LMFDB - Embedded newform 2254.4.a.z.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 200x^{12} + 15521x^{10} - 598294x^{8} + 12167812x^{6} - 125559722x^{4} + 539505876x^{2} - 324615200 $ x^14 - 200x^12 + 15521x^10 - 598294x^8 + 12167812x^6 - 125559722x^4 + 539505876x^2 - 324615200
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.40048$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -3.40048 q^{3} +4.00000 q^{4} -0.220651 q^{5} +6.80097 q^{6} -8.00000 q^{8} -15.4367 q^{9} +0.441301 q^{10} +49.1322 q^{11} -13.6019 q^{12} +37.0988 q^{13} +0.750319 q^{15} +16.0000 q^{16} -42.4366 q^{17} +30.8734 q^{18} -1.64187 q^{19} -0.882602 q^{20} -98.2644 q^{22} +23.0000 q^{23} +27.2039 q^{24} -124.951 q^{25} -74.1976 q^{26} +144.305 q^{27} +42.1069 q^{29} -1.50064 q^{30} -141.778 q^{31} -32.0000 q^{32} -167.073 q^{33} +84.8731 q^{34} -61.7468 q^{36} -211.538 q^{37} +3.28374 q^{38} -126.154 q^{39} +1.76520 q^{40} +221.663 q^{41} -207.246 q^{43} +196.529 q^{44} +3.40612 q^{45} -46.0000 q^{46} -118.778 q^{47} -54.4078 q^{48} +249.903 q^{50} +144.305 q^{51} +148.395 q^{52} +434.789 q^{53} -288.611 q^{54} -10.8410 q^{55} +5.58316 q^{57} -84.2138 q^{58} +69.1542 q^{59} +3.00127 q^{60} -103.321 q^{61} +283.555 q^{62} +64.0000 q^{64} -8.18587 q^{65} +334.147 q^{66} +8.86339 q^{67} -169.746 q^{68} -78.2112 q^{69} -172.114 q^{71} +123.494 q^{72} +291.160 q^{73} +423.077 q^{74} +424.895 q^{75} -6.56748 q^{76} +252.308 q^{78} -381.665 q^{79} -3.53041 q^{80} -73.9173 q^{81} -443.327 q^{82} -258.138 q^{83} +9.36365 q^{85} +414.493 q^{86} -143.184 q^{87} -393.058 q^{88} +26.6642 q^{89} -6.81223 q^{90} +92.0000 q^{92} +482.113 q^{93} +237.556 q^{94} +0.362280 q^{95} +108.816 q^{96} +35.9974 q^{97} -758.439 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) $ 14 * q - 28 * q^2 + 56 * q^4 - 112 * q^8 + 22 * q^9 - 92 * q^11 - 268 * q^15 + 224 * q^16 - 44 * q^18 + 184 * q^22 + 322 * q^23 + 130 * q^25 + 196 * q^29 + 536 * q^30 - 448 * q^32 + 88 * q^36 + 628 * q^37 + 388 * q^39 + 640 * q^43 - 368 * q^44 - 644 * q^46 - 260 * q^50 + 656 * q^51 - 1260 * q^53 + 380 * q^57 - 392 * q^58 - 1072 * q^60 + 896 * q^64 - 204 * q^65 - 1564 * q^67 - 900 * q^71 - 176 * q^72 - 1256 * q^74 - 776 * q^78 - 404 * q^79 - 4278 * q^81 + 1512 * q^85 - 1280 * q^86 + 736 * q^88 + 1288 * q^92 - 2712 * q^93 - 4752 * q^95 + 1800 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.40048	−0.654424	−0.327212	−	0.944951i	$-0.606109\pi$
$3$	−3.40048	−0.654424	−0.327212	+	0.944951i	$0.606109\pi$
$4$	4.00000	0.500000
$5$	−0.220651	−0.0197356	−0.00986779	−	0.999951i	$-0.503141\pi$
$5$	−0.220651	−0.0197356	−0.00986779	+	0.999951i	$0.503141\pi$
$6$	6.80097	0.462747
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−15.4367	−0.571730
$10$	0.441301	0.0139552
$11$	49.1322	1.34672	0.673360	−	0.739315i	$-0.264850\pi$
$11$	49.1322	1.34672	0.673360	+	0.739315i	$0.264850\pi$
$12$	−13.6019	−0.327212
$13$	37.0988	0.791489	0.395744	−	0.918361i	$-0.370487\pi$
$13$	37.0988	0.791489	0.395744	+	0.918361i	$0.370487\pi$
$14$	0	0
$15$	0.750319	0.0129154
$16$	16.0000	0.250000
$17$	−42.4366	−0.605434	−0.302717	−	0.953081i	$-0.597894\pi$
$17$	−42.4366	−0.605434	−0.302717	+	0.953081i	$0.597894\pi$
$18$	30.8734	0.404274
$19$	−1.64187	−0.0198248	−0.00991239	−	0.999951i	$-0.503155\pi$
$19$	−1.64187	−0.0198248	−0.00991239	+	0.999951i	$0.503155\pi$
$20$	−0.882602	−0.00986779
$21$	0	0
$22$	−98.2644	−0.952275
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	27.2039	0.231374
$25$	−124.951	−0.999611
$26$	−74.1976	−0.559667
$27$	144.305	1.02858
$28$	0	0
$29$	42.1069	0.269623	0.134811	−	0.990871i	$-0.456957\pi$
$29$	42.1069	0.269623	0.134811	+	0.990871i	$0.456957\pi$
$30$	−1.50064	−0.00913259
$31$	−141.778	−0.821420	−0.410710	−	0.911766i	$-0.634719\pi$
$31$	−141.778	−0.821420	−0.410710	+	0.911766i	$0.634719\pi$
$32$	−32.0000	−0.176777
$33$	−167.073	−0.881325
$34$	84.8731	0.428106
$35$	0	0
$36$	−61.7468	−0.285865
$37$	−211.538	−0.939911	−0.469955	−	0.882690i	$-0.655730\pi$
$37$	−211.538	−0.939911	−0.469955	+	0.882690i	$0.655730\pi$
$38$	3.28374	0.0140182
$39$	−126.154	−0.517969
$40$	1.76520	0.00697758
$41$	221.663	0.844342	0.422171	−	0.906516i	$-0.361268\pi$
$41$	221.663	0.844342	0.422171	+	0.906516i	$0.361268\pi$
$42$	0	0
$43$	−207.246	−0.734995	−0.367497	−	0.930025i	$-0.619785\pi$
$43$	−207.246	−0.734995	−0.367497	+	0.930025i	$0.619785\pi$
$44$	196.529	0.673360
$45$	3.40612	0.0112834
$46$	−46.0000	−0.147442
$47$	−118.778	−0.368629	−0.184314	−	0.982867i	$-0.559006\pi$
$47$	−118.778	−0.368629	−0.184314	+	0.982867i	$0.559006\pi$
$48$	−54.4078	−0.163606
$49$	0	0
$50$	249.903	0.706831
$51$	144.305	0.396210
$52$	148.395	0.395744
$53$	434.789	1.12685	0.563423	−	0.826168i	$-0.309484\pi$
$53$	434.789	1.12685	0.563423	+	0.826168i	$0.309484\pi$
$54$	−288.611	−0.727314
$55$	−10.8410	−0.0265783
$56$	0	0
$57$	5.58316	0.0129738
$58$	−84.2138	−0.190652
$59$	69.1542	0.152595	0.0762975	−	0.997085i	$-0.475690\pi$
$59$	69.1542	0.152595	0.0762975	+	0.997085i	$0.475690\pi$
$60$	3.00127	0.00645772
$61$	−103.321	−0.216868	−0.108434	−	0.994104i	$-0.534584\pi$
$61$	−103.321	−0.216868	−0.108434	+	0.994104i	$0.534584\pi$
$62$	283.555	0.580832
$63$	0	0
$64$	64.0000	0.125000
$65$	−8.18587	−0.0156205
$66$	334.147	0.623191
$67$	8.86339	0.0161617	0.00808086	−	0.999967i	$-0.497428\pi$
$67$	8.86339	0.0161617	0.00808086	+	0.999967i	$0.497428\pi$
$68$	−169.746	−0.302717
