.984050i	$0.443072\pi$
$864$	0	0
$865$	−60986.9	−2.39725
$866$	0	0
$867$	73893.1	2.89451
$868$	0	0
$869$	61398.0	2.39676
$870$	0	0
$871$	1411.74	0.0549195
$872$	0	0
$873$	12555.4	0.486755
$874$	0	0
$875$	−8671.69	−0.335036
$876$	0	0
$877$	−35816.6	−1.37907	−0.689534	−	0.724253i	$-0.742185\pi$
$877$	−35816.6	−1.37907	−0.689534	+	0.724253i	$0.742185\pi$
$878$	0	0
$879$	−18951.2	−0.727198
$880$	0	0
$881$	5301.13	0.202724	0.101362	−	0.994850i	$-0.467680\pi$
$881$	5301.13	0.202724	0.101362	+	0.994850i	$0.467680\pi$
$882$	0	0
$883$	−16117.7	−0.614275	−0.307138	−	0.951665i	$-0.599371\pi$
$883$	−16117.7	−0.614275	−0.307138	+	0.951665i	$0.599371\pi$
$884$	0	0
$885$	−11420.5	−0.433779
$886$	0	0
$887$	30081.6	1.13871	0.569357	−	0.822090i	$-0.307192\pi$
$887$	30081.6	1.13871	0.569357	+	0.822090i	$0.307192\pi$
$888$	0	0
$889$	4845.64	0.182809
$890$	0	0
$891$	−49855.3	−1.87454
$892$	0	0
$893$	8457.37	0.316926
$894$	0	0
$895$	52681.9	1.96756
$896$	0	0
$897$	−258.943	−0.00963862
$898$	0	0
$899$	−16017.1	−0.594216
$900$	0	0
$901$	73256.2	2.70868
$902$	0	0
$903$	−40969.8	−1.50985
$904$	0	0
$905$	−37705.2	−1.38493
$906$	0	0
$907$	−3409.96	−0.124836	−0.0624178	−	0.998050i	$-0.519881\pi$
$907$	−3409.96	−0.124836	−0.0624178	+	0.998050i	$0.519881\pi$
$908$	0	0
$909$	−1447.58	−0.0528199
$910$	0	0
$911$	36180.8	1.31583	0.657917	−	0.753091i	$-0.271438\pi$
$911$	36180.8	1.31583	0.657917	+	0.753091i	$0.271438\pi$
$912$	0	0
$913$	54601.7	1.97925
$914$	0	0
$915$	14863.2	0.537009
$916$	0	0
$917$	−10147.7	−0.365437
$918$	0	0
$919$	13302.5	0.477486	0.238743	−	0.971083i	$-0.423265\pi$
$919$	13302.5	0.477486	0.238743	+	0.971083i	$0.423265\pi$
$920$	0	0
$921$	56560.8	2.02361
$922$	0	0
$923$	209.073	0.00745581
$924$	0	0
$925$	24066.2	0.855451
$926$	0	0
$927$	−8249.32	−0.292279
$928$	0	0
$929$	19908.6	0.703101	0.351550	−	0.936169i	$-0.385655\pi$
$929$	19908.6	0.703101	0.351550	+	0.936169i	$0.385655\pi$
$930$	0	0
$931$	887.290	0.0312350
$932$	0	0
$933$	−1994.69	−0.0699927
$934$	0	0
$935$	106796.	3.73539
$936$	0	0
$937$	16038.2	0.559172	0.279586	−	0.960121i	$-0.409803\pi$
$937$	16038.2	0.559172	0.279586	+	0.960121i	$0.409803\pi$
$938$	0	0
$939$	−47669.0	−1.65668
$940$	0	0
$941$	30691.9	1.06326	0.531630	−	0.846976i	$-0.321580\pi$
$941$	30691.9	1.06326	0.531630	+	0.846976i	$0.321580\pi$
$942$	0	0
$943$	3614.24	0.124810
$944$	0	0
$945$	−29627.0	−1.01986
$946$	0	0
$947$	−1181.10	−0.0405288	−0.0202644	−	0.999795i	$-0.506451\pi$
$947$	−1181.10	−0.0405288	−0.0202644	+	0.999795i	$0.506451\pi$
$948$	0	0
$949$	−1371.62	−0.0469175
$950$	0	0
$951$	23850.2	0.813245
$952$	0	0
$953$	−10571.8	−0.359344	−0.179672	−	0.983727i	$-0.557504\pi$
$953$	−10571.8	−0.359344	−0.179672	+	0.983727i	$0.557504\pi$
$954$	0	0
$955$	−43323.0	−1.46796
$956$	0	0
$957$	−26296.5	−0.888240
$958$	0	0
$959$	−29157.5	−0.981799
$960$	0	0
$961$	12758.0	0.428251
$962$	0	0
$963$	782.681	0.0261906
$964$	0	0
$965$	13334.9	0.444834
$966$	0	0
$967$	48057.2	1.59815	0.799077	−	0.601228i	$-0.205322\pi$
$967$	48057.2	1.59815	0.799077	+	0.601228i	$0.205322\pi$
$968$	0	0
$969$	−13821.3	−0.458207
$970$	0	0
$971$	−2506.90	−0.0828529	−0.0414265	−	0.999142i	$-0.513190\pi$
$971$	−2506.90	−0.0828529	−0.0414265	+	0.999142i	$0.513190\pi$
$972$	0	0
$973$	−4218.94	−0.139006
$974$	0	0
$975$	1076.26	0.0353517
$976$	0	0
$977$	7266.61	0.237952	0.118976	−	0.992897i	$-0.462039\pi$
$977$	7266.61	0.237952	0.118976	+	0.992897i	$0.462039\pi$
$978$	0	0
$979$	76250.8	2.48926
$980$	0	0
$981$	4330.78	0.140949
$982$	0	0
$983$	1178.54	0.0382397	0.0191199	−	0.999817i	$-0.493914\pi$
$983$	1178.54	0.0382397	0.0191199	+	0.999817i	$0.493914\pi$
$984$	0	0
$985$	25372.0	0.820729
$986$	0	0
$987$	−59531.2	−1.91986
$988$	0	0
$989$	−7736.94	−0.248757
$990$	0	0
$991$	6000.25	0.192335	0.0961677	−	0.995365i	$-0.469342\pi$
$991$	6000.25	0.192335	0.0961677	+	0.995365i	$0.469342\pi$
$992$	0	0
$993$	−46542.1	−1.48738
$994$	0	0
$995$	81466.8	2.59565
$996$	0	0
$997$	23508.2	0.746752	0.373376	−	0.927680i	$-0.378200\pi$
$997$	23508.2	0.746752	0.373376	+	0.927680i	$0.378200\pi$
$998$	0	0
$999$	−25290.2	−0.800947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.w.1.3		3
4.3	odd	2		1472.4.a.p.1.1		3
8.3	odd	2		368.4.a.k.1.3		3
8.5	even	2		92.4.a.a.1.1	✓	3
24.5	odd	2		828.4.a.f.1.1		3
40.13	odd	4		2300.4.c.b.1749.1		6
40.29	even	2		2300.4.a.b.1.3		3
40.37	odd	4		2300.4.c.b.1749.6		6
184.45	odd	2		2116.4.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.a.a.1.1	✓	3	8.5	even	2
368.4.a.k.1.3		3	8.3	odd	2
828.4.a.f.1.1		3	24.5	odd	2
1472.4.a.p.1.1		3	4.3	odd	2
1472.4.a.w.1.3		3	1.1	even	1	trivial
2116.4.a.a.1.1		3	184.45	odd	2
2300.4.a.b.1.3		3	40.29	even	2
2300.4.c.b.1749.1		6	40.13	odd	4
2300.4.c.b.1749.6		6	40.37	odd	4

