LMFDB - Embedded newform 675.4.a.x.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,26,0,0,30] 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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.3173728.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 29x^{2} + 118 $ x^4 - 29*x^2 + 118
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.90965$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.90965 q^{2} +16.1047 q^{4} -2.10469 q^{7} +39.7912 q^{8} +40.3052 q^{11} -61.7328 q^{13} -10.3333 q^{14} +66.5234 q^{16} +99.2210 q^{17} +134.523 q^{19} +197.884 q^{22} +76.4985 q^{23} -303.087 q^{26} -33.8953 q^{28} +236.691 q^{29} +243.628 q^{31} +8.27738 q^{32} +487.141 q^{34} -57.4187 q^{37} +660.463 q^{38} -411.383 q^{41} -102.884 q^{43} +649.102 q^{44} +375.581 q^{46} -260.442 q^{47} -338.570 q^{49} -994.187 q^{52} +75.4706 q^{53} -83.7480 q^{56} +1162.07 q^{58} -39.2772 q^{59} -675.851 q^{61} +1196.13 q^{62} -491.548 q^{64} +601.176 q^{67} +1597.92 q^{68} +222.192 q^{71} +297.419 q^{73} -281.906 q^{74} +2166.46 q^{76} -84.8297 q^{77} -95.6999 q^{79} -2019.75 q^{82} -3.08384 q^{83} -505.126 q^{86} +1603.79 q^{88} +1342.73 q^{89} +129.928 q^{91} +1231.99 q^{92} -1278.68 q^{94} -1482.57 q^{97} -1662.26 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 26 q^{4} + 30 q^{7} + 22 q^{13} + 74 q^{16} + 346 q^{19} + 100 q^{22} - 174 q^{28} + 744 q^{31} + 796 q^{34} - 76 q^{37} + 280 q^{43} + 1656 q^{46} - 778 q^{49} - 2440 q^{52} + 2420 q^{58} + 178 q^{61}+ \cdots - 2050 q^{97}+O(q^{100}) $ 4 * q + 26 * q^4 + 30 * q^7 + 22 * q^13 + 74 * q^16 + 346 * q^19 + 100 * q^22 - 174 * q^28 + 744 * q^31 + 796 * q^34 - 76 * q^37 + 280 * q^43 + 1656 * q^46 - 778 * q^49 - 2440 * q^52 + 2420 * q^58 + 178 * q^61 + 70 * q^64 + 138 * q^67 + 1036 * q^73 + 4094 * q^76 + 2076 * q^79 - 5236 * q^82 + 6492 * q^88 + 2748 * q^91 - 1196 * q^94 - 2050 * q^97

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 $

$2$	4.90965	1.73582	0.867912	−	0.496718i	$-0.165462\pi$
$2$	4.90965	1.73582	0.867912	+	0.496718i	$0.165462\pi$
$3$	0	0
$3$	0	0
$4$	16.1047	2.01309
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.10469	−0.113642	−0.0568212	−	0.998384i	$-0.518096\pi$
$7$	−2.10469	−0.113642	−0.0568212	+	0.998384i	$0.518096\pi$
$8$	39.7912	1.75854
$9$	0	0
$10$	0	0
$11$	40.3052	1.10477	0.552385	−	0.833589i	$-0.313718\pi$
$11$	40.3052	1.10477	0.552385	+	0.833589i	$0.313718\pi$
$12$	0	0
$13$	−61.7328	−1.31705	−0.658523	−	0.752561i	$-0.728818\pi$
$13$	−61.7328	−1.31705	−0.658523	+	0.752561i	$0.728818\pi$
$14$	−10.3333	−0.197263
$15$	0	0
$16$	66.5234	1.03943
$17$	99.2210	1.41557	0.707783	−	0.706430i	$-0.249695\pi$
$17$	99.2210	1.41557	0.707783	+	0.706430i	$0.249695\pi$
$18$	0	0
$19$	134.523	1.62430	0.812152	−	0.583445i	$-0.198296\pi$
$19$	134.523	1.62430	0.812152	+	0.583445i	$0.198296\pi$
$20$	0	0
$21$	0	0
$22$	197.884	1.91769
$23$	76.4985	0.693524	0.346762	−	0.937953i	$-0.387281\pi$
$23$	76.4985	0.693524	0.346762	+	0.937953i	$0.387281\pi$
$24$	0	0
$25$	0	0
$26$	−303.087	−2.28616
$27$	0	0
$28$	−33.8953	−0.228772
$29$	236.691	1.51560	0.757801	−	0.652486i	$-0.226274\pi$
$29$	236.691	1.51560	0.757801	+	0.652486i	$0.226274\pi$
$30$	0	0
$31$	243.628	1.41151	0.705756	−	0.708455i	$-0.250607\pi$
$31$	243.628	1.41151	0.705756	+	0.708455i	$0.250607\pi$
$32$	8.27738	0.0457265
$33$	0	0
$34$	487.141	2.45717
$35$	0	0
$36$	0	0
$37$	−57.4187	−0.255124	−0.127562	−	0.991831i	$-0.540715\pi$
$37$	−57.4187	−0.255124	−0.127562	+	0.991831i	$0.540715\pi$
$38$	660.463	2.81951
$39$	0	0
$40$	0	0
$41$	−411.383	−1.56701	−0.783503	−	0.621389i	$-0.786569\pi$
$41$	−411.383	−1.56701	−0.783503	+	0.621389i	$0.786569\pi$
$42$	0	0
$43$	−102.884	−0.364877	−0.182439	−	0.983217i	$-0.558399\pi$
$43$	−102.884	−0.364877	−0.182439	+	0.983217i	$0.558399\pi$
$44$	649.102	2.22400
$45$	0	0
$46$	375.581	1.20384
$47$	−260.442	−0.808283	−0.404142	−	0.914696i	$-0.632430\pi$
$47$	−260.442	−0.808283	−0.404142	+	0.914696i	$0.632430\pi$
$48$	0	0
$49$	−338.570	−0.987085
$50$	0	0
$51$	0	0
$52$	−994.187	−2.65133
$53$	75.4706	0.195598	0.0977989	−	0.995206i	$-0.468820\pi$
$53$	75.4706	0.195598	0.0977989	+	0.995206i	$0.468820\pi$
$54$	0	0
$55$	0	0
$56$	−83.7480	−0.199845
$57$	0	0
$58$	1162.07	2.63082
$59$	−39.2772	−0.0866688	−0.0433344	−	0.999061i	$-0.513798\pi$
$59$	−39.2772	−0.0866688	−0.0433344	+	0.999061i	$0.513798\pi$
$60$	0	0
$61$	−675.851	−1.41859	−0.709294	−	0.704912i	$-0.750986\pi$
$61$	−675.851	−1.41859	−0.709294	+	0.704912i	$0.750986\pi$
$62$	1196.13	2.45014
$63$	0	0
$64$	−491.548	−0.960055
$65$	0	0
$66$	0	0
$67$	601.176	1.09620	0.548100	−	0.836413i	$-0.315351\pi$
$67$	601.176	1.09620	0.548100	+	0.836413i	$0.315351\pi$
$68$	1597.92	2.84966