$69$	−78.2112	−0.136457
$70$	0	0
$71$	−172.114	−0.287692	−0.143846	−	0.989600i	$-0.545947\pi$
$71$	−172.114	−0.287692	−0.143846	+	0.989600i	$0.545947\pi$
$72$	123.494	0.202137
$73$	291.160	0.466818	0.233409	−	0.972379i	$-0.425012\pi$
$73$	291.160	0.466818	0.233409	+	0.972379i	$0.425012\pi$
$74$	423.077	0.664617
$75$	424.895	0.654169
$76$	−6.56748	−0.00991239
$77$	0	0
$78$	252.308	0.366259
$79$	−381.665	−0.543552	−0.271776	−	0.962361i	$-0.587611\pi$
$79$	−381.665	−0.543552	−0.271776	+	0.962361i	$0.587611\pi$
$80$	−3.53041	−0.00493390
$81$	−73.9173	−0.101395
$82$	−443.327	−0.597040
$83$	−258.138	−0.341377	−0.170689	−	0.985325i	$-0.554599\pi$
$83$	−258.138	−0.341377	−0.170689	+	0.985325i	$0.554599\pi$
$84$	0	0
$85$	9.36365	0.0119486
$86$	414.493	0.519720
$87$	−143.184	−0.176447
$88$	−393.058	−0.476137
$89$	26.6642	0.0317573	0.0158786	−	0.999874i	$-0.494945\pi$
$89$	26.6642	0.0317573	0.0158786	+	0.999874i	$0.494945\pi$
$90$	−6.81223	−0.00797858
$91$	0	0
$92$	92.0000	0.104257
$93$	482.113	0.537557
$94$	237.556	0.260660
$95$	0.362280	0.000391254 0
$96$	108.816	0.115687
$97$	35.9974	0.0376802	0.0188401	−	0.999823i	$-0.494003\pi$
$97$	35.9974	0.0376802	0.0188401	+	0.999823i	$0.494003\pi$
$98$	0	0
$99$	−758.439	−0.769960
$100$	−499.805	−0.499805
$101$	982.629	0.968072	0.484036	−	0.875048i	$-0.339170\pi$
$101$	982.629	0.968072	0.484036	+	0.875048i	$0.339170\pi$
$102$	−288.610	−0.280163
$103$	804.577	0.769682	0.384841	−	0.922983i	$-0.374256\pi$
$103$	804.577	0.769682	0.384841	+	0.922983i	$0.374256\pi$
$104$	−296.790	−0.279834
$105$	0	0
$106$	−869.578	−0.796801
$107$	−8.37581	−0.00756748	−0.00378374	−	0.999993i	$-0.501204\pi$
$107$	−8.37581	−0.00756748	−0.00378374	+	0.999993i	$0.501204\pi$
$108$	577.221	0.514289
$109$	199.927	0.175684	0.0878420	−	0.996134i	$-0.472003\pi$
$109$	199.927	0.175684	0.0878420	+	0.996134i	$0.472003\pi$
$110$	21.6821	0.0187937
$111$	719.333	0.615100
$112$	0	0
$113$	658.773	0.548426	0.274213	−	0.961669i	$-0.411583\pi$
$113$	658.773	0.548426	0.274213	+	0.961669i	$0.411583\pi$
$114$	−11.1663	−0.00917387
$115$	−5.07496	−0.00411515
$116$	168.428	0.134811
$117$	−572.683	−0.452518
$118$	−138.308	−0.107901
$119$	0	0
$120$	−6.00255	−0.00456629
$121$	1082.97	0.813655
$122$	206.643	0.153349
$123$	−753.763	−0.552557
$124$	−567.111	−0.410710
$125$	55.1519	0.0394635
$126$	0	0
$127$	−726.840	−0.507847	−0.253924	−	0.967224i	$-0.581721\pi$
$127$	−726.840	−0.507847	−0.253924	+	0.967224i	$0.581721\pi$
$128$	−128.000	−0.0883883
$129$	704.738	0.480998
$130$	16.3717	0.0110454
$131$	−2710.39	−1.80769	−0.903847	−	0.427856i	$-0.859269\pi$
$131$	−2710.39	−1.80769	−0.903847	+	0.427856i	$0.859269\pi$
$132$	−668.293	−0.440663
$133$	0	0
$134$	−17.7268	−0.0114281
$135$	−31.8411	−0.0202996
$136$	339.492	0.214053
$137$	412.910	0.257499	0.128749	−	0.991677i	$-0.458904\pi$
$137$	412.910	0.257499	0.128749	+	0.991677i	$0.458904\pi$
$138$	156.422	0.0964895
$139$	2731.84	1.66699	0.833496	−	0.552526i	$-0.186336\pi$
$139$	2731.84	1.66699	0.833496	+	0.552526i	$0.186336\pi$
$140$	0	0
$141$	403.903	0.241240
$142$	344.227	0.203429
$143$	1822.75	1.06591
$144$	−246.987	−0.142932
$145$	−9.29091	−0.00532116
$146$	−582.320	−0.330090
$147$	0	0
$148$	−846.154	−0.469955
$149$	−3007.78	−1.65374	−0.826869	−	0.562394i	$-0.809880\pi$
$149$	−3007.78	−1.65374	−0.826869	+	0.562394i	$0.809880\pi$
$150$	−849.790	−0.462567
$151$	−849.535	−0.457843	−0.228921	−	0.973445i	$-0.573520\pi$
$151$	−849.535	−0.457843	−0.228921	+	0.973445i	$0.573520\pi$
$152$	13.1350	0.00700912
$153$	655.080	0.346145
$154$	0	0
$155$	31.2833	0.0162112
$156$	−504.616	−0.258985
$157$	3894.97	1.97995	0.989977	−	0.141232i	$-0.0451062\pi$
$157$	3894.97	1.97995	0.989977	+	0.141232i	$0.0451062\pi$
$158$	763.329	0.384349
$159$	−1478.49	−0.737435
$160$	7.06082	0.00348879
$161$	0	0
$162$	147.835	0.0716974
$163$	−1444.79	−0.694264	−0.347132	−	0.937816i	$-0.612844\pi$
$163$	−1444.79	−0.694264	−0.347132	+	0.937816i	$0.612844\pi$
$164$	886.654	0.422171
$165$	36.8648	0.0173935
$166$	516.276	0.241390
$167$	1456.75	0.675012	0.337506	−	0.941323i	$-0.390417\pi$
$167$	1456.75	0.675012	0.337506	+	0.941323i	$0.390417\pi$
$168$	0	0
$169$	−820.679	−0.373545
$170$	−18.7273	−0.00844893
$171$	25.3451	0.0113344
$172$	−828.985	−0.367497
$173$	−2121.68	−0.932419	−0.466210	−	0.884674i	$-0.654381\pi$
$173$	−2121.68	−0.932419	−0.466210	+	0.884674i	$0.654381\pi$
$174$	286.368	0.124767
$175$	0	0
$176$	786.115	0.336680
$177$	−235.158	−0.0998618
$178$	−53.3284	−0.0224558
$179$	−3665.58	−1.53060	−0.765302	−	0.643672i	$-0.777410\pi$
$179$	−3665.58	−1.53060	−0.765302	+	0.643672i	$0.777410\pi$
$180$	13.6245	0.00564171
$181$	−1278.28	−0.524939	−0.262470	−	0.964940i	$-0.584537\pi$
$181$	−1278.28	−0.524939	−0.262470	+	0.964940i	$0.584537\pi$
$182$	0	0
$183$	351.343	0.141923
$184$	−184.000	−0.0737210
$185$	46.6761	0.0185497
$186$	−964.225	−0.380110
$187$	−2085.00	−0.815350
$188$	−475.112	−0.184314
$189$	0	0
$190$	−0.724559	−0.000276658 0
$191$	280.250	0.106169	0.0530843	−	0.998590i	$-0.483095\pi$
$191$	280.250	0.106169	0.0530843	+	0.998590i	$0.483095\pi$
$192$	−217.631	−0.0818030
$193$	3120.93	1.16399	0.581994	−	0.813193i	$-0.302273\pi$
$193$	3120.93	1.16399	0.581994	+	0.813193i	$0.302273\pi$
$194$	−71.9948	−0.0266439
$195$	27.8359	0.0102224
$196$	0	0
$197$	2800.74	1.01292	0.506459	−	0.862264i	$-0.330954\pi$
$197$	2800.74	1.01292	0.506459	+	0.862264i	$0.330954\pi$
