LMFDB - Embedded newform 2254.4.a.z.1.12

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			1.12
Root			$5.47743$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +5.47743 q^{3} +4.00000 q^{4} +6.20172 q^{5} -10.9549 q^{6} -8.00000 q^{8} +3.00226 q^{9} -12.4034 q^{10} -33.0895 q^{11} +21.9097 q^{12} -34.5381 q^{13} +33.9695 q^{15} +16.0000 q^{16} +131.015 q^{17} -6.00452 q^{18} -13.5939 q^{19} +24.8069 q^{20} +66.1789 q^{22} +23.0000 q^{23} -43.8195 q^{24} -86.5386 q^{25} +69.0761 q^{26} -131.446 q^{27} +66.8864 q^{29} -67.9390 q^{30} -308.319 q^{31} -32.0000 q^{32} -181.245 q^{33} -262.030 q^{34} +12.0090 q^{36} +200.638 q^{37} +27.1878 q^{38} -189.180 q^{39} -49.6138 q^{40} +300.162 q^{41} -141.539 q^{43} -132.358 q^{44} +18.6192 q^{45} -46.0000 q^{46} +247.281 q^{47} +87.6389 q^{48} +173.077 q^{50} +717.626 q^{51} -138.152 q^{52} +190.653 q^{53} +262.892 q^{54} -205.212 q^{55} -74.4597 q^{57} -133.773 q^{58} +818.525 q^{59} +135.878 q^{60} -539.384 q^{61} +616.637 q^{62} +64.0000 q^{64} -214.196 q^{65} +362.490 q^{66} -465.724 q^{67} +524.061 q^{68} +125.981 q^{69} -214.039 q^{71} -24.0181 q^{72} -1054.49 q^{73} -401.276 q^{74} -474.009 q^{75} -54.3757 q^{76} +378.360 q^{78} +719.679 q^{79} +99.2276 q^{80} -801.047 q^{81} -600.324 q^{82} +3.76807 q^{83} +812.520 q^{85} +283.077 q^{86} +366.366 q^{87} +264.716 q^{88} -1068.75 q^{89} -37.2384 q^{90} +92.0000 q^{92} -1688.79 q^{93} -494.562 q^{94} -84.3057 q^{95} -175.278 q^{96} -1226.84 q^{97} -99.3431 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.47743	1.05413	0.527066	−	0.849824i	$-0.323292\pi$
$3$	5.47743	1.05413	0.527066	+	0.849824i	$0.323292\pi$
$4$	4.00000	0.500000
$5$	6.20172	0.554699	0.277350	−	0.960769i	$-0.410544\pi$
$5$	6.20172	0.554699	0.277350	+	0.960769i	$0.410544\pi$
$6$	−10.9549	−0.745384
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	3.00226	0.111195
$10$	−12.4034	−0.392232
$11$	−33.0895	−0.906986	−0.453493	−	0.891260i	$-0.649822\pi$
$11$	−33.0895	−0.906986	−0.453493	+	0.891260i	$0.649822\pi$
$12$	21.9097	0.527066
$13$	−34.5381	−0.736857	−0.368428	−	0.929656i	$-0.620104\pi$
$13$	−34.5381	−0.736857	−0.368428	+	0.929656i	$0.620104\pi$
$14$	0	0
$15$	33.9695	0.584726
$16$	16.0000	0.250000
$17$	131.015	1.86917	0.934583	−	0.355744i	$-0.115772\pi$
$17$	131.015	1.86917	0.934583	+	0.355744i	$0.115772\pi$
$18$	−6.00452	−0.0786266
$19$	−13.5939	−0.164140	−0.0820699	−	0.996627i	$-0.526153\pi$
$19$	−13.5939	−0.164140	−0.0820699	+	0.996627i	$0.526153\pi$
$20$	24.8069	0.277350
$21$	0	0
$22$	66.1789	0.641336
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−43.8195	−0.372692
$25$	−86.5386	−0.692309
$26$	69.0761	0.521036
$27$	−131.446	−0.936918
$28$	0	0
$29$	66.8864	0.428293	0.214146	−	0.976802i	$-0.431303\pi$
$29$	66.8864	0.428293	0.214146	+	0.976802i	$0.431303\pi$
$30$	−67.9390	−0.413464
$31$	−308.319	−1.78631	−0.893156	−	0.449747i	$-0.851514\pi$
$31$	−308.319	−1.78631	−0.893156	+	0.449747i	$0.851514\pi$
$32$	−32.0000	−0.176777
$33$	−181.245	−0.956083
$34$	−262.030	−1.32170
$35$	0	0
$36$	12.0090	0.0555974
$37$	200.638	0.891478	0.445739	−	0.895163i	$-0.352941\pi$
$37$	200.638	0.891478	0.445739	+	0.895163i	$0.352941\pi$
$38$	27.1878	0.116064
$39$	−189.180	−0.776744
$40$	−49.6138	−0.196116
$41$	300.162	1.14335	0.571675	−	0.820480i	$-0.306294\pi$
$41$	300.162	1.14335	0.571675	+	0.820480i	$0.306294\pi$
$42$	0	0
$43$	−141.539	−0.501964	−0.250982	−	0.967992i	$-0.580753\pi$
$43$	−141.539	−0.501964	−0.250982	+	0.967992i	$0.580753\pi$
$44$	−132.358	−0.453493
$45$	18.6192	0.0616797
$46$	−46.0000	−0.147442
$47$	247.281	0.767439	0.383720	−	0.923450i	$-0.374643\pi$
$47$	247.281	0.767439	0.383720	+	0.923450i	$0.374643\pi$
$48$	87.6389	0.263533