$69$	0	0
$70$	0	0
$71$	222.192	0.371400	0.185700	−	0.982607i	$-0.440545\pi$
$71$	222.192	0.371400	0.185700	+	0.982607i	$0.440545\pi$
$72$	0	0
$73$	297.419	0.476852	0.238426	−	0.971161i	$-0.423368\pi$
$73$	297.419	0.476852	0.238426	+	0.971161i	$0.423368\pi$
$74$	−281.906	−0.442850
$75$	0	0
$76$	2166.46	3.26986
$77$	−84.8297	−0.125549
$78$	0	0
$79$	−95.6999	−0.136292	−0.0681461	−	0.997675i	$-0.521708\pi$
$79$	−95.6999	−0.136292	−0.0681461	+	0.997675i	$0.521708\pi$
$80$	0	0
$81$	0	0
$82$	−2019.75	−2.72005
$83$	−3.08384	−0.00407826	−0.00203913	−	0.999998i	$-0.500649\pi$
$83$	−3.08384	−0.00407826	−0.00203913	+	0.999998i	$0.500649\pi$
$84$	0	0
$85$	0	0
$86$	−505.126	−0.633363
$87$	0	0
$88$	1603.79	1.94278
$89$	1342.73	1.59920	0.799601	−	0.600532i	$-0.205044\pi$
$89$	1342.73	1.59920	0.799601	+	0.600532i	$0.205044\pi$
$90$	0	0
$91$	129.928	0.149672
$92$	1231.99	1.39612
$93$	0	0
$94$	−1278.68	−1.40304
$95$	0	0
$96$	0	0
$97$	−1482.57	−1.55188	−0.775941	−	0.630806i	$-0.782724\pi$
$97$	−1482.57	−1.55188	−0.775941	+	0.630806i	$0.782724\pi$
$98$	−1662.26	−1.71341
$99$	0	0
$100$	0	0
$101$	−1535.82	−1.51306	−0.756532	−	0.653957i	$-0.773108\pi$
$101$	−1535.82	−1.51306	−0.756532	+	0.653957i	$0.773108\pi$
$102$	0	0
$103$	356.581	0.341116	0.170558	−	0.985348i	$-0.445443\pi$
$103$	356.581	0.341116	0.170558	+	0.985348i	$0.445443\pi$
$104$	−2456.42	−2.31608
$105$	0	0
$106$	370.534	0.339523
$107$	−1580.23	−1.42773	−0.713864	−	0.700285i	$-0.753056\pi$
$107$	−1580.23	−1.42773	−0.713864	+	0.700285i	$0.753056\pi$
$108$	0	0
$109$	1189.98	1.04568	0.522842	−	0.852430i	$-0.324872\pi$
$109$	1189.98	1.04568	0.522842	+	0.852430i	$0.324872\pi$
$110$	0	0
$111$	0	0
$112$	−140.011	−0.118123
$113$	−369.022	−0.307209	−0.153605	−	0.988132i	$-0.549088\pi$
$113$	−369.022	−0.307209	−0.153605	+	0.988132i	$0.549088\pi$
$114$	0	0
$115$	0	0
$116$	3811.84	3.05104
$117$	0	0
$118$	−192.837	−0.150442
$119$	−208.829	−0.160868
$120$	0	0
$121$	293.506	0.220516
$122$	−3318.20	−2.46242
$123$	0	0
$124$	3923.55	2.84150
$125$	0	0
$126$	0	0
$127$	1414.84	0.988560	0.494280	−	0.869303i	$-0.335432\pi$
$127$	1414.84	0.988560	0.494280	+	0.869303i	$0.335432\pi$
$128$	−2479.55	−1.71221
$129$	0	0
$130$	0	0
$131$	−155.161	−0.103484	−0.0517422	−	0.998660i	$-0.516477\pi$
$131$	−155.161	−0.103484	−0.0517422	+	0.998660i	$0.516477\pi$
$132$	0	0
$133$	−283.130	−0.184590
$134$	2951.57	1.90281
$135$	0	0
$136$	3948.12	2.48933
$137$	−2447.63	−1.52639	−0.763194	−	0.646169i	$-0.776370\pi$
$137$	−2447.63	−1.52639	−0.763194	+	0.646169i	$0.776370\pi$
$138$	0	0
$139$	1463.98	0.893329	0.446665	−	0.894702i	$-0.352612\pi$
$139$	1463.98	0.893329	0.446665	+	0.894702i	$0.352612\pi$
$140$	0	0
$141$	0	0
$142$	1090.89	0.644685
$143$	−2488.15	−1.45503
$144$	0	0
$145$	0	0
$146$	1460.22	0.827732
$147$	0	0
$148$	−924.711	−0.513586
$149$	−3363.34	−1.84923	−0.924616	−	0.380899i	$-0.875614\pi$
$149$	−3363.34	−1.84923	−0.924616	+	0.380899i	$0.875614\pi$
$150$	0	0
$151$	2004.07	1.08006	0.540029	−	0.841647i	$-0.318413\pi$
$151$	2004.07	1.08006	0.540029	+	0.841647i	$0.318413\pi$
$152$	5352.85	2.85640
$153$	0	0
$154$	−416.485	−0.217930
$155$	0	0
$156$	0	0
$157$	951.144	0.483500	0.241750	−	0.970339i	$-0.422279\pi$
$157$	951.144	0.483500	0.241750	+	0.970339i	$0.422279\pi$
$158$	−469.853	−0.236579
$159$	0	0
$160$	0	0
$161$	−161.005	−0.0788137
$162$	0	0
$163$	−3152.95	−1.51508	−0.757539	−	0.652790i	$-0.773598\pi$
$163$	−3152.95	−1.51508	−0.757539	+	0.652790i	$0.773598\pi$
$164$	−6625.19	−3.15452
$165$	0	0
$166$	−15.1406	−0.00707914
$167$	−1886.55	−0.874165	−0.437083	−	0.899421i	$-0.643988\pi$
$167$	−1886.55	−0.874165	−0.437083	+	0.899421i	$0.643988\pi$
$168$	0	0
$169$	1613.94	0.734610
$170$	0	0
$171$	0	0
$172$	−1656.92	−0.734529
$173$	−1036.73	−0.455615	−0.227808	−	0.973706i	$-0.573156\pi$
$173$	−1036.73	−0.455615	−0.227808	+	0.973706i	$0.573156\pi$
$174$	0	0
$175$	0	0
$176$	2681.24	1.14833
$177$	0	0
$178$	6592.33	2.77593
$179$	3790.25	1.58266	0.791332	−	0.611387i	$-0.209388\pi$
$179$	3790.25	1.58266	0.791332	+	0.611387i	$0.209388\pi$
$180$	0	0
$181$	946.498	0.388689	0.194344	−	0.980933i	$-0.437742\pi$
$181$	946.498	0.388689	0.194344	+	0.980933i	$0.437742\pi$
$182$	637.902	0.259805
$183$	0	0
$184$	3043.97	1.21959
$185$	0	0
$186$	0	0
$187$	3999.12	1.56387
$188$	−4194.33	−1.62714
$189$	0	0
$190$	0	0
$191$	2618.92	0.992137	0.496068	−	0.868284i	$-0.334777\pi$
$191$	2618.92	0.992137	0.496068	+	0.868284i	$0.334777\pi$
$192$	0	0
$193$	−2587.20	−0.964927	−0.482463	−	0.875916i	$-0.660258\pi$
$193$	−2587.20	−0.964927	−0.482463	+	0.875916i	$0.660258\pi$
$194$	−7278.92	−2.69379
$195$	0	0
$196$	−5452.57	−1.98709
$197$	628.328	0.227241	0.113621	−	0.993524i	$-0.463755\pi$