$198$	1516.88	0.544444
$199$	350.245	0.124765	0.0623824	−	0.998052i	$-0.480130\pi$
$199$	350.245	0.124765	0.0623824	+	0.998052i	$0.480130\pi$
$200$	999.611	0.353416
$201$	−30.1398	−0.0105766
$202$	−1965.26	−0.684530
$203$	0	0
$204$	577.219	0.198105
$205$	−48.9101	−0.0166636
$206$	−1609.15	−0.544248
$207$	−355.044	−0.119214
$208$	593.581	0.197872
$209$	−80.6687	−0.0266984
$210$	0	0
$211$	2523.65	0.823390	0.411695	−	0.911322i	$-0.364937\pi$
$211$	2523.65	0.823390	0.411695	+	0.911322i	$0.364937\pi$
$212$	1739.16	0.563423
$213$	585.270	0.188272
$214$	16.7516	0.00535101
$215$	45.7290	0.0145055
$216$	−1154.44	−0.363657
$217$	0	0
$218$	−399.855	−0.124227
$219$	−990.085	−0.305496
$220$	−43.3642	−0.0132892
$221$	−1574.35	−0.479194
$222$	−1438.67	−0.434941
$223$	1396.55	0.419371	0.209686	−	0.977769i	$-0.432756\pi$
$223$	1396.55	0.419371	0.209686	+	0.977769i	$0.432756\pi$
$224$	0	0
$225$	1928.84	0.571507
$226$	−1317.55	−0.387796
$227$	4869.56	1.42381	0.711903	−	0.702278i	$-0.247834\pi$
$227$	4869.56	1.42381	0.711903	+	0.702278i	$0.247834\pi$
$228$	22.3326	0.00648690
$229$	−290.940	−0.0839557	−0.0419779	−	0.999119i	$-0.513366\pi$
$229$	−290.940	−0.0839557	−0.0419779	+	0.999119i	$0.513366\pi$
$230$	10.1499	0.00290985
$231$	0	0
$232$	−336.855	−0.0953260
$233$	2157.21	0.606538	0.303269	−	0.952905i	$-0.401922\pi$
$233$	2157.21	0.606538	0.303269	+	0.952905i	$0.401922\pi$
$234$	1145.37	0.319978
$235$	26.2084	0.00727511
$236$	276.617	0.0762975
$237$	1297.84	0.355713
$238$	0	0
$239$	850.858	0.230282	0.115141	−	0.993349i	$-0.463268\pi$
$239$	850.858	0.230282	0.115141	+	0.993349i	$0.463268\pi$
$240$	12.0051	0.00322886
$241$	1046.09	0.279604	0.139802	−	0.990179i	$-0.455353\pi$
$241$	1046.09	0.279604	0.139802	+	0.990179i	$0.455353\pi$
$242$	−2165.95	−0.575341
$243$	−3644.89	−0.962222
$244$	−413.285	−0.108434
$245$	0	0
$246$	1507.53	0.390717
$247$	−60.9114	−0.0156911
$248$	1134.22	0.290416
$249$	877.794	0.223405
$250$	−110.304	−0.0279049
$251$	−321.243	−0.0807835	−0.0403917	−	0.999184i	$-0.512861\pi$
$251$	−321.243	−0.0807835	−0.0403917	+	0.999184i	$0.512861\pi$
$252$	0	0
$253$	1130.04	0.280811
$254$	1453.68	0.359102
$255$	−31.8409	−0.00781944
$256$	256.000	0.0625000
$257$	−4863.37	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$257$	−4863.37	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$258$	−1409.48	−0.340117
$259$	0	0
$260$	−32.7435	−0.00781025
$261$	−649.992	−0.154151
$262$	5420.78	1.27823
$263$	−5659.03	−1.32681	−0.663404	−	0.748261i	$-0.730889\pi$
$263$	−5659.03	−1.32681	−0.663404	+	0.748261i	$0.730889\pi$
$264$	1336.59	0.311596
$265$	−95.9364	−0.0222390
$266$	0	0
$267$	−90.6712	−0.0207827
$268$	35.4536	0.00808086
$269$	1846.82	0.418597	0.209298	−	0.977852i	$-0.432882\pi$
$269$	1846.82	0.418597	0.209298	+	0.977852i	$0.432882\pi$
$270$	63.6821	0.0143540
$271$	6954.32	1.55884	0.779419	−	0.626503i	$-0.215515\pi$
$271$	6954.32	1.55884	0.779419	+	0.626503i	$0.215515\pi$
$272$	−678.985	−0.151358
$273$	0	0
$274$	−825.820	−0.182079
$275$	−6139.13	−1.34620
$276$	−312.845	−0.0682284
$277$	−2368.41	−0.513732	−0.256866	−	0.966447i	$-0.582690\pi$
$277$	−2368.41	−0.513732	−0.256866	+	0.966447i	$0.582690\pi$
$278$	−5463.68	−1.17874
$279$	2188.58	0.469630
$280$	0	0
$281$	−2537.83	−0.538770	−0.269385	−	0.963033i	$-0.586820\pi$
$281$	−2537.83	−0.538770	−0.269385	+	0.963033i	$0.586820\pi$
$282$	−807.806	−0.170582
$283$	−1213.90	−0.254978	−0.127489	−	0.991840i	$-0.540692\pi$
$283$	−1213.90	−0.254978	−0.127489	+	0.991840i	$0.540692\pi$
$284$	−688.454	−0.143846
$285$	−1.23193	−0.000256046 0
$286$	−3645.49	−0.753715
$287$	0	0
$288$	493.974	0.101068
$289$	−3112.14	−0.633450
$290$	18.5818	0.00376263
$291$	−122.409	−0.0246588
$292$	1164.64	0.233409
$293$	−6666.15	−1.32915	−0.664575	−	0.747222i	$-0.731387\pi$
$293$	−6666.15	−1.32915	−0.664575	+	0.747222i	$0.731387\pi$
$294$	0	0
$295$	−15.2589	−0.00301155
$296$	1692.31	0.332309
$297$	7090.04	1.38521
$298$	6015.56	1.16937
$299$	853.272	0.165037
$300$	1699.58	0.327084
$301$	0	0
$302$	1699.07	0.323744
$303$	−3341.42	−0.633529
$304$	−26.2699	−0.00495620
$305$	22.7979	0.00428001
$306$	−1310.16	−0.244761
$307$	−7841.95	−1.45786	−0.728931	−	0.684587i	$-0.759983\pi$
$307$	−7841.95	−1.45786	−0.728931	+	0.684587i	$0.759983\pi$
$308$	0	0
$309$	−2735.95	−0.503698
$310$	−62.5666	−0.0114630
$311$	3854.41	0.702776	0.351388	−	0.936230i	$-0.385710\pi$
$311$	3854.41	0.702776	0.351388	+	0.936230i	$0.385710\pi$
$312$	1009.23	0.183130
$313$	−118.929	−0.0214769	−0.0107384	−	0.999942i	$-0.503418\pi$
$313$	−118.929	−0.0214769	−0.0107384	+	0.999942i	$0.503418\pi$
$314$	−7789.94	−1.40004
$315$	0	0
$316$	−1526.66	−0.271776
$317$	−9647.74	−1.70937	−0.854686	−	0.519145i	$-0.826250\pi$
$317$	−9647.74	−1.70937	−0.854686	+	0.519145i	$0.826250\pi$
$318$	2956.99	0.521445
$319$	2068.81	0.363106
$320$	−14.1216	−0.00246695
$321$	28.4818	0.00495234
$322$	0	0
$323$	69.6753	0.0120026
$324$	−295.669	−0.0506977
$325$	−4635.54	−0.791181
$326$	2889.59	0.490919
$327$	−679.850	−0.114972
$328$	−1773.31	−0.298520
$329$	0	0
$330$	−73.7296	−0.0122990
$331$	−4187.63	−0.695387	−0.347693	−	0.937608i	$-0.613035\pi$
$331$	−4187.63	−0.695387	−0.347693	+	0.937608i	$0.613035\pi$
$332$	−1032.55	−0.170689
$333$	3265.46	0.537375
$334$	−2913.51	−0.477305
$335$	−1.95571	−0.000318961 0
$336$	0	0
$337$	−1460.68	−0.236107	−0.118054	−	0.993007i	$-0.537665\pi$