$49$	0	0
$50$	173.077	0.489536
$51$	717.626	1.97035
$52$	−138.152	−0.368428
$53$	190.653	0.494117	0.247058	−	0.969001i	$-0.420536\pi$
$53$	190.653	0.494117	0.247058	+	0.969001i	$0.420536\pi$
$54$	262.892	0.662501
$55$	−205.212	−0.503104
$56$	0	0
$57$	−74.4597	−0.173025
$58$	−133.773	−0.302849
$59$	818.525	1.80615	0.903076	−	0.429481i	$-0.141304\pi$
$59$	818.525	1.80615	0.903076	+	0.429481i	$0.141304\pi$
$60$	135.878	0.292363
$61$	−539.384	−1.13215	−0.566075	−	0.824354i	$-0.691539\pi$
$61$	−539.384	−1.13215	−0.566075	+	0.824354i	$0.691539\pi$
$62$	616.637	1.26311
$63$	0	0
$64$	64.0000	0.125000
$65$	−214.196	−0.408734
$66$	362.490	0.676053
$67$	−465.724	−0.849213	−0.424606	−	0.905378i	$-0.639587\pi$
$67$	−465.724	−0.849213	−0.424606	+	0.905378i	$0.639587\pi$
$68$	524.061	0.934583
$69$	125.981	0.219802
$70$	0	0
$71$	−214.039	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$71$	−214.039	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$72$	−24.0181	−0.0393133
$73$	−1054.49	−1.69067	−0.845333	−	0.534240i	$-0.820598\pi$
$73$	−1054.49	−1.69067	−0.845333	+	0.534240i	$0.820598\pi$
$74$	−401.276	−0.630370
$75$	−474.009	−0.729785
$76$	−54.3757	−0.0820699
$77$	0	0
$78$	378.360	0.549241
$79$	719.679	1.02494	0.512470	−	0.858705i	$-0.328731\pi$
$79$	719.679	1.02494	0.512470	+	0.858705i	$0.328731\pi$
$80$	99.2276	0.138675
$81$	−801.047	−1.09883
$82$	−600.324	−0.808471
$83$	3.76807	0.00498313	0.00249156	−	0.999997i	$-0.499207\pi$
$83$	3.76807	0.00498313	0.00249156	+	0.999997i	$0.499207\pi$
$84$	0	0
$85$	812.520	1.03683
$86$	283.077	0.354942
$87$	366.366	0.451477
$88$	264.716	0.320668
$89$	−1068.75	−1.27289	−0.636445	−	0.771322i	$-0.719596\pi$
$89$	−1068.75	−1.27289	−0.636445	+	0.771322i	$0.719596\pi$
$90$	−37.2384	−0.0436141
$91$	0	0
$92$	92.0000	0.104257
$93$	−1688.79	−1.88301
$94$	−494.562	−0.542661
$95$	−84.3057	−0.0910482
$96$	−175.278	−0.186346
$97$	−1226.84	−1.28420	−0.642099	−	0.766622i	$-0.721936\pi$
$97$	−1226.84	−1.28420	−0.642099	+	0.766622i	$0.721936\pi$
$98$	0	0
$99$	−99.3431	−0.100852
$100$	−346.154	−0.346154
$101$	−1784.33	−1.75790	−0.878949	−	0.476915i	$-0.841755\pi$
$101$	−1784.33	−1.75790	−0.878949	+	0.476915i	$0.841755\pi$
$102$	−1435.25	−1.39325
$103$	−1084.01	−1.03700	−0.518499	−	0.855078i	$-0.673509\pi$
$103$	−1084.01	−1.03700	−0.518499	+	0.855078i	$0.673509\pi$
$104$	276.305	0.260518
$105$	0	0
$106$	−381.306	−0.349393
$107$	−1564.66	−1.41366	−0.706829	−	0.707385i	$-0.749875\pi$
$107$	−1564.66	−1.41366	−0.706829	+	0.707385i	$0.749875\pi$
$108$	−525.784	−0.468459
$109$	−1508.66	−1.32572	−0.662858	−	0.748745i	$-0.730657\pi$
$109$	−1508.66	−1.32572	−0.662858	+	0.748745i	$0.730657\pi$
$110$	410.423	0.355748
$111$	1098.98	0.939735
$112$	0	0
$113$	2038.04	1.69666	0.848332	−	0.529464i	$-0.177607\pi$
$113$	2038.04	1.69666	0.848332	+	0.529464i	$0.177607\pi$
$114$	148.919	0.122347
$115$	142.640	0.115663
$116$	267.546	0.214146
$117$	−103.692	−0.0819346
$118$	−1637.05	−1.27714
$119$	0	0
$120$	−271.756	−0.206732
$121$	−236.088	−0.177376
$122$	1078.77	0.800550
$123$	1644.12	1.20524
$124$	−1233.27	−0.893156
$125$	−1311.90	−0.938722
$126$	0	0
$127$	1823.60	1.27416	0.637079	−	0.770799i	$-0.280143\pi$
$127$	1823.60	1.27416	0.637079	+	0.770799i	$0.280143\pi$
$128$	−128.000	−0.0883883
$129$	−775.268	−0.529136
$130$	428.391	0.289018
$131$	−2049.89	−1.36717	−0.683586	−	0.729870i	$-0.739581\pi$
$131$	−2049.89	−1.36717	−0.683586	+	0.729870i	$0.739581\pi$
$132$	−724.981	−0.478042
$133$	0	0
$134$	931.448	0.600484
$135$	−815.192	−0.519708
$136$	−1048.12	−0.660850
$137$	−1188.62	−0.741243	−0.370621	−	0.928784i	$-0.620855\pi$
$137$	−1188.62	−0.741243	−0.370621	+	0.928784i	$0.620855\pi$
$138$	−251.962	−0.155423