$197$	628.328	0.227241	0.113621	+	0.993524i	$0.463755\pi$
$198$	0	0
$199$	−2892.44	−1.03035	−0.515175	−	0.857085i	$-0.672273\pi$
$199$	−2892.44	−1.03035	−0.515175	+	0.857085i	$0.672273\pi$
$200$	0	0
$201$	0	0
$202$	−7540.32	−2.62641
$203$	−498.161	−0.172237
$204$	0	0
$205$	0	0
$206$	1750.69	0.592118
$207$	0	0
$208$	−4106.68	−1.36898
$209$	5421.99	1.79448
$210$	0	0
$211$	−2864.38	−0.934560	−0.467280	−	0.884109i	$-0.654766\pi$
$211$	−2864.38	−0.934560	−0.467280	+	0.884109i	$0.654766\pi$
$212$	1215.43	0.393755
$213$	0	0
$214$	−7758.39	−2.47828
$215$	0	0
$216$	0	0
$217$	−512.761	−0.160408
$218$	5842.39	1.81512
$219$	0	0
$220$	0	0
$221$	−6125.19	−1.86437
$222$	0	0
$223$	2244.32	0.673950	0.336975	−	0.941514i	$-0.390596\pi$
$223$	2244.32	0.673950	0.336975	+	0.941514i	$0.390596\pi$
$224$	−17.4213	−0.00519647
$225$	0	0
$226$	−1811.77	−0.533261
$227$	−5959.64	−1.74253	−0.871267	−	0.490809i	$-0.836701\pi$
$227$	−5959.64	−1.74253	−0.871267	+	0.490809i	$0.836701\pi$
$228$	0	0
$229$	2181.56	0.629527	0.314764	−	0.949170i	$-0.398075\pi$
$229$	2181.56	0.629527	0.314764	+	0.949170i	$0.398075\pi$
$230$	0	0
$231$	0	0
$232$	9418.23	2.66525
$233$	−4013.04	−1.12834	−0.564170	−	0.825659i	$-0.690804\pi$
$233$	−4013.04	−1.12834	−0.564170	+	0.825659i	$0.690804\pi$
$234$	0	0
$235$	0	0
$236$	−632.547	−0.174472
$237$	0	0
$238$	−1025.28	−0.279239
$239$	6489.56	1.75638	0.878190	−	0.478312i	$-0.158751\pi$
$239$	6489.56	1.75638	0.878190	+	0.478312i	$0.158751\pi$
$240$	0	0
$241$	1886.47	0.504224	0.252112	−	0.967698i	$-0.418875\pi$
$241$	1886.47	0.504224	0.252112	+	0.967698i	$0.418875\pi$
$242$	1441.01	0.382777
$243$	0	0
$244$	−10884.4	−2.85574
$245$	0	0
$246$	0	0
$247$	−8304.51	−2.13928
$248$	9694.25	2.48220
$249$	0	0
$250$	0	0
$251$	−6307.08	−1.58605	−0.793026	−	0.609187i	$-0.791496\pi$
$251$	−6307.08	−1.58605	−0.793026	+	0.609187i	$0.791496\pi$
$252$	0	0
$253$	3083.29	0.766184
$254$	6946.39	1.71597
$255$	0	0
$256$	−8241.35	−2.01205
$257$	−4384.12	−1.06410	−0.532050	−	0.846713i	$-0.678578\pi$
$257$	−4384.12	−1.06410	−0.532050	+	0.846713i	$0.678578\pi$
$258$	0	0
$259$	120.848	0.0289929
$260$	0	0
$261$	0	0
$262$	−761.785	−0.179631
$263$	2927.40	0.686353	0.343177	−	0.939271i	$-0.388497\pi$
$263$	2927.40	0.686353	0.343177	+	0.939271i	$0.388497\pi$
$264$	0	0
$265$	0	0
$266$	−1390.07	−0.320416
$267$	0	0
$268$	9681.76	2.20674
$269$	−1869.35	−0.423703	−0.211852	−	0.977302i	$-0.567949\pi$
$269$	−1869.35	−0.423703	−0.211852	+	0.977302i	$0.567949\pi$
$270$	0	0
$271$	−111.800	−0.0250603	−0.0125302	−	0.999921i	$-0.503989\pi$
$271$	−111.800	−0.0250603	−0.0125302	+	0.999921i	$0.503989\pi$
$272$	6600.52	1.47138
$273$	0	0
$274$	−12017.0	−2.64954
$275$	0	0
$276$	0	0
$277$	4454.02	0.966124	0.483062	−	0.875586i	$-0.339525\pi$
$277$	4454.02	0.966124	0.483062	+	0.875586i	$0.339525\pi$
$278$	7187.61	1.55066
$279$	0	0
$280$	0	0
$281$	2236.64	0.474828	0.237414	−	0.971409i	$-0.423700\pi$
$281$	2236.64	0.474828	0.237414	+	0.971409i	$0.423700\pi$
$282$	0	0
$283$	−1629.67	−0.342309	−0.171155	−	0.985244i	$-0.554750\pi$
$283$	−1629.67	−0.342309	−0.171155	+	0.985244i	$0.554750\pi$
$284$	3578.34	0.747660
$285$	0	0
$286$	−12216.0	−2.52568
$287$	865.832	0.178078
$288$	0	0
$289$	4931.81	1.00383
$290$	0	0
$291$	0	0
$292$	4789.84	0.959945
$293$	−2540.47	−0.506538	−0.253269	−	0.967396i	$-0.581506\pi$
$293$	−2540.47	−0.506538	−0.253269	+	0.967396i	$0.581506\pi$
$294$	0	0
$295$	0	0
$296$	−2284.76	−0.448645
$297$	0	0
$298$	−16512.8	−3.20994
$299$	−4722.47	−0.913403
$300$	0	0
$301$	216.539	0.0414655
$302$	9839.27	1.87479
$303$	0	0
$304$	8948.96	1.68835
$305$	0	0
$306$	0	0
$307$	4752.32	0.883482	0.441741	−	0.897143i	$-0.354361\pi$
$307$	4752.32	0.883482	0.441741	+	0.897143i	$0.354361\pi$
$308$	−1366.16	−0.252740
$309$	0	0
$310$	0	0
$311$	−7665.82	−1.39771	−0.698857	−	0.715262i	$-0.746308\pi$
$311$	−7665.82	−1.39771	−0.698857	+	0.715262i	$0.746308\pi$
$312$	0	0
$313$	−7864.29	−1.42018	−0.710089	−	0.704112i	$-0.751345\pi$
$313$	−7864.29	−1.42018	−0.710089	+	0.704112i	$0.751345\pi$
$314$	4669.78	0.839271
$315$	0	0
$316$	−1541.22	−0.274368
$317$	3845.65	0.681367	0.340684	−	0.940178i	$-0.389342\pi$
$317$	3845.65	0.681367	0.340684	+	0.940178i	$0.389342\pi$
$318$	0	0
$319$	9539.88	1.67439
$320$	0	0
$321$	0	0
$322$	−790.481	−0.136807
$323$	13347.5	2.29931
$324$	0	0
$325$	0	0
$326$	−15479.9	−2.62991
$327$	0	0
$328$	−16369.4	−2.75564
$329$	548.148	0.0918553
$330$	0	0
$331$	−700.894	−0.116389	−0.0581943	−	0.998305i	$-0.518534\pi$
$331$	−700.894	−0.116389	−0.0581943	+	0.998305i	$0.518534\pi$
$332$	−49.6643	−0.00820989
$333$	0	0
$334$	−9262.30	−1.51740
$335$	0	0
$336$	0	0
$337$	12070.9	1.95118	0.975588	−	0.219610i	$-0.0704786\pi$