$337$	−1460.68	−0.236107	−0.118054	+	0.993007i	$0.537665\pi$
$338$	1641.36	0.264136
$339$	−2240.15	−0.358903
$340$	37.4546	0.00597430
$341$	−6965.85	−1.10622
$342$	−50.6901	−0.00801464
$343$	0	0
$344$	1657.97	0.259860
$345$	17.2573	0.00269305
$346$	4243.36	0.659320
$347$	−4543.81	−0.702953	−0.351476	−	0.936197i	$-0.614320\pi$
$347$	−4543.81	−0.702953	−0.351476	+	0.936197i	$0.614320\pi$
$348$	−572.736	−0.0882237
$349$	−5077.83	−0.778825	−0.389412	−	0.921063i	$-0.627322\pi$
$349$	−5077.83	−0.778825	−0.389412	+	0.921063i	$0.627322\pi$
$350$	0	0
$351$	5353.56	0.814107
$352$	−1572.23	−0.238069
$353$	−7926.46	−1.19514	−0.597568	−	0.801818i	$-0.703866\pi$
$353$	−7926.46	−1.19514	−0.597568	+	0.801818i	$0.703866\pi$
$354$	470.315	0.0706129
$355$	37.9769	0.00567777
$356$	106.657	0.0158786
$357$	0	0
$358$	7331.15	1.08230
$359$	−4822.99	−0.709047	−0.354524	−	0.935047i	$-0.615357\pi$
$359$	−4822.99	−0.709047	−0.354524	+	0.935047i	$0.615357\pi$
$360$	−27.2489	−0.00398929
$361$	−6856.30	−0.999607
$362$	2556.57	0.371188
$363$	−3682.64	−0.532475
$364$	0	0
$365$	−64.2446	−0.00921292
$366$	−702.685	−0.100355
$367$	−3207.38	−0.456195	−0.228098	−	0.973638i	$-0.573251\pi$
$367$	−3207.38	−0.456195	−0.228098	+	0.973638i	$0.573251\pi$
$368$	368.000	0.0521286
$369$	−3421.75	−0.482735
$370$	−93.3521	−0.0131166
$371$	0	0
$372$	1928.45	0.268778
$373$	−7101.61	−0.985811	−0.492906	−	0.870083i	$-0.664065\pi$
$373$	−7101.61	−0.985811	−0.492906	+	0.870083i	$0.664065\pi$
$374$	4170.00	0.576539
$375$	−187.543	−0.0258258
$376$	950.225	0.130330
$377$	1562.12	0.213403
$378$	0	0
$379$	−10846.0	−1.46998	−0.734991	−	0.678077i	$-0.762814\pi$
$379$	−10846.0	−1.46998	−0.734991	+	0.678077i	$0.762814\pi$
$380$	1.44912	0.000195627 0
$381$	2471.61	0.332347
$382$	−560.500	−0.0750725
$383$	3407.42	0.454598	0.227299	−	0.973825i	$-0.427010\pi$
$383$	3407.42	0.454598	0.227299	+	0.973825i	$0.427010\pi$
$384$	435.262	0.0578434
$385$	0	0
$386$	−6241.86	−0.823063
$387$	3199.20	0.420218
$388$	143.990	0.0188401
$389$	−4925.24	−0.641952	−0.320976	−	0.947087i	$-0.604011\pi$
$389$	−4925.24	−0.641952	−0.320976	+	0.947087i	$0.604011\pi$
$390$	−55.6718	−0.00722834
$391$	−976.041	−0.126242
$392$	0	0
$393$	9216.64	1.18300
$394$	−5601.49	−0.716241
$395$	84.2145	0.0107273
$396$	−3033.76	−0.384980
$397$	−7891.95	−0.997696	−0.498848	−	0.866689i	$-0.666243\pi$
$397$	−7891.95	−0.997696	−0.498848	+	0.866689i	$0.666243\pi$
$398$	−700.490	−0.0882220
$399$	0	0
$400$	−1999.22	−0.249903
$401$	−12355.0	−1.53860	−0.769298	−	0.638890i	$-0.779394\pi$
$401$	−12355.0	−1.53860	−0.769298	+	0.638890i	$0.779394\pi$
$402$	60.2796	0.00747879
$403$	−5259.78	−0.650145
$404$	3930.52	0.484036
$405$	16.3099	0.00200110
$406$	0	0
$407$	−10393.4	−1.26580
$408$	−1154.44	−0.140081
$409$	−5203.83	−0.629126	−0.314563	−	0.949237i	$-0.601858\pi$
$409$	−5203.83	−0.629126	−0.314563	+	0.949237i	$0.601858\pi$
$410$	97.8203	0.0117829
$411$	−1404.09	−0.168513
$412$	3218.31	0.384841
$413$	0	0
$414$	710.088	0.0842969
$415$	56.9582	0.00673728
$416$	−1187.16	−0.139917
$417$	−9289.59	−1.09092
$418$	161.337	0.0188786
$419$	3863.85	0.450505	0.225252	−	0.974300i	$-0.427679\pi$
$419$	3863.85	0.450505	0.225252	+	0.974300i	$0.427679\pi$
$420$	0	0
$421$	837.811	0.0969891	0.0484945	−	0.998823i	$-0.484558\pi$
$421$	837.811	0.0969891	0.0484945	+	0.998823i	$0.484558\pi$
$422$	−5047.30	−0.582225
$423$	1833.54	0.210756
$424$	−3478.31	−0.398400
$425$	5302.50	0.605198
$426$	−1170.54	−0.133129
$427$	0	0
$428$	−33.5032	−0.00378374
$429$	−6198.22	−0.697559
$430$	−91.4580	−0.0102570
$431$	−4882.25	−0.545637	−0.272818	−	0.962066i	$-0.587956\pi$
$431$	−4882.25	−0.545637	−0.272818	+	0.962066i	$0.587956\pi$
$432$	2308.89	0.257144
$433$	5190.45	0.576067	0.288034	−	0.957620i	$-0.406999\pi$
$433$	5190.45	0.576067	0.288034	+	0.957620i	$0.406999\pi$
$434$	0	0
$435$	31.5936	0.00348229
$436$	799.709	0.0878420
$437$	−37.7630	−0.00413375
$438$	1980.17	0.216019
$439$	−1001.29	−0.108859	−0.0544294	−	0.998518i	$-0.517334\pi$
$439$	−1001.29	−0.108859	−0.0544294	+	0.998518i	$0.517334\pi$
$440$	86.7284	0.00939685
$441$	0	0
$442$	3148.69	0.338841
$443$	9544.03	1.02359	0.511795	−	0.859107i	$-0.328981\pi$
$443$	9544.03	1.02359	0.511795	+	0.859107i	$0.328981\pi$
$444$	2877.33	0.307550
$445$	−5.88347	−0.000626749 0
$446$	−2793.10	−0.296540
$447$	10227.9	1.08225
$448$	0	0
$449$	−8241.63	−0.866251	−0.433125	−	0.901334i	$-0.642589\pi$
$449$	−8241.63	−0.866251	−0.433125	+	0.901334i	$0.642589\pi$
$450$	−3857.67	−0.404116
$451$	10890.8	1.13709
$452$	2635.09	0.274213
$453$	2888.83	0.299623
$454$	−9739.11	−1.00678
$455$	0	0
$456$	−44.6652	−0.00458693
$457$	9053.23	0.926678	0.463339	−	0.886181i	$-0.346651\pi$
$457$	9053.23	0.926678	0.463339	+	0.886181i	$0.346651\pi$
$458$	581.880	0.0593657
$459$	−6123.82	−0.622735
$460$	−20.2998	−0.00205758
$461$	1160.76	0.117271	0.0586357	−	0.998279i	$-0.481325\pi$
$461$	1160.76	0.117271	0.0586357	+	0.998279i	$0.481325\pi$
$462$	0	0
$463$	−16766.1	−1.68291	−0.841456	−	0.540325i	$-0.818301\pi$
$463$	−16766.1	−1.68291	−0.841456	+	0.540325i	$0.818301\pi$
$464$	673.711	0.0674057
$465$	−106.378	−0.0106090
$466$	−4314.41	−0.428887
$467$	−4184.19	−0.414606	−0.207303	−	0.978277i	$-0.566469\pi$
$467$	−4184.19	−0.414606	−0.207303	+	0.978277i	$0.566469\pi$
$468$	−2290.73	−0.226259
$469$	0	0
$470$	−52.4169	−0.00514428
$471$	−13244.8	−1.29573
$472$	−553.233	−0.0539505
$473$	−10182.5	−0.989832
$474$	−2595.69	−0.251527