$139$	1766.39	1.07787	0.538934	−	0.842348i	$-0.318827\pi$
$139$	1766.39	1.07787	0.538934	+	0.842348i	$0.318827\pi$
$140$	0	0
$141$	1354.46	0.808982
$142$	428.077	0.252982
$143$	1142.85	0.668319
$144$	48.0362	0.0277987
$145$	414.811	0.237574
$146$	2108.98	1.19548
$147$	0	0
$148$	802.552	0.445739
$149$	−959.466	−0.527534	−0.263767	−	0.964586i	$-0.584965\pi$
$149$	−959.466	−0.527534	−0.263767	+	0.964586i	$0.584965\pi$
$150$	948.019	0.516036
$151$	−2428.87	−1.30900	−0.654499	−	0.756063i	$-0.727120\pi$
$151$	−2428.87	−1.30900	−0.654499	+	0.756063i	$0.727120\pi$
$152$	108.751	0.0580322
$153$	393.342	0.207842
$154$	0	0
$155$	−1912.11	−0.990866
$156$	−756.720	−0.388372
$157$	2885.04	1.46657	0.733285	−	0.679921i	$-0.237986\pi$
$157$	2885.04	1.46657	0.733285	+	0.679921i	$0.237986\pi$
$158$	−1439.36	−0.724741
$159$	1044.29	0.520865
$160$	−198.455	−0.0980579
$161$	0	0
$162$	1602.09	0.776991
$163$	91.4630	0.0439505	0.0219753	−	0.999759i	$-0.493004\pi$
$163$	91.4630	0.0439505	0.0219753	+	0.999759i	$0.493004\pi$
$164$	1200.65	0.571675
$165$	−1124.03	−0.530338
$166$	−7.53614	−0.00352360
$167$	−1957.04	−0.906829	−0.453415	−	0.891300i	$-0.649794\pi$
$167$	−1957.04	−0.906829	−0.453415	+	0.891300i	$0.649794\pi$
$168$	0	0
$169$	−1004.12	−0.457042
$170$	−1625.04	−0.733146
$171$	−40.8125	−0.0182515
$172$	−566.155	−0.250982
$173$	−1428.78	−0.627907	−0.313954	−	0.949438i	$-0.601654\pi$
$173$	−1428.78	−0.627907	−0.313954	+	0.949438i	$0.601654\pi$
$174$	−732.731	−0.319243
$175$	0	0
$176$	−529.431	−0.226746
$177$	4483.42	1.90392
$178$	2137.50	0.900069
$179$	3461.98	1.44559	0.722794	−	0.691063i	$-0.242857\pi$
$179$	3461.98	1.44559	0.722794	+	0.691063i	$0.242857\pi$
$180$	74.4768	0.0308398
$181$	−3521.15	−1.44600	−0.722998	−	0.690850i	$-0.757237\pi$
$181$	−3521.15	−1.44600	−0.722998	+	0.690850i	$0.757237\pi$
$182$	0	0
$183$	−2954.44	−1.19343
$184$	−184.000	−0.0737210
$185$	1244.30	0.494502
$186$	3377.59	1.33149
$187$	−4335.22	−1.69531
$188$	989.124	0.383720
$189$	0	0
$190$	168.611	0.0643808
$191$	20.3079	0.00769333	0.00384667	−	0.999993i	$-0.498776\pi$
$191$	20.3079	0.00769333	0.00384667	+	0.999993i	$0.498776\pi$
$192$	350.556	0.131767
$193$	3213.22	1.19841	0.599203	−	0.800597i	$-0.295484\pi$
$193$	3213.22	1.19841	0.599203	+	0.800597i	$0.295484\pi$
$194$	2453.69	0.908065
$195$	−1173.24	−0.430859
$196$	0	0
$197$	−2414.42	−0.873200	−0.436600	−	0.899656i	$-0.643817\pi$
$197$	−2414.42	−0.873200	−0.436600	+	0.899656i	$0.643817\pi$
$198$	198.686	0.0713132
$199$	3260.30	1.16139	0.580694	−	0.814122i	$-0.302781\pi$
$199$	3260.30	1.16139	0.580694	+	0.814122i	$0.302781\pi$
$200$	692.309	0.244768
$201$	−2550.97	−0.895182
$202$	3568.67	1.24302
$203$	0	0
$204$	2870.51	0.985175
$205$	1861.52	0.634216
$206$	2168.02	0.733269
$207$	69.0520	0.0231857
$208$	−552.609	−0.184214
$209$	449.815	0.148873
$210$	0	0
$211$	3217.94	1.04991	0.524957	−	0.851129i	$-0.324081\pi$
$211$	3217.94	1.04991	0.524957	+	0.851129i	$0.324081\pi$
$212$	762.612	0.247058
$213$	−1172.38	−0.377138
$214$	3129.32	0.999607
$215$	−877.784	−0.278439
$216$	1051.57	0.331251
$217$	0	0
$218$	3017.31	0.937423
$219$	−5775.89	−1.78219
$220$	−820.847	−0.251552
$221$	−4525.01	−1.37731
$222$	−2197.96	−0.664493
$223$	−5062.61	−1.52026	−0.760128	−	0.649773i	$-0.774864\pi$
$223$	−5062.61	−1.52026	−0.760128	+	0.649773i	$0.774864\pi$
$224$	0	0
$225$	−259.811	−0.0769812
$226$	−4076.09	−1.19972
$227$	−4694.40	−1.37259	−0.686295	−	0.727323i	$-0.740764\pi$
$227$	−4694.40	−1.37259	−0.686295	+	0.727323i	$0.740764\pi$
$228$	−297.839	−0.0865126
$229$	−2771.67	−0.799813	−0.399907	−	0.916556i	$-0.630957\pi$
$229$	−2771.67	−0.799813	−0.399907	+	0.916556i	$0.630957\pi$
$230$	−285.279	−0.0817859
$231$	0	0
$232$	−535.091	−0.151424