$337$	12070.9	1.95118	0.975588	+	0.219610i	$0.0704786\pi$
$338$	7923.88	1.27515
$339$	0	0
$340$	0	0
$341$	9819.47	1.55940
$342$	0	0
$343$	1434.49	0.225817
$344$	−4093.89	−0.641651
$345$	0	0
$346$	−5090.01	−0.790868
$347$	8064.60	1.24764	0.623819	−	0.781569i	$-0.285580\pi$
$347$	8064.60	1.24764	0.623819	+	0.781569i	$0.285580\pi$
$348$	0	0
$349$	8726.16	1.33840	0.669199	−	0.743083i	$-0.266637\pi$
$349$	8726.16	1.33840	0.669199	+	0.743083i	$0.266637\pi$
$350$	0	0
$351$	0	0
$352$	333.621	0.0505173
$353$	6125.68	0.923618	0.461809	−	0.886979i	$-0.347201\pi$
$353$	6125.68	0.923618	0.461809	+	0.886979i	$0.347201\pi$
$354$	0	0
$355$	0	0
$356$	21624.2	3.21933
$357$	0	0
$358$	18608.8	2.74723
$359$	−567.297	−0.0834005	−0.0417002	−	0.999130i	$-0.513277\pi$
$359$	−567.297	−0.0834005	−0.0417002	+	0.999130i	$0.513277\pi$
$360$	0	0
$361$	11237.6	1.63837
$362$	4646.98	0.674695
$363$	0	0
$364$	2092.45	0.301303
$365$	0	0
$366$	0	0
$367$	−380.770	−0.0541581	−0.0270791	−	0.999633i	$-0.508621\pi$
$367$	−380.770	−0.0541581	−0.0270791	+	0.999633i	$0.508621\pi$
$368$	5088.95	0.720869
$369$	0	0
$370$	0	0
$371$	−158.842	−0.0222282
$372$	0	0
$373$	1049.75	0.145721	0.0728604	−	0.997342i	$-0.476787\pi$
$373$	1049.75	0.145721	0.0728604	+	0.997342i	$0.476787\pi$
$374$	19634.3	2.71461
$375$	0	0
$376$	−10363.3	−1.42140
$377$	−14611.6	−1.99612
$378$	0	0
$379$	3574.95	0.484520	0.242260	−	0.970211i	$-0.422111\pi$
$379$	3574.95	0.484520	0.242260	+	0.970211i	$0.422111\pi$
$380$	0	0
$381$	0	0
$382$	12858.0	1.72217
$383$	−14221.2	−1.89731	−0.948656	−	0.316310i	$-0.897556\pi$
$383$	−14221.2	−1.89731	−0.948656	+	0.316310i	$0.897556\pi$
$384$	0	0
$385$	0	0
$386$	−12702.3	−1.67494
$387$	0	0
$388$	−23876.4	−3.12407
$389$	7982.31	1.04041	0.520205	−	0.854042i	$-0.325856\pi$
$389$	7982.31	1.04041	0.520205	+	0.854042i	$0.325856\pi$
$390$	0	0
$391$	7590.26	0.981729
$392$	−13472.1	−1.73583
$393$	0	0
$394$	3084.87	0.394451
$395$	0	0
$396$	0	0
$397$	2355.10	0.297731	0.148865	−	0.988857i	$-0.452438\pi$
$397$	2355.10	0.297731	0.148865	+	0.988857i	$0.452438\pi$
$398$	−14200.9	−1.78851
$399$	0	0
$400$	0	0
$401$	−6419.98	−0.799498	−0.399749	−	0.916625i	$-0.630903\pi$
$401$	−6419.98	−0.799498	−0.399749	+	0.916625i	$0.630903\pi$
$402$	0	0
$403$	−15039.8	−1.85903
$404$	−24733.8	−3.04593
$405$	0	0
$406$	−2445.80	−0.298973
$407$	−2314.27	−0.281853
$408$	0	0
$409$	−8435.21	−1.01979	−0.509895	−	0.860236i	$-0.670316\pi$
$409$	−8435.21	−1.01979	−0.509895	+	0.860236i	$0.670316\pi$
$410$	0	0
$411$	0	0
$412$	5742.63	0.686697
$413$	82.6662	0.00984925
$414$	0	0
$415$	0	0
$416$	−510.986	−0.0602239
$417$	0	0
$418$	26620.1	3.11491
$419$	−9663.61	−1.12673	−0.563363	−	0.826210i	$-0.690493\pi$
$419$	−9663.61	−1.12673	−0.563363	+	0.826210i	$0.690493\pi$
$420$	0	0
$421$	−3723.76	−0.431081	−0.215540	−	0.976495i	$-0.569151\pi$
$421$	−3723.76	−0.431081	−0.215540	+	0.976495i	$0.569151\pi$
$422$	−14063.1	−1.62223
$423$	0	0
$424$	3003.07	0.343966
$425$	0	0
$426$	0	0
$427$	1422.46	0.161212
$428$	−25449.2	−2.87414
$429$	0	0
$430$	0	0
$431$	2671.60	0.298577	0.149288	−	0.988794i	$-0.452302\pi$
$431$	2671.60	0.298577	0.149288	+	0.988794i	$0.452302\pi$
$432$	0	0
$433$	−2723.52	−0.302273	−0.151137	−	0.988513i	$-0.548293\pi$
$433$	−2723.52	−0.302273	−0.151137	+	0.988513i	$0.548293\pi$
$434$	−2517.48	−0.278440
$435$	0	0
$436$	19164.3	2.10505
$437$	10290.8	1.12649
$438$	0	0
$439$	8281.91	0.900396	0.450198	−	0.892929i	$-0.351353\pi$
$439$	8281.91	0.900396	0.450198	+	0.892929i	$0.351353\pi$
$440$	0	0
$441$	0	0
$442$	−30072.6	−3.23621
$443$	−6286.69	−0.674243	−0.337121	−	0.941461i	$-0.609453\pi$
$443$	−6286.69	−0.674243	−0.337121	+	0.941461i	$0.609453\pi$
$444$	0	0
$445$	0	0
$446$	11018.8	1.16986
$447$	0	0
$448$	1034.56	0.109103
$449$	−4737.77	−0.497971	−0.248986	−	0.968507i	$-0.580097\pi$
$449$	−4737.77	−0.497971	−0.248986	+	0.968507i	$0.580097\pi$
$450$	0	0
$451$	−16580.9	−1.73118
$452$	−5942.98	−0.618439
$453$	0	0
$454$	−29259.8	−3.02473
$455$	0	0
$456$	0	0
$457$	−13151.5	−1.34617	−0.673085	−	0.739565i	$-0.735031\pi$
$457$	−13151.5	−1.34617	−0.673085	+	0.739565i	$0.735031\pi$
$458$	10710.7	1.09275
$459$	0	0
$460$	0	0
$461$	18163.7	1.83507	0.917533	−	0.397658i	$-0.130177\pi$
$461$	18163.7	1.83507	0.917533	+	0.397658i	$0.130177\pi$
$462$	0	0
$463$	−12788.0	−1.28361	−0.641803	−	0.766869i	$-0.721813\pi$
$463$	−12788.0	−1.28361	−0.641803	+	0.766869i	$0.721813\pi$
$464$	15745.5	1.57536
$465$	0	0
$466$	−19702.6	−1.95860
$467$	3978.90	0.394265	0.197132	−	0.980377i	$-0.436837\pi$
$467$	3978.90	0.394265	0.197132	+	0.980377i	$0.436837\pi$
$468$	0	0
$469$	−1265.29	−0.124575
$470$	0	0
$471$	0	0
$472$	−1562.89	−0.152410
$473$	−4146.77	−0.403105
$474$	0	0
$475$	0	0
$476$	−3363.13	−0.323842