$475$	205.154	0.0198171
$476$	0	0
$477$	−6711.71	−0.644252
$478$	−1701.72	−0.162834
$479$	−13344.5	−1.27291	−0.636456	−	0.771313i	$-0.719600\pi$
$479$	−13344.5	−1.27291	−0.636456	+	0.771313i	$0.719600\pi$
$480$	−24.0102	−0.00228315
$481$	−7847.82	−0.743929
$482$	−2092.18	−0.197710
$483$	0	0
$484$	4331.90	0.406827
$485$	−7.94284	−0.000743641 0
$486$	7289.78	0.680393
$487$	−4118.17	−0.383187	−0.191594	−	0.981474i	$-0.561366\pi$
$487$	−4118.17	−0.383187	−0.191594	+	0.981474i	$0.561366\pi$
$488$	826.571	0.0766744
$489$	4913.00	0.454343
$490$	0	0
$491$	−12559.9	−1.15442	−0.577209	−	0.816597i	$-0.695858\pi$
$491$	−12559.9	−1.15442	−0.577209	+	0.816597i	$0.695858\pi$
$492$	−3015.05	−0.276279
$493$	−1786.87	−0.163239
$494$	121.823	0.0110953
$495$	167.350	0.0151956
$496$	−2268.44	−0.205355
$497$	0	0
$498$	−1755.59	−0.157971
$499$	−5030.52	−0.451297	−0.225648	−	0.974209i	$-0.572450\pi$
$499$	−5030.52	−0.451297	−0.225648	+	0.974209i	$0.572450\pi$
$500$	220.608	0.0197317
$501$	−4953.67	−0.441744
$502$	642.485	0.0571225
$503$	13705.6	1.21492	0.607460	−	0.794350i	$-0.292189\pi$
$503$	13705.6	1.21492	0.607460	+	0.794350i	$0.292189\pi$
$504$	0	0
$505$	−216.818	−0.0191055
$506$	−2260.08	−0.198563
$507$	2790.71	0.244457
$508$	−2907.36	−0.253924
$509$	−282.533	−0.0246032	−0.0123016	−	0.999924i	$-0.503916\pi$
$509$	−282.533	−0.0246032	−0.0123016	+	0.999924i	$0.503916\pi$
$510$	63.6819	0.00552918
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−236.931	−0.0203913
$514$	9726.74	0.834685
$515$	−177.530	−0.0151901
$516$	2818.95	0.240499
$517$	−5835.83	−0.496440
$518$	0	0
$519$	7214.75	0.610197
$520$	65.4869	0.00552268
$521$	16799.8	1.41269	0.706346	−	0.707867i	$-0.250342\pi$
$521$	16799.8	1.41269	0.706346	+	0.707867i	$0.250342\pi$
$522$	1299.98	0.109001
$523$	3898.08	0.325910	0.162955	−	0.986633i	$-0.447897\pi$
$523$	3898.08	0.325910	0.162955	+	0.986633i	$0.447897\pi$
$524$	−10841.6	−0.903847
$525$	0	0
$526$	11318.1	0.938196
$527$	6016.55	0.497315
$528$	−2673.17	−0.220331
$529$	529.000	0.0434783
$530$	191.873	0.0157253
$531$	−1067.51	−0.0872431
$532$	0	0
$533$	8223.45	0.668287
$534$	181.342	0.0146956
$535$	1.84813	0.000149349 0
$536$	−70.9071	−0.00571403
$537$	12464.7	1.00166
$538$	−3693.64	−0.295992
$539$	0	0
$540$	−127.364	−0.0101498
$541$	7184.02	0.570916	0.285458	−	0.958391i	$-0.407854\pi$
$541$	7184.02	0.570916	0.285458	+	0.958391i	$0.407854\pi$
$542$	−13908.6	−1.10226
$543$	4346.78	0.343533
$544$	1357.97	0.107027
$545$	−44.1141	−0.00346723
$546$	0	0
$547$	−16248.4	−1.27007	−0.635037	−	0.772482i	$-0.719015\pi$
$547$	−16248.4	−1.27007	−0.635037	+	0.772482i	$0.719015\pi$
$548$	1651.64	0.128749
$549$	1594.94	0.123990
$550$	12278.3	0.951904
$551$	−69.1341	−0.00534521
$552$	625.689	0.0482447
$553$	0	0
$554$	4736.82	0.363264
$555$	−158.721	−0.0121394
$556$	10927.4	0.833496
$557$	−22275.0	−1.69448	−0.847238	−	0.531213i	$-0.821736\pi$
$557$	−22275.0	−1.69448	−0.847238	+	0.531213i	$0.821736\pi$
$558$	−4377.16	−0.332079
$559$	−7688.59	−0.581740
$560$	0	0
$561$	7090.02	0.533584
$562$	5075.67	0.380968
$563$	15899.3	1.19019	0.595095	−	0.803656i	$-0.297115\pi$
$563$	15899.3	1.19019	0.595095	+	0.803656i	$0.297115\pi$
$564$	1615.61	0.120620
$565$	−145.359	−0.0108235
$566$	2427.80	0.180297
$567$	0	0
$568$	1376.91	0.101714
$569$	−18584.4	−1.36924	−0.684622	−	0.728899i	$-0.740033\pi$
$569$	−18584.4	−1.36924	−0.684622	+	0.728899i	$0.740033\pi$
$570$	2.46385	0.000181052 0
$571$	14066.0	1.03090	0.515449	−	0.856920i	$-0.327625\pi$
$571$	14066.0	1.03090	0.515449	+	0.856920i	$0.327625\pi$
$572$	7290.98	0.532957
$573$	−952.987	−0.0694792
$574$	0	0
$575$	−2873.88	−0.208433
$576$	−987.949	−0.0714662
$577$	−3524.26	−0.254275	−0.127138	−	0.991885i	$-0.540579\pi$
$577$	−3524.26	−0.254275	−0.127138	+	0.991885i	$0.540579\pi$
$578$	6224.28	0.447917
$579$	−10612.7	−0.761741
$580$	−37.1637	−0.00266058
$581$	0	0
$582$	244.817	0.0174364
$583$	21362.1	1.51755
$584$	−2329.28	−0.165045
$585$	126.363	0.00893070
$586$	13332.3	0.939850
$587$	298.215	0.0209688	0.0104844	−	0.999945i	$-0.496663\pi$
$587$	298.215	0.0209688	0.0104844	+	0.999945i	$0.496663\pi$
$588$	0	0
$589$	232.781	0.0162845
$590$	30.5178	0.00212949
$591$	−9523.89	−0.662877
$592$	−3384.62	−0.234978
$593$	16813.1	1.16430	0.582150	−	0.813082i	$-0.302212\pi$
$593$	16813.1	1.16430	0.582150	+	0.813082i	$0.302212\pi$
$594$	−14180.1	−0.979488
$595$	0	0
$596$	−12031.1	−0.826869
$597$	−1191.00	−0.0816490
$598$	−1706.54	−0.116699
$599$	12432.1	0.848013	0.424007	−	0.905659i	$-0.360623\pi$
$599$	12432.1	0.848013	0.424007	+	0.905659i	$0.360623\pi$
$600$	−3399.16	−0.231284
$601$	7094.18	0.481494	0.240747	−	0.970588i	$-0.422608\pi$
$601$	7094.18	0.481494	0.240747	+	0.970588i	$0.422608\pi$
$602$	0	0
$603$	−136.822	−0.00924014
$604$	−3398.14	−0.228921
$605$	−238.959	−0.0160580
$606$	6682.83	0.447973
$607$	−6159.43	−0.411867	−0.205934	−	0.978566i	$-0.566023\pi$
$607$	−6159.43	−0.411867	−0.205934	+	0.978566i	$0.566023\pi$
$608$	52.5399	0.00350456
$609$	0	0
$610$	−45.5958	−0.00302643
$611$	−4406.52	−0.291766
$612$	2620.32	0.173072
$613$	−8427.93	−0.555303	−0.277652	−	0.960682i	$-0.589556\pi$
$613$	−8427.93	−0.555303	−0.277652	+	0.960682i	$0.589556\pi$
$614$	15683.9	1.03086
$615$	166.318	0.0109050
$616$	0	0
$617$	−7542.03	−0.492108	−0.246054	−	0.969256i	$-0.579134\pi$
$617$	−7542.03	−0.492108	−0.246054	+	0.969256i	$0.579134\pi$
$618$	5471.90	0.356169