$233$	−4911.35	−1.38092	−0.690458	−	0.723372i	$-0.742591\pi$
$233$	−4911.35	−1.38092	−0.690458	+	0.723372i	$0.742591\pi$
$234$	207.385	0.0579365
$235$	1533.57	0.425698
$236$	3274.10	0.903076
$237$	3941.99	1.08042
$238$	0	0
$239$	2841.87	0.769143	0.384571	−	0.923095i	$-0.374349\pi$
$239$	2841.87	0.769143	0.384571	+	0.923095i	$0.374349\pi$
$240$	543.512	0.146182
$241$	544.531	0.145545	0.0727725	−	0.997349i	$-0.476815\pi$
$241$	544.531	0.145545	0.0727725	+	0.997349i	$0.476815\pi$
$242$	472.176	0.125424
$243$	−838.641	−0.221394
$244$	−2157.54	−0.566075
$245$	0	0
$246$	−3288.23	−0.852236
$247$	469.508	0.120948
$248$	2466.55	0.631557
$249$	20.6394	0.00525287
$250$	2623.81	0.663777
$251$	88.4254	0.0222365	0.0111182	−	0.999938i	$-0.496461\pi$
$251$	88.4254	0.0222365	0.0111182	+	0.999938i	$0.496461\pi$
$252$	0	0
$253$	−761.057	−0.189120
$254$	−3647.19	−0.900966
$255$	4450.52	1.09295
$256$	256.000	0.0625000
$257$	984.317	0.238911	0.119455	−	0.992840i	$-0.461885\pi$
$257$	984.317	0.238911	0.119455	+	0.992840i	$0.461885\pi$
$258$	1550.54	0.374156
$259$	0	0
$260$	−856.782	−0.204367
$261$	200.810	0.0476239
$262$	4099.77	0.966736
$263$	−1161.57	−0.272340	−0.136170	−	0.990685i	$-0.543479\pi$
$263$	−1161.57	−0.272340	−0.136170	+	0.990685i	$0.543479\pi$
$264$	1449.96	0.338026
$265$	1182.38	0.274086
$266$	0	0
$267$	−5854.00	−1.34179
$268$	−1862.90	−0.424606
$269$	4147.30	0.940020	0.470010	−	0.882661i	$-0.344250\pi$
$269$	4147.30	0.940020	0.470010	+	0.882661i	$0.344250\pi$
$270$	1630.38	0.367489
$271$	3680.51	0.824999	0.412500	−	0.910958i	$-0.364656\pi$
$271$	3680.51	0.824999	0.412500	+	0.910958i	$0.364656\pi$
$272$	2096.24	0.467292
$273$	0	0
$274$	2377.23	0.524138
$275$	2863.52	0.627914
$276$	503.924	0.109901
$277$	−4958.58	−1.07557	−0.537784	−	0.843083i	$-0.680738\pi$
$277$	−4958.58	−1.07557	−0.537784	+	0.843083i	$0.680738\pi$
$278$	−3532.79	−0.762168
$279$	−925.653	−0.198629
$280$	0	0
$281$	−2739.24	−0.581528	−0.290764	−	0.956795i	$-0.593909\pi$
$281$	−2739.24	−0.581528	−0.290764	+	0.956795i	$0.593909\pi$
$282$	−2708.93	−0.572037
$283$	1295.94	0.272211	0.136105	−	0.990694i	$-0.456541\pi$
$283$	1295.94	0.272211	0.136105	+	0.990694i	$0.456541\pi$
$284$	−856.155	−0.178885
$285$	−461.779	−0.0959769
$286$	−2285.69	−0.472573
$287$	0	0
$288$	−96.0723	−0.0196567
$289$	12252.0	2.49379
$290$	−829.622	−0.167990
$291$	−6719.96	−1.35371
$292$	−4217.96	−0.845333
$293$	9016.11	1.79770	0.898851	−	0.438254i	$-0.144403\pi$
$293$	9016.11	1.79770	0.898851	+	0.438254i	$0.144403\pi$
$294$	0	0
$295$	5076.27	1.00187
$296$	−1605.10	−0.315185
$297$	4349.48	0.849772
$298$	1918.93	0.373023
$299$	−794.376	−0.153645
$300$	−1896.04	−0.364893
$301$	0	0
$302$	4857.74	0.925601
$303$	−9773.56	−1.85306
$304$	−217.503	−0.0410350
$305$	−3345.11	−0.628002
$306$	−786.683	−0.146966
$307$	1380.95	0.256726	0.128363	−	0.991727i	$-0.459028\pi$
$307$	1380.95	0.256726	0.128363	+	0.991727i	$0.459028\pi$
$308$	0	0
$309$	−5937.60	−1.09313
$310$	3824.22	0.700648
$311$	−298.237	−0.0543777	−0.0271889	−	0.999630i	$-0.508656\pi$
$311$	−298.237	−0.0543777	−0.0271889	+	0.999630i	$0.508656\pi$
$312$	1513.44	0.274621
$313$	−8201.17	−1.48102	−0.740508	−	0.672048i	$-0.765415\pi$
$313$	−8201.17	−1.48102	−0.740508	+	0.672048i	$0.765415\pi$
$314$	−5770.09	−1.03702
$315$	0	0
$316$	2878.72	0.512470
$317$	−3381.81	−0.599184	−0.299592	−	0.954067i	$-0.596851\pi$
$317$	−3381.81	−0.599184	−0.299592	+	0.954067i	$0.596851\pi$
$318$	−2088.58	−0.368307
$319$	−2213.23	−0.388455
$320$	396.910	0.0693374
$321$	−8570.31	−1.49018
$322$	0	0
$323$	−1781.01	−0.306805
$324$	−3204.19	−0.549415
$325$	2988.88	0.510132
$326$	−182.926	−0.0310777
$327$	−8263.56	−1.39748
$328$	−2401.29	−0.404236
$329$	0	0
$330$	2248.07	0.375006