$477$	0	0
$478$	31861.5	3.04877
$479$	1095.93	0.104539	0.0522694	−	0.998633i	$-0.483355\pi$
$479$	1095.93	0.104539	0.0522694	+	0.998633i	$0.483355\pi$
$480$	0	0
$481$	3544.62	0.336010
$482$	9261.90	0.875244
$483$	0	0
$484$	4726.83	0.443917
$485$	0	0
$486$	0	0
$487$	15345.9	1.42790	0.713951	−	0.700196i	$-0.246904\pi$
$487$	15345.9	1.42790	0.713951	+	0.700196i	$0.246904\pi$
$488$	−26892.9	−2.49464
$489$	0	0
$490$	0	0
$491$	−2148.29	−0.197457	−0.0987283	−	0.995114i	$-0.531477\pi$
$491$	−2148.29	−0.197457	−0.0987283	+	0.995114i	$0.531477\pi$
$492$	0	0
$493$	23484.7	2.14544
$494$	−40772.3	−3.71342
$495$	0	0
$496$	16207.0	1.46717
$497$	−467.645	−0.0422068
$498$	0	0
$499$	771.203	0.0691860	0.0345930	−	0.999401i	$-0.488987\pi$
$499$	771.203	0.0691860	0.0345930	+	0.999401i	$0.488987\pi$
$500$	0	0
$501$	0	0
$502$	−30965.6	−2.75311
$503$	−4031.11	−0.357333	−0.178666	−	0.983910i	$-0.557178\pi$
$503$	−4031.11	−0.357333	−0.178666	+	0.983910i	$0.557178\pi$
$504$	0	0
$505$	0	0
$506$	15137.9	1.32996
$507$	0	0
$508$	22785.6	1.99006
$509$	−7337.75	−0.638979	−0.319489	−	0.947590i	$-0.603511\pi$
$509$	−7337.75	−0.638979	−0.319489	+	0.947590i	$0.603511\pi$
$510$	0	0
$511$	−625.973	−0.0541907
$512$	−20625.7	−1.78035
$513$	0	0
$514$	−21524.5	−1.84709
$515$	0	0
$516$	0	0
$517$	−10497.1	−0.892967
$518$	593.324	0.0503266
$519$	0	0
$520$	0	0
$521$	−6972.35	−0.586304	−0.293152	−	0.956066i	$-0.594704\pi$
$521$	−6972.35	−0.586304	−0.293152	+	0.956066i	$0.594704\pi$
$522$	0	0
$523$	−9040.75	−0.755878	−0.377939	−	0.925830i	$-0.623367\pi$
$523$	−9040.75	−0.755878	−0.377939	+	0.925830i	$0.623367\pi$
$524$	−2498.81	−0.208323
$525$	0	0
$526$	14372.5	1.19139
$527$	24173.0	1.99809
$528$	0	0
$529$	−6314.97	−0.519025
$530$	0	0
$531$	0	0
$532$	−4559.71	−0.371595
$533$	25395.8	2.06382
$534$	0	0
$535$	0	0
$536$	23921.5	1.92771
$537$	0	0
$538$	−9177.85	−0.735475
$539$	−13646.1	−1.09050
$540$	0	0
$541$	22956.9	1.82439	0.912196	−	0.409755i	$-0.134386\pi$
$541$	22956.9	1.82439	0.912196	+	0.409755i	$0.134386\pi$
$542$	−548.898	−0.0435003
$543$	0	0
$544$	821.290	0.0647289
$545$	0	0
$546$	0	0
$547$	−734.019	−0.0573755	−0.0286877	−	0.999588i	$-0.509133\pi$
$547$	−734.019	−0.0573755	−0.0286877	+	0.999588i	$0.509133\pi$
$548$	−39418.3	−3.07275
$549$	0	0
$550$	0	0
$551$	31840.5	2.46180
$552$	0	0
$553$	201.418	0.0154886
$554$	21867.7	1.67702
$555$	0	0
$556$	23576.9	1.79835
$557$	8428.97	0.641197	0.320599	−	0.947215i	$-0.396116\pi$
$557$	8428.97	0.641197	0.320599	+	0.947215i	$0.396116\pi$
$558$	0	0
$559$	6351.34	0.480560
$560$	0	0
$561$	0	0
$562$	10981.1	0.824218
$563$	−9091.27	−0.680553	−0.340276	−	0.940325i	$-0.610521\pi$
$563$	−9091.27	−0.680553	−0.340276	+	0.940325i	$0.610521\pi$
$564$	0	0
$565$	0	0
$566$	−8001.09	−0.594189
$567$	0	0
$568$	8841.30	0.653121
$569$	−6766.12	−0.498507	−0.249253	−	0.968438i	$-0.580185\pi$
$569$	−6766.12	−0.498507	−0.249253	+	0.968438i	$0.580185\pi$
$570$	0	0
$571$	−3864.13	−0.283203	−0.141602	−	0.989924i	$-0.545225\pi$
$571$	−3864.13	−0.283203	−0.141602	+	0.989924i	$0.545225\pi$
$572$	−40070.9	−2.92911
$573$	0	0
$574$	4250.93	0.309112
$575$	0	0
$576$	0	0
$577$	3572.62	0.257765	0.128882	−	0.991660i	$-0.458861\pi$
$577$	3572.62	0.257765	0.128882	+	0.991660i	$0.458861\pi$
$578$	24213.5	1.74247
$579$	0	0
$580$	0	0
$581$	6.49052	0.000463463 0
$582$	0	0
$583$	3041.85	0.216090
$584$	11834.6	0.838564
$585$	0	0
$586$	−12472.8	−0.879262
$587$	−10186.2	−0.716236	−0.358118	−	0.933676i	$-0.616581\pi$
$587$	−10186.2	−0.716236	−0.358118	+	0.933676i	$0.616581\pi$
$588$	0	0
$589$	32773.7	2.29273
$590$	0	0
$591$	0	0
$592$	−3819.69	−0.265183
$593$	2691.78	0.186405	0.0932025	−	0.995647i	$-0.470290\pi$
$593$	2691.78	0.186405	0.0932025	+	0.995647i	$0.470290\pi$
$594$	0	0
$595$	0	0
$596$	−54165.6	−3.72266
$597$	0	0
$598$	−23185.7	−1.58551
$599$	4236.48	0.288978	0.144489	−	0.989506i	$-0.453846\pi$
$599$	4236.48	0.288978	0.144489	+	0.989506i	$0.453846\pi$
$600$	0	0
$601$	10749.6	0.729595	0.364798	−	0.931087i	$-0.381138\pi$
$601$	10749.6	0.729595	0.364798	+	0.931087i	$0.381138\pi$
$602$	1063.13	0.0719768
$603$	0	0
$604$	32274.9	2.17425
$605$	0	0
$606$	0	0
$607$	19499.1	1.30386	0.651932	−	0.758278i	$-0.273959\pi$
$607$	19499.1	1.30386	0.651932	+	0.758278i	$0.273959\pi$
$608$	1113.50	0.0742738
$609$	0	0
$610$	0	0
$611$	16077.8	1.06455
$612$	0	0
$613$	−1509.82	−0.0994796	−0.0497398	−	0.998762i	$-0.515839\pi$
$613$	−1509.82	−0.0994796	−0.0497398	+	0.998762i	$0.515839\pi$
$614$	23332.2	1.53357
$615$	0	0
$616$	−3375.48	−0.220782
$617$	27905.1	1.82077	0.910387	−	0.413758i	$-0.135784\pi$
$617$	27905.1	1.82077	0.910387	+	0.413758i	$0.135784\pi$
$618$	0	0
$619$	1448.88	0.0940801	0.0470401	−	0.998893i	$-0.485021\pi$