$619$	4722.63	0.306654	0.153327	−	0.988176i	$-0.451001\pi$
$619$	4722.63	0.306654	0.153327	+	0.988176i	$0.451001\pi$
$620$	125.133	0.00810560
$621$	3319.02	0.214473
$622$	−7708.82	−0.496938
$623$	0	0
$624$	−2018.46	−0.129492
$625$	15606.7	0.998832
$626$	237.858	0.0151864
$627$	274.313	0.0174721
$628$	15579.9	0.989977
$629$	8976.96	0.569054
$630$	0	0
$631$	8273.12	0.521946	0.260973	−	0.965346i	$-0.415957\pi$
$631$	8273.12	0.521946	0.260973	+	0.965346i	$0.415957\pi$
$632$	3053.32	0.192175
$633$	−8581.63	−0.538846
$634$	19295.5	1.20871
$635$	160.378	0.0100227
$636$	−5913.97	−0.368717
$637$	0	0
$638$	−4137.61	−0.256755
$639$	2656.87	0.164482
$640$	28.2433	0.00174440
$641$	1256.63	0.0774319	0.0387159	−	0.999250i	$-0.487673\pi$
$641$	1256.63	0.0774319	0.0387159	+	0.999250i	$0.487673\pi$
$642$	−56.9636	−0.00350183
$643$	7773.92	0.476786	0.238393	−	0.971169i	$-0.423379\pi$
$643$	7773.92	0.476786	0.238393	+	0.971169i	$0.423379\pi$
$644$	0	0
$645$	−155.501	−0.00949277
$646$	−139.351	−0.00848712
$647$	−5750.18	−0.349402	−0.174701	−	0.984622i	$-0.555896\pi$
$647$	−5750.18	−0.349402	−0.174701	+	0.984622i	$0.555896\pi$
$648$	591.338	0.0358487
$649$	3397.70	0.205503
$650$	9271.09	0.559449
$651$	0	0
$652$	−5779.18	−0.347132
$653$	2579.42	0.154580	0.0772899	−	0.997009i	$-0.475373\pi$
$653$	2579.42	0.154580	0.0772899	+	0.997009i	$0.475373\pi$
$654$	1359.70	0.0812973
$655$	598.049	0.0356759
$656$	3546.61	0.211085
$657$	−4494.55	−0.266893
$658$	0	0
$659$	−19349.4	−1.14377	−0.571885	−	0.820334i	$-0.693788\pi$
$659$	−19349.4	−1.14377	−0.571885	+	0.820334i	$0.693788\pi$
$660$	147.459	0.00869673
$661$	−22301.3	−1.31229	−0.656143	−	0.754637i	$-0.727813\pi$
$661$	−22301.3	−1.31229	−0.656143	+	0.754637i	$0.727813\pi$
$662$	8375.26	0.491713
$663$	5353.54	0.313596
$664$	2065.10	0.120695
$665$	0	0
$666$	−6530.91	−0.379982
$667$	968.459	0.0562202
$668$	5827.01	0.337506
$669$	−4748.94	−0.274446
$670$	3.91142	0.000225539 0
$671$	−5076.41	−0.292060
$672$	0	0
$673$	12650.6	0.724582	0.362291	−	0.932065i	$-0.381995\pi$
$673$	12650.6	0.724582	0.362291	+	0.932065i	$0.381995\pi$
$674$	2921.35	0.166953
$675$	−18031.1	−1.02818
$676$	−3282.72	−0.186773
$677$	−25600.3	−1.45332	−0.726660	−	0.686997i	$-0.758928\pi$
$677$	−25600.3	−1.45332	−0.726660	+	0.686997i	$0.758928\pi$
$678$	4480.30	0.253783
$679$	0	0
$680$	−74.9092	−0.00422446
$681$	−16558.9	−0.931772
$682$	13931.7	0.782217
$683$	6962.45	0.390059	0.195030	−	0.980797i	$-0.437520\pi$
$683$	6962.45	0.390059	0.195030	+	0.980797i	$0.437520\pi$
$684$	101.380	0.00566721
$685$	−91.1088	−0.00508188
$686$	0	0
$687$	989.337	0.0549426
$688$	−3315.94	−0.183749
$689$	16130.1	0.891886
$690$	−34.5147	−0.00190428
$691$	−18379.3	−1.01184	−0.505919	−	0.862581i	$-0.668846\pi$
$691$	−18379.3	−1.01184	−0.505919	+	0.862581i	$0.668846\pi$
$692$	−8486.73	−0.466210
$693$	0	0
$694$	9087.62	0.497063
$695$	−602.782	−0.0328991
$696$	1145.47	0.0623836
$697$	−9406.63	−0.511193
$698$	10155.7	0.550712
$699$	−7335.55	−0.396933
$700$	0	0
$701$	−25533.9	−1.37575	−0.687876	−	0.725828i	$-0.741457\pi$
$701$	−25533.9	−1.37575	−0.687876	+	0.725828i	$0.741457\pi$
$702$	−10707.1	−0.575661
$703$	347.319	0.0186335
$704$	3144.46	0.168340
$705$	−89.1214	−0.00476100
$706$	15852.9	0.845089
$707$	0	0
$708$	−940.631	−0.0499309
$709$	18440.0	0.976771	0.488385	−	0.872628i	$-0.337586\pi$
$709$	18440.0	0.976771	0.488385	+	0.872628i	$0.337586\pi$
$710$	−75.9539	−0.00401479
$711$	5891.64	0.310765
$712$	−213.314	−0.0112279
$713$	−3260.89	−0.171278
$714$	0	0
$715$	−402.190	−0.0210364
$716$	−14662.3	−0.765302
$717$	−2893.33	−0.150702
$718$	9645.99	0.501372
$719$	−15988.6	−0.829308	−0.414654	−	0.909979i	$-0.636097\pi$
$719$	−15988.6	−0.829308	−0.414654	+	0.909979i	$0.636097\pi$
$720$	54.4979	0.00282085
$721$	0	0
$722$	13712.6	0.706829
$723$	−3557.22	−0.182980
$724$	−5113.13	−0.262470
$725$	−5261.31	−0.269518
$726$	7365.28	0.376517
$727$	24613.1	1.25564	0.627820	−	0.778359i	$-0.283948\pi$
$727$	24613.1	1.25564	0.627820	+	0.778359i	$0.283948\pi$
$728$	0	0
$729$	14390.2	0.731096
$730$	128.489	0.00651452
$731$	8794.82	0.444991
$732$	1405.37	0.0709617
$733$	−25613.2	−1.29065	−0.645325	−	0.763908i	$-0.723278\pi$
$733$	−25613.2	−1.29065	−0.645325	+	0.763908i	$0.723278\pi$
$734$	6414.75	0.322579
$735$	0	0
$736$	−736.000	−0.0368605
$737$	435.478	0.0217653
$738$	6843.50	0.341345
$739$	2818.95	0.140320	0.0701602	−	0.997536i	$-0.477649\pi$
$739$	2818.95	0.140320	0.0701602	+	0.997536i	$0.477649\pi$
$740$	186.704	0.00927484
$741$	207.128	0.0102686
$742$	0	0
$743$	−23601.8	−1.16536	−0.582682	−	0.812700i	$-0.697997\pi$
$743$	−23601.8	−1.16536	−0.582682	+	0.812700i	$0.697997\pi$
$744$	−3856.90	−0.190055
$745$	663.668	0.0326375
$746$	14203.2	0.697074
$747$	3984.80	0.195175
$748$	−8340.01	−0.407675
$749$	0	0
$750$	375.086	0.0182616
$751$	8940.49	0.434411	0.217206	−	0.976126i	$-0.430306\pi$
$751$	8940.49	0.434411	0.217206	+	0.976126i	$0.430306\pi$
$752$	−1900.45	−0.0921572
$753$	1092.38	0.0528666
$754$	−3124.23	−0.150899
$755$	187.450	0.00903579
$756$	0	0
$757$	−25082.5	−1.20428	−0.602139	−	0.798392i	$-0.705685\pi$
$757$	−25082.5	−1.20428	−0.602139	+	0.798392i	$0.705685\pi$
$758$	21692.1	1.03943
$759$	−3842.69	−0.183769
$760$	−2.89824	−0.000138329 0
$761$	4364.44	0.207898	0.103949	−	0.994583i	$-0.466852\pi$
$761$	4364.44	0.207898	0.103949	+	0.994583i	$0.466852\pi$
$762$	−4943.22	−0.235005
$763$	0	0
$764$	1121.00	0.0530843