$331$	−5397.39	−0.896276	−0.448138	−	0.893964i	$-0.647913\pi$
$331$	−5397.39	−0.896276	−0.448138	+	0.893964i	$0.647913\pi$
$332$	15.0723	0.00249156
$333$	602.367	0.0991277
$334$	3914.08	0.641225
$335$	−2888.29	−0.471057
$336$	0	0
$337$	−6266.75	−1.01297	−0.506486	−	0.862248i	$-0.669056\pi$
$337$	−6266.75	−1.01297	−0.506486	+	0.862248i	$0.669056\pi$
$338$	2008.24	0.323178
$339$	11163.2	1.78851
$340$	3250.08	0.518413
$341$	10202.1	1.62016
$342$	81.6249	0.0129058
$343$	0	0
$344$	1132.31	0.177471
$345$	781.299	0.121924
$346$	2857.56	0.443998
$347$	2727.90	0.422022	0.211011	−	0.977484i	$-0.432324\pi$
$347$	2727.90	0.422022	0.211011	+	0.977484i	$0.432324\pi$
$348$	1465.46	0.225739
$349$	10329.3	1.58428	0.792142	−	0.610337i	$-0.208966\pi$
$349$	10329.3	1.58428	0.792142	+	0.610337i	$0.208966\pi$
$350$	0	0
$351$	4539.89	0.690374
$352$	1058.86	0.160334
$353$	5759.95	0.868474	0.434237	−	0.900799i	$-0.357018\pi$
$353$	5759.95	0.868474	0.434237	+	0.900799i	$0.357018\pi$
$354$	−8966.83	−1.34628
$355$	−1327.41	−0.198455
$356$	−4275.00	−0.636445
$357$	0	0
$358$	−6923.95	−1.02219
$359$	11297.9	1.66094	0.830471	−	0.557063i	$-0.188072\pi$
$359$	11297.9	1.66094	0.830471	+	0.557063i	$0.188072\pi$
$360$	−148.954	−0.0218071
$361$	−6674.21	−0.973058
$362$	7042.31	1.02247
$363$	−1293.16	−0.186978
$364$	0	0
$365$	−6539.65	−0.937811
$366$	5908.88	0.843886
$367$	−1522.31	−0.216524	−0.108262	−	0.994122i	$-0.534528\pi$
$367$	−1522.31	−0.216524	−0.108262	+	0.994122i	$0.534528\pi$
$368$	368.000	0.0521286
$369$	901.164	0.127135
$370$	−2488.60	−0.349666
$371$	0	0
$372$	−6755.18	−0.941505
$373$	−10906.0	−1.51391	−0.756956	−	0.653466i	$-0.773314\pi$
$373$	−10906.0	−1.51391	−0.756956	+	0.653466i	$0.773314\pi$
$374$	8670.44	1.19876
$375$	−7185.87	−0.989537
$376$	−1978.25	−0.271331
$377$	−2310.13	−0.315590
$378$	0	0
$379$	−2611.22	−0.353904	−0.176952	−	0.984220i	$-0.556624\pi$
$379$	−2611.22	−0.353904	−0.176952	+	0.984220i	$0.556624\pi$
$380$	−337.223	−0.0455241
$381$	9988.63	1.34313
$382$	−40.6158	−0.00544001
$383$	5232.93	0.698147	0.349073	−	0.937095i	$-0.386496\pi$
$383$	5232.93	0.698147	0.349073	+	0.937095i	$0.386496\pi$
$384$	−701.111	−0.0931730
$385$	0	0
$386$	−6426.43	−0.847401
$387$	−424.936	−0.0558158
$388$	−4907.38	−0.642099
$389$	−8167.82	−1.06459	−0.532294	−	0.846560i	$-0.678670\pi$
$389$	−8167.82	−1.06459	−0.532294	+	0.846560i	$0.678670\pi$
$390$	2346.48	0.304664
$391$	3013.35	0.389748
$392$	0	0
$393$	−11228.1	−1.44118
$394$	4828.84	0.617445
$395$	4463.25	0.568533
$396$	−397.373	−0.0504261
$397$	2607.28	0.329611	0.164806	−	0.986326i	$-0.447300\pi$
$397$	2607.28	0.329611	0.164806	+	0.986326i	$0.447300\pi$
$398$	−6520.59	−0.821226
$399$	0	0
$400$	−1384.62	−0.173077
$401$	−130.007	−0.0161901	−0.00809507	−	0.999967i	$-0.502577\pi$
$401$	−130.007	−0.0161901	−0.00809507	+	0.999967i	$0.502577\pi$
$402$	5101.94	0.632990
$403$	10648.7	1.31626
$404$	−7137.33	−0.878949
$405$	−4967.88	−0.609520
$406$	0	0
$407$	−6639.00	−0.808558
$408$	−5741.01	−0.696624
$409$	8857.36	1.07083	0.535414	−	0.844590i	$-0.320156\pi$
$409$	8857.36	1.07083	0.535414	+	0.844590i	$0.320156\pi$
$410$	−3723.04	−0.448458
$411$	−6510.56	−0.781368
$412$	−4336.05	−0.518499
$413$	0	0
$414$	−138.104	−0.0163948
$415$	23.3685	0.00276414
$416$	1105.22	0.130259
$417$	9675.30	1.13622
$418$	−899.630	−0.105269
$419$	3417.59	0.398473	0.199236	−	0.979951i	$-0.436154\pi$
$419$	3417.59	0.398473	0.199236	+	0.979951i	$0.436154\pi$
$420$	0	0
$421$	9191.52	1.06406	0.532028	−	0.846727i	$-0.321430\pi$
$421$	9191.52	1.06406	0.532028	+	0.846727i	$0.321430\pi$
$422$	−6435.87	−0.742401
$423$	742.402	0.0853352
$424$	−1525.22	−0.174697
$425$	−11337.9	−1.29404
$426$	2344.77	0.266677
$427$	0	0
$428$	−6258.64	−0.706829