$619$	1448.88	0.0940801	0.0470401	+	0.998893i	$0.485021\pi$
$620$	0	0
$621$	0	0
$622$	−37636.5	−2.42619
$623$	−2826.02	−0.181737
$624$	0	0
$625$	0	0
$626$	−38610.9	−2.46518
$627$	0	0
$628$	15317.9	0.973327
$629$	−5697.15	−0.361145
$630$	0	0
$631$	7476.02	0.471657	0.235829	−	0.971795i	$-0.424220\pi$
$631$	7476.02	0.471657	0.235829	+	0.971795i	$0.424220\pi$
$632$	−3808.01	−0.239675
$633$	0	0
$634$	18880.8	1.18273
$635$	0	0
$636$	0	0
$637$	20900.9	1.30004
$638$	46837.5	2.90645
$639$	0	0
$640$	0	0
$641$	50.3581	0.00310301	0.00155150	−	0.999999i	$-0.499506\pi$
$641$	50.3581	0.00310301	0.00155150	+	0.999999i	$0.499506\pi$
$642$	0	0
$643$	16728.8	1.02600	0.513002	−	0.858387i	$-0.328533\pi$
$643$	16728.8	1.02600	0.513002	+	0.858387i	$0.328533\pi$
$644$	−2592.94	−0.158659
$645$	0	0
$646$	65531.8	3.99120
$647$	13149.5	0.799010	0.399505	−	0.916731i	$-0.369182\pi$
$647$	13149.5	0.799010	0.399505	+	0.916731i	$0.369182\pi$
$648$	0	0
$649$	−1583.07	−0.0957490
$650$	0	0
$651$	0	0
$652$	−50777.2	−3.04998
$653$	−4131.96	−0.247620	−0.123810	−	0.992306i	$-0.539511\pi$
$653$	−4131.96	−0.247620	−0.123810	+	0.992306i	$0.539511\pi$
$654$	0	0
$655$	0	0
$656$	−27366.6	−1.62879
$657$	0	0
$658$	2691.22	0.159445
$659$	4141.95	0.244837	0.122419	−	0.992479i	$-0.460935\pi$
$659$	4141.95	0.244837	0.122419	+	0.992479i	$0.460935\pi$
$660$	0	0
$661$	−25593.3	−1.50600	−0.753000	−	0.658021i	$-0.771394\pi$
$661$	−25593.3	−1.50600	−0.753000	+	0.658021i	$0.771394\pi$
$662$	−3441.15	−0.202030
$663$	0	0
$664$	−122.710	−0.00717178
$665$	0	0
$666$	0	0
$667$	18106.5	1.05111
$668$	−30382.3	−1.75977
$669$	0	0
$670$	0	0
$671$	−27240.3	−1.56721
$672$	0	0
$673$	−27185.3	−1.55708	−0.778541	−	0.627593i	$-0.784040\pi$
$673$	−27185.3	−1.55708	−0.778541	+	0.627593i	$0.784040\pi$
$674$	59264.1	3.38690
$675$	0	0
$676$	25992.0	1.47883
$677$	3359.82	0.190736	0.0953680	−	0.995442i	$-0.469597\pi$
$677$	3359.82	0.190736	0.0953680	+	0.995442i	$0.469597\pi$
$678$	0	0
$679$	3120.35	0.176360
$680$	0	0
$681$	0	0
$682$	48210.2	2.70684
$683$	25176.1	1.41045	0.705223	−	0.708985i	$-0.250847\pi$
$683$	25176.1	1.41045	0.705223	+	0.708985i	$0.250847\pi$
$684$	0	0
$685$	0	0
$686$	7042.86	0.391979
$687$	0	0
$688$	−6844.22	−0.379264
$689$	−4659.01	−0.257611
$690$	0	0
$691$	−5980.72	−0.329258	−0.164629	−	0.986356i	$-0.552643\pi$
$691$	−5980.72	−0.329258	−0.164629	+	0.986356i	$0.552643\pi$
$692$	−16696.3	−0.917193
$693$	0	0
$694$	39594.4	2.16568
$695$	0	0
$696$	0	0
$697$	−40817.8	−2.21820
$698$	42842.4	2.32322
$699$	0	0
$700$	0	0
$701$	8195.53	0.441570	0.220785	−	0.975322i	$-0.429138\pi$
$701$	8195.53	0.441570	0.220785	+	0.975322i	$0.429138\pi$
$702$	0	0
$703$	−7724.17	−0.414399
$704$	−19811.9	−1.06064
$705$	0	0
$706$	30075.0	1.60324
$707$	3232.41	0.171948
$708$	0	0
$709$	6700.56	0.354929	0.177465	−	0.984127i	$-0.443210\pi$
$709$	6700.56	0.354929	0.177465	+	0.984127i	$0.443210\pi$
$710$	0	0
$711$	0	0
$712$	53428.8	2.81226
$713$	18637.2	0.978918
$714$	0	0
$715$	0	0
$716$	61040.8	3.18604
$717$	0	0
$718$	−2785.23	−0.144769
$719$	319.573	0.0165759	0.00828794	−	0.999966i	$-0.497362\pi$
$719$	319.573	0.0165759	0.00828794	+	0.999966i	$0.497362\pi$
$720$	0	0
$721$	−750.492	−0.0387653
$722$	55172.5	2.84392
$723$	0	0
$724$	15243.1	0.782464
$725$	0	0
$726$	0	0
$727$	4869.19	0.248402	0.124201	−	0.992257i	$-0.460363\pi$
$727$	4869.19	0.248402	0.124201	+	0.992257i	$0.460363\pi$
$728$	5170.00	0.263204
$729$	0	0
$730$	0	0
$731$	−10208.3	−0.516508
$732$	0	0
$733$	−26969.4	−1.35899	−0.679495	−	0.733681i	$-0.737801\pi$
$733$	−26969.4	−1.35899	−0.679495	+	0.733681i	$0.737801\pi$
$734$	−1869.45	−0.0940090
$735$	0	0
$736$	633.208	0.0317124
$737$	24230.5	1.21105
$738$	0	0
$739$	−29544.7	−1.47066	−0.735331	−	0.677708i	$-0.762973\pi$
$739$	−29544.7	−1.47066	−0.735331	+	0.677708i	$0.762973\pi$
$740$	0	0
$741$	0	0
$742$	−779.859	−0.0385842
$743$	37991.6	1.87588	0.937939	−	0.346800i	$-0.112732\pi$
$743$	37991.6	1.87588	0.937939	+	0.346800i	$0.112732\pi$
$744$	0	0
$745$	0	0
$746$	5153.90	0.252946
$747$	0	0
$748$	64404.6	3.14821
$749$	3325.89	0.162250
$750$	0	0
$751$	−18817.9	−0.914349	−0.457174	−	0.889377i	$-0.651138\pi$
$751$	−18817.9	−0.914349	−0.457174	+	0.889377i	$0.651138\pi$
$752$	−17325.5	−0.840153
$753$	0	0
$754$	−71738.0	−3.46491
$755$	0	0
$756$	0	0
$757$	6255.42	0.300339	0.150170	−	0.988660i	$-0.452018\pi$
$757$	6255.42	0.300339	0.150170	+	0.988660i	$0.452018\pi$
$758$	17551.8	0.841041
$759$	0	0
$760$	0	0
$761$	4900.19	0.233419	0.116709	−	0.993166i	$-0.462765\pi$
$761$	4900.19	0.233419	0.116709	+	0.993166i	$0.462765\pi$
$762$	0	0
$763$	−2504.54	−0.118834
$764$	42176.8	1.99726
$765$	0	0
$766$	−69821.3	−3.29340
$767$	2424.69	0.114147
$768$	0	0