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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+6.13366 q^{3} -14.8525 q^{5} -19.8565 q^{7} +10.6218 q^{9} +55.2128 q^{11} +1.83551 q^{13} -91.1001 q^{15} -130.231 q^{17} +17.3027 q^{19} -121.793 q^{21} -23.0000 q^{23} +95.5963 q^{25} -100.458 q^{27} -77.6495 q^{29} +206.274 q^{31} +338.657 q^{33} +294.918 q^{35} +251.748 q^{37} +11.2584 q^{39} -157.141 q^{41} +336.389 q^{43} -157.760 q^{45} +488.790 q^{47} +51.2805 q^{49} -798.794 q^{51} -562.510 q^{53} -820.047 q^{55} +106.129 q^{57} +125.362 q^{59} -163.153 q^{61} -210.912 q^{63} -27.2618 q^{65} +769.127 q^{67} -141.074 q^{69} +113.905 q^{71} -747.272 q^{73} +586.356 q^{75} -1096.33 q^{77} +1112.03 q^{79} -902.966 q^{81} +988.932 q^{83} +1934.26 q^{85} -476.276 q^{87} +1381.04 q^{89} -36.4467 q^{91} +1265.22 q^{93} -256.988 q^{95} +1182.04 q^{97} +586.460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 10 q^{5} - 46 q^{7} - 31 q^{9} + 64 q^{11} + 44 q^{13} - 134 q^{15} - 88 q^{17} + 94 q^{19} + 6 q^{21} - 69 q^{23} + 181 q^{25} - 20 q^{27} - 308 q^{29} - 140 q^{31} + 510 q^{33} - 192 q^{35}+ \cdots + 18 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 10 * q^5 - 46 * q^7 - 31 * q^9 + 64 * q^11 + 44 * q^13 - 134 * q^15 - 88 * q^17 + 94 * q^19 + 6 * q^21 - 69 * q^23 + 181 * q^25 - 20 * q^27 - 308 * q^29 - 140 * q^31 + 510 * q^33 - 192 * q^35 - 26 * q^37 + 180 * q^39 + 584 * q^41 + 478 * q^43 - 640 * q^45 - 28 * q^47 + 695 * q^49 - 334 * q^51 - 356 * q^53 - 1004 * q^55 + 520 * q^57 - 144 * q^59 + 1052 * q^61 + 120 * q^63 + 150 * q^65 + 2008 * q^67 - 92 * q^69 + 360 * q^71 - 252 * q^73 + 88 * q^75 + 576 * q^77 - 720 * q^79 - 325 * q^81 + 1404 * q^83 + 1632 * q^85 - 156 * q^87 + 534 * q^89 + 1526 * q^91 + 2342 * q^93 - 108 * q^95 + 736 * q^97 + 18 * q^99

Introduction

L-functions

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