$765$	−144.544	−0.00683136
$766$	−6814.84	−0.321450
$767$	2565.54	0.120777
$768$	−870.524	−0.0409015
$769$	−13115.8	−0.615042	−0.307521	−	0.951541i	$-0.599499\pi$
$769$	−13115.8	−0.615042	−0.307521	+	0.951541i	$0.599499\pi$
$770$	0	0
$771$	16537.8	0.772496
$772$	12483.7	0.581994
$773$	−18034.1	−0.839121	−0.419561	−	0.907727i	$-0.637816\pi$
$773$	−18034.1	−0.839121	−0.419561	+	0.907727i	$0.637816\pi$
$774$	−6398.40	−0.297139
$775$	17715.3	0.821100
$776$	−287.979	−0.0133220
$777$	0	0
$778$	9850.47	0.453929
$779$	−363.943	−0.0167389
$780$	111.344	0.00511121
$781$	−8456.32	−0.387440
$782$	1952.08	0.0892664
$783$	6076.25	0.277328
$784$	0	0
$785$	−859.428	−0.0390755
$786$	−18433.3	−0.836505
$787$	−17241.3	−0.780921	−0.390461	−	0.920620i	$-0.627684\pi$
$787$	−17241.3	−0.780921	−0.390461	+	0.920620i	$0.627684\pi$
$788$	11203.0	0.506459
$789$	19243.4	0.868295
$790$	−168.429	−0.00758536
$791$	0	0
$792$	6067.52	0.272222
$793$	−3833.10	−0.171648
$794$	15783.9	0.705478
$795$	326.230	0.0145537
$796$	1400.98	0.0623824
$797$	20475.3	0.910004	0.455002	−	0.890490i	$-0.349639\pi$
$797$	20475.3	0.910004	0.455002	+	0.890490i	$0.349639\pi$
$798$	0	0
$799$	5040.53	0.223180
$800$	3998.44	0.176708
$801$	−411.607	−0.0181566
$802$	24709.9	1.08795
$803$	14305.3	0.628673
$804$	−120.559	−0.00528831
$805$	0	0
$806$	10519.6	0.459722
$807$	−6280.08	−0.273939
$808$	−7861.03	−0.342265
$809$	−10690.3	−0.464587	−0.232294	−	0.972646i	$-0.574623\pi$
$809$	−10690.3	−0.464587	−0.232294	+	0.972646i	$0.574623\pi$
$810$	−32.6198	−0.00141499
$811$	16130.1	0.698402	0.349201	−	0.937048i	$-0.386453\pi$
$811$	16130.1	0.698402	0.349201	+	0.937048i	$0.386453\pi$
$812$	0	0
$813$	−23648.1	−1.02014
$814$	20786.7	0.895054
$815$	318.795	0.0137017
$816$	2308.88	0.0990526
$817$	340.272	0.0145711
$818$	10407.7	0.444859
$819$	0	0
$820$	−195.641	−0.00833179
$821$	398.560	0.0169426	0.00847128	−	0.999964i	$-0.497303\pi$
$821$	398.560	0.0169426	0.00847128	+	0.999964i	$0.497303\pi$
$822$	2808.19	0.119157
$823$	11687.1	0.495004	0.247502	−	0.968887i	$-0.420390\pi$
$823$	11687.1	0.495004	0.247502	+	0.968887i	$0.420390\pi$
$824$	−6436.61	−0.272124
$825$	20876.0	0.880982
$826$	0	0
$827$	13169.7	0.553755	0.276877	−	0.960905i	$-0.410700\pi$
$827$	13169.7	0.553755	0.276877	+	0.960905i	$0.410700\pi$
$828$	−1420.18	−0.0596069
$829$	1207.96	0.0506083	0.0253042	−	0.999680i	$-0.491945\pi$
$829$	1207.96	0.0506083	0.0253042	+	0.999680i	$0.491945\pi$
$830$	−113.916	−0.00476397
$831$	8053.74	0.336199
$832$	2374.32	0.0989361
$833$	0	0
$834$	18579.2	0.771396
$835$	−321.433	−0.0133217
$836$	−322.675	−0.0133492
$837$	−20459.3	−0.844894
$838$	−7727.70	−0.318555
$839$	−16022.5	−0.659308	−0.329654	−	0.944102i	$-0.606932\pi$
$839$	−16022.5	−0.659308	−0.329654	+	0.944102i	$0.606932\pi$
$840$	0	0
$841$	−22616.0	−0.927304
$842$	−1675.62	−0.0685816
$843$	8629.86	0.352584
$844$	10094.6	0.411695
$845$	181.083	0.00737214
$846$	−3667.08	−0.149027
$847$	0	0
$848$	6956.62	0.281712
$849$	4127.85	0.166864
$850$	−10605.0	−0.427940
$851$	−4865.38	−0.195985
$852$	2341.08	0.0941362
$853$	24877.7	0.998588	0.499294	−	0.866433i	$-0.333593\pi$
$853$	24877.7	0.998588	0.499294	+	0.866433i	$0.333593\pi$
$854$	0	0
$855$	−5.59240	−0.000223691 0
$856$	67.0065	0.00267551
$857$	−20654.2	−0.823260	−0.411630	−	0.911351i	$-0.635040\pi$
$857$	−20654.2	−0.823260	−0.411630	+	0.911351i	$0.635040\pi$
$858$	12396.4	0.493249
$859$	31241.4	1.24091	0.620455	−	0.784242i	$-0.286948\pi$
$859$	31241.4	1.24091	0.620455	+	0.784242i	$0.286948\pi$
$860$	182.916	0.00725277
$861$	0	0
$862$	9764.49	0.385823
$863$	41099.2	1.62113	0.810565	−	0.585649i	$-0.199160\pi$
$863$	41099.2	1.62113	0.810565	+	0.585649i	$0.199160\pi$
$864$	−4617.77	−0.181828
$865$	468.150	0.0184018
$866$	−10380.9	−0.407341
$867$	10582.8	0.414545
$868$	0	0
$869$	−18752.0	−0.732013
$870$	−63.1872	−0.00246235
$871$	328.821	0.0127918
$872$	−1599.42	−0.0621137
$873$	−555.681	−0.0215429
$874$	75.5260	0.00292301
$875$	0	0
$876$	−3960.34	−0.152748
$877$	22061.8	0.849457	0.424728	−	0.905321i	$-0.360370\pi$
$877$	22061.8	0.849457	0.424728	+	0.905321i	$0.360370\pi$
$878$	2002.58	0.0769748
$879$	22668.1	0.869827
$880$	−173.457	−0.00664458
$881$	−25185.5	−0.963135	−0.481568	−	0.876409i	$-0.659932\pi$
$881$	−25185.5	−0.963135	−0.481568	+	0.876409i	$0.659932\pi$
$882$	0	0
$883$	42938.0	1.63644	0.818221	−	0.574903i	$-0.194960\pi$
$883$	42938.0	1.63644	0.818221	+	0.574903i	$0.194960\pi$
$884$	−6297.38	−0.239597
$885$	51.8877	0.00197083
$886$	−19088.1	−0.723788
$887$	−10101.2	−0.382375	−0.191187	−	0.981554i	$-0.561234\pi$
$887$	−10101.2	−0.382375	−0.191187	+	0.981554i	$0.561234\pi$
$888$	−5754.67	−0.217471
$889$	0	0
$890$	11.7669	0.000443178 0
$891$	−3631.72	−0.136551
$892$	5586.19	0.209686
$893$	195.018	0.00730799
$894$	−20455.8	−0.765263
$895$	808.811	0.0302073
$896$	0	0
$897$	−2901.54	−0.108004
$898$	16483.3	0.612532
$899$	−5969.82	−0.221473
$900$	7715.34	0.285754
$901$	−18450.9	−0.682231
$902$	−21781.6	−0.804045
$903$	0	0
$904$	−5270.19	−0.193898
$905$	282.054	0.0103600
$906$	−5777.67	−0.211865
$907$	−18777.1	−0.687413	−0.343706	−	0.939077i	$-0.611682\pi$
$907$	−18777.1	−0.687413	−0.343706	+	0.939077i	$0.611682\pi$
$908$	19478.2	0.711903
$909$	−15168.6	−0.553475
$910$	0	0
$911$	−18212.5	−0.662356	−0.331178	−	0.943568i	$-0.607446\pi$
$911$	−18212.5	−0.662356	−0.331178	+	0.943568i	$0.607446\pi$
$912$	89.3305	0.00324345
$913$	−12682.9	−0.459740