$769$	21378.4	1.00250	0.501252	−	0.865301i	$-0.332873\pi$
$769$	21378.4	1.00250	0.501252	+	0.865301i	$0.332873\pi$
$770$	0	0
$771$	0	0
$772$	−41666.1	−1.94248
$773$	7694.82	0.358038	0.179019	−	0.983846i	$-0.442708\pi$
$773$	7694.82	0.358038	0.179019	+	0.983846i	$0.442708\pi$
$774$	0	0
$775$	0	0
$776$	−58993.4	−2.72904
$777$	0	0
$778$	39190.4	1.80597
$779$	−55340.6	−2.54529
$780$	0	0
$781$	8955.50	0.410311
$782$	37265.5	1.70411
$783$	0	0
$784$	−22522.9	−1.02600
$785$	0	0
$786$	0	0
$787$	−26712.1	−1.20989	−0.604945	−	0.796267i	$-0.706805\pi$
$787$	−26712.1	−1.20989	−0.604945	+	0.796267i	$0.706805\pi$
$788$	10119.0	0.457456
$789$	0	0
$790$	0	0
$791$	776.675	0.0349120
$792$	0	0
$793$	41722.2	1.86835
$794$	11562.7	0.516808
$795$	0	0
$796$	−46581.8	−2.07418
$797$	−12894.2	−0.573070	−0.286535	−	0.958070i	$-0.592503\pi$
$797$	−12894.2	−0.573070	−0.286535	+	0.958070i	$0.592503\pi$
$798$	0	0
$799$	−25841.3	−1.14418
$800$	0	0
$801$	0	0
$802$	−31519.9	−1.38779
$803$	11987.5	0.526812
$804$	0	0
$805$	0	0
$806$	−73840.4	−3.22695
$807$	0	0
$808$	−61111.9	−2.66078
$809$	13581.6	0.590241	0.295120	−	0.955460i	$-0.404640\pi$
$809$	13581.6	0.590241	0.295120	+	0.955460i	$0.404640\pi$
$810$	0	0
$811$	29019.1	1.25647	0.628235	−	0.778023i	$-0.283777\pi$
$811$	29019.1	1.25647	0.628235	+	0.778023i	$0.283777\pi$
$812$	−8022.72	−0.346727
$813$	0	0
$814$	−11362.3	−0.489247
$815$	0	0
$816$	0	0
$817$	−13840.4	−0.592672
$818$	−41414.0	−1.77018
$819$	0	0
$820$	0	0
$821$	7950.00	0.337950	0.168975	−	0.985620i	$-0.445954\pi$
$821$	7950.00	0.337950	0.168975	+	0.985620i	$0.445954\pi$
$822$	0	0
$823$	−10499.7	−0.444708	−0.222354	−	0.974966i	$-0.571374\pi$
$823$	−10499.7	−0.444708	−0.222354	+	0.974966i	$0.571374\pi$
$824$	14188.8	0.599867
$825$	0	0
$826$	405.862	0.0170966
$827$	−27097.0	−1.13936	−0.569682	−	0.821865i	$-0.692934\pi$
$827$	−27097.0	−1.13936	−0.569682	+	0.821865i	$0.692934\pi$
$828$	0	0
$829$	325.331	0.0136299	0.00681497	−	0.999977i	$-0.497831\pi$
$829$	325.331	0.0136299	0.00681497	+	0.999977i	$0.497831\pi$
$830$	0	0
$831$	0	0
$832$	30344.7	1.26444
$833$	−33593.3	−1.39728
$834$	0	0
$835$	0	0
$836$	87319.4	3.61245
$837$	0	0
$838$	−47444.9	−1.95580
$839$	30523.3	1.25600	0.627999	−	0.778214i	$-0.283874\pi$
$839$	30523.3	1.25600	0.627999	+	0.778214i	$0.283874\pi$
$840$	0	0
$841$	31633.8	1.29705
$842$	−18282.4	−0.748281
$843$	0	0
$844$	−46130.0	−1.88135
$845$	0	0
$846$	0	0
$847$	−617.739	−0.0250599
$848$	5020.56	0.203310
$849$	0	0
$850$	0	0
$851$	−4392.45	−0.176935
$852$	0	0
$853$	32915.4	1.32122	0.660612	−	0.750728i	$-0.270297\pi$
$853$	32915.4	1.32122	0.660612	+	0.750728i	$0.270297\pi$
$854$	6983.76	0.279835
$855$	0	0
$856$	−62879.3	−2.51071
$857$	−18915.5	−0.753959	−0.376979	−	0.926222i	$-0.623037\pi$
$857$	−18915.5	−0.753959	−0.376979	+	0.926222i	$0.623037\pi$
$858$	0	0
$859$	−31407.3	−1.24750	−0.623750	−	0.781624i	$-0.714392\pi$
$859$	−31407.3	−1.24750	−0.623750	+	0.781624i	$0.714392\pi$
$860$	0	0
$861$	0	0
$862$	13116.6	0.518277
$863$	−39038.2	−1.53983	−0.769916	−	0.638146i	$-0.779702\pi$
$863$	−39038.2	−1.53983	−0.769916	+	0.638146i	$0.779702\pi$
$864$	0	0
$865$	0	0
$866$	−13371.6	−0.524693
$867$	0	0
$868$	−8257.85	−0.322914
$869$	−3857.20	−0.150571
$870$	0	0
$871$	−37112.3	−1.44375
$872$	47350.8	1.83888
$873$	0	0
$874$	50524.5	1.95540
$875$	0	0
$876$	0	0
$877$	−47360.9	−1.82356	−0.911781	−	0.410677i	$-0.865293\pi$
$877$	−47360.9	−1.82356	−0.911781	+	0.410677i	$0.865293\pi$
$878$	40661.3	1.56293
$879$	0	0
$880$	0	0
$881$	−5167.57	−0.197616	−0.0988081	−	0.995107i	$-0.531503\pi$
$881$	−5167.57	−0.197616	−0.0988081	+	0.995107i	$0.531503\pi$
$882$	0	0
$883$	31450.1	1.19862	0.599308	−	0.800518i	$-0.295442\pi$
$883$	31450.1	1.19862	0.599308	+	0.800518i	$0.295442\pi$
$884$	−98644.3	−3.75313
$885$	0	0
$886$	−30865.4	−1.17037
$887$	−7264.18	−0.274980	−0.137490	−	0.990503i	$-0.543904\pi$
$887$	−7264.18	−0.274980	−0.137490	+	0.990503i	$0.543904\pi$
$888$	0	0
$889$	−2977.80	−0.112342
$890$	0	0
$891$	0	0
$892$	36144.1	1.35672
$893$	−35035.5	−1.31290
$894$	0	0
$895$	0	0
$896$	5218.68	0.194580
$897$	0	0
$898$	−23260.8	−0.864391
$899$	57664.6	2.13929
$900$	0	0
$901$	7488.27	0.276882
$902$	−81406.2	−3.00502
$903$	0	0
$904$	−14683.8	−0.540239
$905$	0	0
$906$	0	0
$907$	9535.80	0.349097	0.174549	−	0.984649i	$-0.444153\pi$
$907$	9535.80	0.349097	0.174549	+	0.984649i	$0.444153\pi$
$908$	−95978.2	−3.50787
$909$	0	0
$910$	0	0
$911$	50779.1	1.84675	0.923373	−	0.383904i	$-0.125421\pi$
$911$	50779.1	1.84675	0.923373	+	0.383904i	$0.125421\pi$
$912$	0	0
$913$	−124.295	−0.00450554
$914$	−64569.1	−2.33671
$915$	0	0
$916$	35133.4	1.26729
$917$	326.564	0.0117602
$918$	0	0
$919$	−44669.7	−1.60339	−0.801696	−	0.597732i	$-0.796069\pi$