$429$	6259.86	0.704496
$430$	1755.57	0.196886
$431$	−3208.01	−0.358525	−0.179263	−	0.983801i	$-0.557371\pi$
$431$	−3208.01	−0.358525	−0.179263	+	0.983801i	$0.557371\pi$
$432$	−2103.14	−0.234230
$433$	8630.72	0.957889	0.478945	−	0.877845i	$-0.341019\pi$
$433$	8630.72	0.957889	0.478945	+	0.877845i	$0.341019\pi$
$434$	0	0
$435$	2272.10	0.250434
$436$	−6034.63	−0.662858
$437$	−312.660	−0.0342255
$438$	11551.8	1.26020
$439$	17516.1	1.90432	0.952162	−	0.305593i	$-0.0988546\pi$
$439$	17516.1	1.90432	0.952162	+	0.305593i	$0.0988546\pi$
$440$	1641.69	0.177874
$441$	0	0
$442$	9050.02	0.973904
$443$	−5727.68	−0.614289	−0.307145	−	0.951663i	$-0.599374\pi$
$443$	−5727.68	−0.614289	−0.307145	+	0.951663i	$0.599374\pi$
$444$	4395.92	0.469868
$445$	−6628.09	−0.706071
$446$	10125.2	1.07498
$447$	−5255.41	−0.556090
$448$	0	0
$449$	−10630.0	−1.11728	−0.558642	−	0.829409i	$-0.688677\pi$
$449$	−10630.0	−1.11728	−0.558642	+	0.829409i	$0.688677\pi$
$450$	519.623	0.0544339
$451$	−9932.19	−1.03700
$452$	8152.18	0.848332
$453$	−13304.0	−1.37986
$454$	9388.80	0.970568
$455$	0	0
$456$	595.678	0.0611736
$457$	−11770.0	−1.20477	−0.602383	−	0.798207i	$-0.705782\pi$
$457$	−11770.0	−1.20477	−0.602383	+	0.798207i	$0.705782\pi$
$458$	5543.34	0.565553
$459$	−17221.4	−1.75126
$460$	570.559	0.0578314
$461$	−7341.33	−0.741692	−0.370846	−	0.928694i	$-0.620932\pi$
$461$	−7341.33	−0.741692	−0.370846	+	0.928694i	$0.620932\pi$
$462$	0	0
$463$	−790.087	−0.0793055	−0.0396528	−	0.999214i	$-0.512625\pi$
$463$	−790.087	−0.0793055	−0.0396528	+	0.999214i	$0.512625\pi$
$464$	1070.18	0.107073
$465$	−10473.4	−1.04450
$466$	9822.71	0.976455
$467$	−11197.3	−1.10953	−0.554765	−	0.832007i	$-0.687192\pi$
$467$	−11197.3	−1.10953	−0.554765	+	0.832007i	$0.687192\pi$
$468$	−414.769	−0.0409673
$469$	0	0
$470$	−3067.14	−0.301014
$471$	15802.6	1.54596
$472$	−6548.20	−0.638571
$473$	4683.44	0.455274
$474$	−7883.98	−0.763973
$475$	1176.40	0.113635
$476$	0	0
$477$	572.390	0.0549432
$478$	−5683.74	−0.543866
$479$	−3193.04	−0.304580	−0.152290	−	0.988336i	$-0.548665\pi$
$479$	−3193.04	−0.304580	−0.152290	+	0.988336i	$0.548665\pi$
$480$	−1087.02	−0.103366
$481$	−6929.65	−0.656891
$482$	−1089.06	−0.102916
$483$	0	0
$484$	−944.352	−0.0886882
$485$	−7608.55	−0.712343
$486$	1677.28	0.156550
$487$	232.509	0.0216345	0.0108172	−	0.999941i	$-0.496557\pi$
$487$	232.509	0.0216345	0.0108172	+	0.999941i	$0.496557\pi$
$488$	4315.08	0.400275
$489$	500.982	0.0463297
$490$	0	0
$491$	10186.1	0.936239	0.468119	−	0.883665i	$-0.344932\pi$
$491$	10186.1	0.936239	0.468119	+	0.883665i	$0.344932\pi$
$492$	6576.46	0.602622
$493$	8763.13	0.800551
$494$	−939.015	−0.0855228
$495$	−616.099	−0.0559426
$496$	−4933.10	−0.446578
$497$	0	0
$498$	−41.2787	−0.00371434
$499$	−10320.8	−0.925893	−0.462947	−	0.886386i	$-0.653208\pi$
$499$	−10320.8	−0.925893	−0.462947	+	0.886386i	$0.653208\pi$
$500$	−5247.62	−0.469361
$501$	−10719.6	−0.955918
$502$	−176.851	−0.0157236
$503$	−3638.41	−0.322522	−0.161261	−	0.986912i	$-0.551556\pi$
$503$	−3638.41	−0.322522	−0.161261	+	0.986912i	$0.551556\pi$
$504$	0	0
$505$	−11065.9	−0.975105
$506$	1522.11	0.133728
$507$	−5500.01	−0.481783
$508$	7294.39	0.637079
$509$	9977.47	0.868848	0.434424	−	0.900709i	$-0.356952\pi$
$509$	9977.47	0.868848	0.434424	+	0.900709i	$0.356952\pi$
$510$	−8901.04	−0.772833
$511$	0	0
$512$	−512.000	−0.0441942
$513$	1786.87	0.153786
$514$	−1968.63	−0.168935
$515$	−6722.74	−0.575222
$516$	−3101.07	−0.264568
$517$	−8182.39	−0.696056
$518$	0	0
$519$	−7826.03	−0.661897
$520$	1713.56	0.144509
$521$	−15983.2	−1.34402	−0.672012	−	0.740540i	$-0.734570\pi$
$521$	−15983.2	−1.34402	−0.672012	+	0.740540i	$0.734570\pi$
$522$	−401.621	−0.0336752