$914$	−18106.5	−0.655261
$915$	−77.5239	−0.00280094
$916$	−1163.76	−0.0419779
$917$	0	0
$918$	12247.6	0.440340
$919$	38684.4	1.38855	0.694276	−	0.719709i	$-0.255725\pi$
$919$	38684.4	1.38855	0.694276	+	0.719709i	$0.255725\pi$
$920$	40.5997	0.00145493
$921$	26666.4	0.954060
$922$	−2321.53	−0.0829234
$923$	−6385.21	−0.227705
$924$	0	0
$925$	26432.0	0.939545
$926$	33532.3	1.19000
$927$	−12420.0	−0.440050
$928$	−1347.42	−0.0476630
$929$	−18022.7	−0.636497	−0.318249	−	0.948007i	$-0.603095\pi$
$929$	−18022.7	−0.636497	−0.318249	+	0.948007i	$0.603095\pi$
$930$	212.757	0.00750169
$931$	0	0
$932$	8628.83	0.303269
$933$	−13106.9	−0.459914
$934$	8368.37	0.293171
$935$	460.057	0.0160914
$936$	4581.46	0.159989
$937$	2386.65	0.0832106	0.0416053	−	0.999134i	$-0.486753\pi$
$937$	2386.65	0.0832106	0.0416053	+	0.999134i	$0.486753\pi$
$938$	0	0
$939$	404.416	0.0140550
$940$	104.834	0.00363755
$941$	−7976.76	−0.276339	−0.138169	−	0.990409i	$-0.544122\pi$
$941$	−7976.76	−0.276339	−0.138169	+	0.990409i	$0.544122\pi$
$942$	26489.6	0.916218
$943$	5098.26	0.176057
$944$	1106.47	0.0381488
$945$	0	0
$946$	20364.9	0.699917
$947$	35356.5	1.21323	0.606617	−	0.794994i	$-0.292526\pi$
$947$	35356.5	1.21323	0.606617	+	0.794994i	$0.292526\pi$
$948$	5191.38	0.177857
$949$	10801.7	0.369481
$950$	−410.308	−0.0140128
$951$	32807.0	1.11865
$952$	0	0
$953$	34207.9	1.16275	0.581377	−	0.813635i	$-0.302514\pi$
$953$	34207.9	1.16275	0.581377	+	0.813635i	$0.302514\pi$
$954$	13423.4	0.455555
$955$	−61.8374	−0.00209530
$956$	3403.43	0.115141
$957$	−7034.94	−0.237625
$958$	26689.0	0.900085
$959$	0	0
$960$	48.0204	0.00161443
$961$	−9690.10	−0.325269
$962$	15695.6	0.526037
$963$	129.295	0.00432655
$964$	4184.36	0.139802
$965$	−688.635	−0.0229720
$966$	0	0
$967$	45253.5	1.50492	0.752458	−	0.658641i	$-0.228868\pi$
$967$	45253.5	1.50492	0.752458	+	0.658641i	$0.228868\pi$
$968$	−8663.80	−0.287670
$969$	−236.930	−0.00785478
$970$	15.8857	0.000525834 0
$971$	16152.1	0.533827	0.266914	−	0.963720i	$-0.413996\pi$
$971$	16152.1	0.533827	0.266914	+	0.963720i	$0.413996\pi$
$972$	−14579.6	−0.481111
$973$	0	0
$974$	8236.34	0.270954
$975$	15763.1	0.517767
$976$	−1653.14	−0.0542170
$977$	−3834.28	−0.125557	−0.0627786	−	0.998027i	$-0.519996\pi$
$977$	−3834.28	−0.125557	−0.0627786	+	0.998027i	$0.519996\pi$
$978$	−9826.00	−0.321269
$979$	1310.07	0.0427682
$980$	0	0
$981$	−3086.22	−0.100444
$982$	25119.7	0.816296
$983$	−27666.0	−0.897669	−0.448835	−	0.893615i	$-0.648161\pi$
$983$	−27666.0	−0.897669	−0.448835	+	0.893615i	$0.648161\pi$
$984$	6030.10	0.195358
$985$	−617.986	−0.0199905
$986$	3573.75	0.115427
$987$	0	0
$988$	−243.646	−0.00784555
$989$	−4766.67	−0.153257
$990$	−334.700	−0.0107449
$991$	38914.4	1.24738	0.623692	−	0.781670i	$-0.285632\pi$
$991$	38914.4	1.24738	0.623692	+	0.781670i	$0.285632\pi$
$992$	4536.88	0.145208
$993$	14240.0	0.455077
$994$	0	0
$995$	−77.2817	−0.00246231
$996$	3511.18	0.111703
$997$	−38140.1	−1.21154	−0.605772	−	0.795639i	$-0.707136\pi$
$997$	−38140.1	−1.21154	−0.605772	+	0.795639i	$0.707136\pi$
$998$	10061.0	0.319115
$999$	−30526.1	−0.966771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.z.1.6	✓	14
7.6	odd	2	inner	2254.4.a.z.1.9	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.z.1.6	✓	14	1.1	even	1	trivial
2254.4.a.z.1.9	yes	14	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.40048 q^{3} +4.00000 q^{4} -0.220651 q^{5} +6.80097 q^{6} -8.00000 q^{8} -15.4367 q^{9} +0.441301 q^{10} +49.1322 q^{11} -13.6019 q^{12} +37.0988 q^{13} +0.750319 q^{15} +16.0000 q^{16} -42.4366 q^{17} +30.8734 q^{18} -1.64187 q^{19} -0.882602 q^{20} -98.2644 q^{22} +23.0000 q^{23} +27.2039 q^{24} -124.951 q^{25} -74.1976 q^{26} +144.305 q^{27} +42.1069 q^{29} -1.50064 q^{30} -141.778 q^{31} -32.0000 q^{32} -167.073 q^{33} +84.8731 q^{34} -61.7468 q^{36} -211.538 q^{37} +3.28374 q^{38} -126.154 q^{39} +1.76520 q^{40} +221.663 q^{41} -207.246 q^{43} +196.529 q^{44} +3.40612 q^{45} -46.0000 q^{46} -118.778 q^{47} -54.4078 q^{48} +249.903 q^{50} +144.305 q^{51} +148.395 q^{52} +434.789 q^{53} -288.611 q^{54} -10.8410 q^{55} +5.58316 q^{57} -84.2138 q^{58} +69.1542 q^{59} +3.00127 q^{60} -103.321 q^{61} +283.555 q^{62} +64.0000 q^{64} -8.18587 q^{65} +334.147 q^{66} +8.86339 q^{67} -169.746 q^{68} -78.2112 q^{69} -172.114 q^{71} +123.494 q^{72} +291.160 q^{73} +423.077 q^{74} +424.895 q^{75} -6.56748 q^{76} +252.308 q^{78} -381.665 q^{79} -3.53041 q^{80} -73.9173 q^{81} -443.327 q^{82} -258.138 q^{83} +9.36365 q^{85} +414.493 q^{86} -143.184 q^{87} -393.058 q^{88} +26.6642 q^{89} -6.81223 q^{90} +92.0000 q^{92} +482.113 q^{93} +237.556 q^{94} +0.362280 q^{95} +108.816 q^{96} +35.9974 q^{97} -758.439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) \) 14 * q - 28 * q^2 + 56 * q^4 - 112 * q^8 + 22 * q^9 - 92 * q^11 - 268 * q^15 + 224 * q^16 - 44 * q^18 + 184 * q^22 + 322 * q^23 + 130 * q^25 + 196 * q^29 + 536 * q^30 - 448 * q^32 + 88 * q^36 + 628 * q^37 + 388 * q^39 + 640 * q^43 - 368 * q^44 - 644 * q^46 - 260 * q^50 + 656 * q^51 - 1260 * q^53 + 380 * q^57 - 392 * q^58 - 1072 * q^60 + 896 * q^64 - 204 * q^65 - 1564 * q^67 - 900 * q^71 - 176 * q^72 - 1256 * q^74 - 776 * q^78 - 404 * q^79 - 4278 * q^81 + 1512 * q^85 - 1280 * q^86 + 736 * q^88 + 1288 * q^92 - 2712 * q^93 - 4752 * q^95 + 1800 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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