$919$	−44669.7	−1.60339	−0.801696	+	0.597732i	$0.796069\pi$
$920$	0	0
$921$	0	0
$922$	89177.2	3.18535
$923$	−13716.6	−0.489151
$924$	0	0
$925$	0	0
$926$	−62784.8	−2.22812
$927$	0	0
$928$	1959.18	0.0693032
$929$	−23431.4	−0.827513	−0.413756	−	0.910388i	$-0.635783\pi$
$929$	−23431.4	−0.827513	−0.413756	+	0.910388i	$0.635783\pi$
$930$	0	0
$931$	−45545.6	−1.60333
$932$	−64628.8	−2.27144
$933$	0	0
$934$	19535.0	0.684375
$935$	0	0
$936$	0	0
$937$	25234.2	0.879792	0.439896	−	0.898049i	$-0.355015\pi$
$937$	25234.2	0.879792	0.439896	+	0.898049i	$0.355015\pi$
$938$	−6212.12	−0.216240
$939$	0	0
$940$	0	0
$941$	−47812.7	−1.65637	−0.828187	−	0.560452i	$-0.810628\pi$
$941$	−47812.7	−1.65637	−0.828187	+	0.560452i	$0.810628\pi$
$942$	0	0
$943$	−31470.2	−1.08676
$944$	−2612.86	−0.0900860
$945$	0	0
$946$	−20359.2	−0.699720
$947$	26024.4	0.893007	0.446504	−	0.894782i	$-0.352669\pi$
$947$	26024.4	0.893007	0.446504	+	0.894782i	$0.352669\pi$
$948$	0	0
$949$	−18360.5	−0.628037
$950$	0	0
$951$	0	0
$952$	−8309.56	−0.282893
$953$	−32403.9	−1.10143	−0.550716	−	0.834692i	$-0.685645\pi$
$953$	−32403.9	−1.10143	−0.550716	+	0.834692i	$0.685645\pi$
$954$	0	0
$955$	0	0
$956$	104512.	3.53574
$957$	0	0
$958$	5380.61	0.181461
$959$	5151.49	0.173462
$960$	0	0
$961$	29563.7	0.992369
$962$	17402.9	0.583254
$963$	0	0
$964$	30381.0	1.01505
$965$	0	0
$966$	0	0
$967$	50840.1	1.69070	0.845350	−	0.534213i	$-0.179392\pi$
$967$	50840.1	1.69070	0.845350	+	0.534213i	$0.179392\pi$
$968$	11679.0	0.387785
$969$	0	0
$970$	0	0
$971$	17336.6	0.572975	0.286488	−	0.958084i	$-0.407512\pi$
$971$	17336.6	0.572975	0.286488	+	0.958084i	$0.407512\pi$
$972$	0	0
$973$	−3081.21	−0.101520
$974$	75342.9	2.47859
$975$	0	0
$976$	−44960.0	−1.47452
$977$	48360.1	1.58360	0.791799	−	0.610781i	$-0.209145\pi$
$977$	48360.1	1.58360	0.791799	+	0.610781i	$0.209145\pi$
$978$	0	0
$979$	54118.9	1.76675
$980$	0	0
$981$	0	0
$982$	−10547.4	−0.342750
$983$	56102.8	1.82035	0.910174	−	0.414226i	$-0.135948\pi$
$983$	56102.8	1.82035	0.910174	+	0.414226i	$0.135948\pi$
$984$	0	0
$985$	0	0
$986$	115302.	3.72410
$987$	0	0
$988$	−133742.	−4.30656
$989$	−7870.50	−0.253051
$990$	0	0
$991$	−46186.7	−1.48049	−0.740247	−	0.672335i	$-0.765291\pi$
$991$	−46186.7	−1.48049	−0.740247	+	0.672335i	$0.765291\pi$
$992$	2016.60	0.0645436
$993$	0	0
$994$	−2295.98	−0.0732635
$995$	0	0
$996$	0	0
$997$	−16073.7	−0.510592	−0.255296	−	0.966863i	$-0.582173\pi$
$997$	−16073.7	−0.510592	−0.255296	+	0.966863i	$0.582173\pi$
$998$	3786.34	0.120095
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.x.1.4	yes	4
3.2	odd	2	inner	675.4.a.x.1.1	yes	4
5.2	odd	4		675.4.b.o.649.7		8
5.3	odd	4		675.4.b.o.649.2		8
5.4	even	2		675.4.a.w.1.1	✓	4
15.2	even	4		675.4.b.o.649.1		8
15.8	even	4		675.4.b.o.649.8		8
15.14	odd	2		675.4.a.w.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.w.1.1	✓	4	5.4	even	2
675.4.a.w.1.4	yes	4	15.14	odd	2
675.4.a.x.1.1	yes	4	3.2	odd	2	inner
675.4.a.x.1.4	yes	4	1.1	even	1	trivial
675.4.b.o.649.1		8	15.2	even	4
675.4.b.o.649.2		8	5.3	odd	4
675.4.b.o.649.7		8	5.2	odd	4
675.4.b.o.649.8		8	15.8	even	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+4.90965 q^{2} +16.1047 q^{4} -2.10469 q^{7} +39.7912 q^{8} +40.3052 q^{11} -61.7328 q^{13} -10.3333 q^{14} +66.5234 q^{16} +99.2210 q^{17} +134.523 q^{19} +197.884 q^{22} +76.4985 q^{23} -303.087 q^{26} -33.8953 q^{28} +236.691 q^{29} +243.628 q^{31} +8.27738 q^{32} +487.141 q^{34} -57.4187 q^{37} +660.463 q^{38} -411.383 q^{41} -102.884 q^{43} +649.102 q^{44} +375.581 q^{46} -260.442 q^{47} -338.570 q^{49} -994.187 q^{52} +75.4706 q^{53} -83.7480 q^{56} +1162.07 q^{58} -39.2772 q^{59} -675.851 q^{61} +1196.13 q^{62} -491.548 q^{64} +601.176 q^{67} +1597.92 q^{68} +222.192 q^{71} +297.419 q^{73} -281.906 q^{74} +2166.46 q^{76} -84.8297 q^{77} -95.6999 q^{79} -2019.75 q^{82} -3.08384 q^{83} -505.126 q^{86} +1603.79 q^{88} +1342.73 q^{89} +129.928 q^{91} +1231.99 q^{92} -1278.68 q^{94} -1482.57 q^{97} -1662.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 26 q^{4} + 30 q^{7} + 22 q^{13} + 74 q^{16} + 346 q^{19} + 100 q^{22} - 174 q^{28} + 744 q^{31} + 796 q^{34} - 76 q^{37} + 280 q^{43} + 1656 q^{46} - 778 q^{49} - 2440 q^{52} + 2420 q^{58} + 178 q^{61}+ \cdots - 2050 q^{97}+O(q^{100}) \) 4 * q + 26 * q^4 + 30 * q^7 + 22 * q^13 + 74 * q^16 + 346 * q^19 + 100 * q^22 - 174 * q^28 + 744 * q^31 + 796 * q^34 - 76 * q^37 + 280 * q^43 + 1656 * q^46 - 778 * q^49 - 2440 * q^52 + 2420 * q^58 + 178 * q^61 + 70 * q^64 + 138 * q^67 + 1036 * q^73 + 4094 * q^76 + 2076 * q^79 - 5236 * q^82 + 6492 * q^88 + 2748 * q^91 - 1196 * q^94 - 2050 * q^97

Introduction

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