$523$	−1082.49	−0.0905051	−0.0452525	−	0.998976i	$-0.514409\pi$
$523$	−1082.49	−0.0905051	−0.0452525	+	0.998976i	$0.514409\pi$
$524$	−8199.55	−0.683586
$525$	0	0
$526$	2323.14	0.192573
$527$	−40394.4	−3.33892
$528$	−2899.92	−0.239021
$529$	529.000	0.0434783
$530$	−2364.75	−0.193808
$531$	2457.43	0.200835
$532$	0	0
$533$	−10367.0	−0.842486
$534$	11708.0	0.948792
$535$	−9703.59	−0.784154
$536$	3725.79	0.300242
$537$	18962.7	1.52384
$538$	−8294.60	−0.664695
$539$	0	0
$540$	−3260.77	−0.259854
$541$	−16237.3	−1.29038	−0.645190	−	0.764022i	$-0.723222\pi$
$541$	−16237.3	−1.29038	−0.645190	+	0.764022i	$0.723222\pi$
$542$	−7361.01	−0.583363
$543$	−19286.9	−1.52427
$544$	−4192.48	−0.330425
$545$	−9356.27	−0.735374
$546$	0	0
$547$	23219.5	1.81498	0.907489	−	0.420077i	$-0.137997\pi$
$547$	23219.5	1.81498	0.907489	+	0.420077i	$0.137997\pi$
$548$	−4754.46	−0.370621
$549$	−1619.37	−0.125889
$550$	−5727.03	−0.444003
$551$	−909.248	−0.0702999
$552$	−1007.85	−0.0777117
$553$	0	0
$554$	9917.16	0.760541
$555$	6815.57	0.521270
$556$	7065.58	0.538934
$557$	49.7701	0.00378604	0.00189302	−	0.999998i	$-0.499397\pi$
$557$	49.7701	0.00378604	0.00189302	+	0.999998i	$0.499397\pi$
$558$	1851.31	0.140452
$559$	4888.47	0.369875
$560$	0	0
$561$	−23745.9	−1.78708
$562$	5478.48	0.411202
$563$	−10025.6	−0.750492	−0.375246	−	0.926925i	$-0.622442\pi$
$563$	−10025.6	−0.750492	−0.375246	+	0.926925i	$0.622442\pi$
$564$	5417.86	0.404491
$565$	12639.4	0.941138
$566$	−2591.88	−0.192482
$567$	0	0
$568$	1712.31	0.126491
$569$	−22242.7	−1.63877	−0.819387	−	0.573240i	$-0.805686\pi$
$569$	−22242.7	−1.63877	−0.819387	+	0.573240i	$0.805686\pi$
$570$	923.558	0.0678659
$571$	8262.28	0.605544	0.302772	−	0.953063i	$-0.402088\pi$
$571$	8262.28	0.605544	0.302772	+	0.953063i	$0.402088\pi$
$572$	4571.38	0.334159
$573$	111.235	0.00810979
$574$	0	0
$575$	−1990.39	−0.144356
$576$	192.145	0.0138994
$577$	6055.19	0.436882	0.218441	−	0.975850i	$-0.429903\pi$
$577$	6055.19	0.436882	0.218441	+	0.975850i	$0.429903\pi$
$578$	−24503.9	−1.76337
$579$	17600.2	1.26328
$580$	1659.24	0.118787
$581$	0	0
$582$	13439.9	0.957220
$583$	−6308.60	−0.448157
$584$	8435.91	0.597741
$585$	−643.071	−0.0454491
$586$	−18032.2	−1.27117
$587$	27901.4	1.96186	0.980931	−	0.194358i	$-0.0622623\pi$
$587$	27901.4	1.96186	0.980931	+	0.194358i	$0.0622623\pi$
$588$	0	0
$589$	4191.26	0.293205
$590$	−10152.5	−0.708429
$591$	−13224.8	−0.920468
$592$	3210.21	0.222869
$593$	−17495.5	−1.21156	−0.605780	−	0.795632i	$-0.707139\pi$
$593$	−17495.5	−1.21156	−0.605780	+	0.795632i	$0.707139\pi$
$594$	−8698.95	−0.600879
$595$	0	0
$596$	−3837.86	−0.263767
$597$	17858.1	1.22426
$598$	1588.75	0.108644
$599$	−14796.2	−1.00928	−0.504638	−	0.863331i	$-0.668374\pi$
$599$	−14796.2	−1.00928	−0.504638	+	0.863331i	$0.668374\pi$
$600$	3792.07	0.258018
$601$	11251.3	0.763645	0.381822	−	0.924236i	$-0.375297\pi$
$601$	11251.3	0.763645	0.381822	+	0.924236i	$0.375297\pi$
$602$	0	0
$603$	−1398.22	−0.0944280
$604$	−9715.48	−0.654499
$605$	−1464.15	−0.0983906
$606$	19547.1	1.31031
$607$	−13458.7	−0.899954	−0.449977	−	0.893040i	$-0.648568\pi$
$607$	−13458.7	−0.899954	−0.449977	+	0.893040i	$0.648568\pi$
$608$	435.005	0.0290161
$609$	0	0
$610$	6690.23	0.444065
$611$	−8540.61	−0.565493
$612$	1573.37	0.103921
$613$	7640.16	0.503398	0.251699	−	0.967806i	$-0.419011\pi$
$613$	7640.16	0.503398	0.251699	+	0.967806i	$0.419011\pi$
$614$	−2761.90	−0.181533
$615$	10196.4	0.668547
$616$	0	0
$617$	9352.41	0.610233	0.305117	−	0.952315i	$-0.401305\pi$
$617$	9352.41	0.610233	0.305117	+	0.952315i	$0.401305\pi$
$618$	11875.2	0.772962
$619$	−21690.8	−1.40844	−0.704221	−	0.709981i	$-0.748704\pi$
$619$	−21690.8	−1.40844	−0.704221	+	0.709981i	$0.748704\pi$
$620$	−7648.43	−0.495433
$621$	−3023.26	−0.195361
$622$	596.474	0.0384509
$623$	0	0
$624$	−3026.88	−0.194186
$625$	2681.26	0.171601
$626$	16402.3	1.04724
$627$	2463.83	0.156931
$628$	11540.2	0.733285
$629$	26286.6	1.66632
$630$	0	0
$631$	−11758.9	−0.741860	−0.370930	−	0.928661i	$-0.620961\pi$
$631$	−11758.9	−0.741860	−0.370930	+	0.928661i	$0.620961\pi$
$632$	−5757.43	−0.362371
$633$	17626.0	1.10675
$634$	6763.62	0.423687
$635$	11309.4	0.706774
$636$	4177.15	0.260432
$637$	0	0
$638$	4426.47	0.274679
$639$	−642.600	−0.0397823
$640$	−793.821	−0.0490289
$641$	14895.7	0.917853	0.458927	−	0.888474i	$-0.348234\pi$
$641$	14895.7	0.917853	0.458927	+	0.888474i	$0.348234\pi$
$642$	17140.6	1.05372
$643$	−3546.00	−0.217482	−0.108741	−	0.994070i	$-0.534682\pi$
$643$	−3546.00	−0.217482	−0.108741	+	0.994070i	$0.534682\pi$
$644$	0	0
$645$	−4808.00	−0.293511
$646$	3562.02	0.216944
$647$	−20297.2	−1.23333	−0.616666	−	0.787225i	$-0.711517\pi$
$647$	−20297.2	−1.23333	−0.616666	+	0.787225i	$0.711517\pi$
$648$	6408.38	0.388495
$649$	−27084.6	−1.63815
$650$	−5977.75	−0.360718
$651$	0	0
$652$	365.852	0.0219753
$653$	22257.7	1.33386	0.666931	−	0.745120i	$-0.267608\pi$
$653$	22257.7	1.33386	0.666931	+	0.745120i	$0.267608\pi$
$654$	16527.1	0.988168
$655$	−12712.8	−0.758369
$656$	4802.59	0.285838
$657$	−3165.85	−0.187993
$658$	0	0
$659$	25374.9	1.49995	0.749974	−	0.661468i	$-0.230066\pi$
$659$	25374.9	1.49995	0.749974	+	0.661468i	$0.230066\pi$
$660$	−4496.13	−0.265169
$661$	−5372.01	−0.316107	−0.158054	−	0.987431i	$-0.550522\pi$
$661$	−5372.01	−0.316107	−0.158054	+	0.987431i	$0.550522\pi$
$662$	10794.8	0.633763
$663$	−24785.4	−1.45187
$664$	−30.1446	−0.00176180
$665$	0	0
$666$	−1204.73	−0.0700939
$667$	1538.39	0.0893052
$668$	−7828.17	−0.453415
$669$	−27730.1	−1.60255
$670$	5776.58	0.333088
$671$	17847.9	1.02684
$672$	0	0
$673$	−27056.0	−1.54967	−0.774837	−	0.632161i	$-0.782168\pi$
$673$	−27056.0	−1.54967	−0.774837	+	0.632161i	$0.782168\pi$
$674$	12533.5	0.716279
$675$	11375.2	0.648637
$676$	−4016.49	−0.228521
$677$	19666.8	1.11648	0.558241	−	0.829679i	$-0.311477\pi$
$677$	19666.8	1.11648	0.558241	+	0.829679i	$0.311477\pi$
$678$	−22326.5	−1.26467
$679$	0	0
$680$	−6500.16	−0.366573
$681$	−25713.2	−1.44689
$682$	−20404.2	−1.14563
$683$	−10452.5	−0.585582	−0.292791	−	0.956177i	$-0.594584\pi$
$683$	−10452.5	−0.585582	−0.292791	+	0.956177i	$0.594584\pi$
$684$	−163.250	−0.00912575
$685$	−7371.47	−0.411167
$686$	0	0
$687$	−15181.6	−0.843109
$688$	−2264.62	−0.125491
$689$	−6584.79	−0.364093
$690$	−1562.60	−0.0862132
$691$	33865.8	1.86442	0.932210	−	0.361917i	$-0.117878\pi$
$691$	33865.8	1.86442	0.932210	+	0.361917i	$0.117878\pi$
$692$	−5715.11	−0.313954
$693$	0	0
$694$	−5455.80	−0.298414
$695$	10954.7	0.597892
$696$	−2930.92	−0.159621
$697$	39325.7	2.13711
$698$	−20658.6	−1.12026
$699$	−26901.6	−1.45567
$700$	0	0
$701$	24826.7	1.33765	0.668825	−	0.743420i	$-0.266797\pi$
$701$	24826.7	1.33765	0.668825	+	0.743420i	$0.266797\pi$
$702$	−9079.78	−0.488168
$703$	−2727.45	−0.146327
$704$	−2117.72	−0.113373
$705$	8400.02	0.448742
$706$	−11519.9	−0.614104
$707$	0	0
$708$	17933.7	0.951961
$709$	−14204.2	−0.752397	−0.376199	−	0.926539i	$-0.622769\pi$
$709$	−14204.2	−0.752397	−0.376199	+	0.926539i	$0.622769\pi$
$710$	2654.82	0.140329
$711$	2160.66	0.113968
$712$	8550.00	0.450035
$713$	−7091.33	−0.372472
$714$	0	0
$715$	7087.62	0.370716
$716$	13847.9	0.722794
$717$	15566.1	0.810778
$718$	−22595.7	−1.17446
$719$	1154.15	0.0598643	0.0299321	−	0.999552i	$-0.490471\pi$
$719$	1154.15	0.0598643	0.0299321	+	0.999552i	$0.490471\pi$
$720$	297.907	0.0154199
$721$	0	0
$722$	13348.4	0.688056
$723$	2982.63	0.153424
$724$	−14084.6	−0.722998
$725$	−5788.25	−0.296511
$726$	2586.31	0.132214
$727$	23346.6	1.19103	0.595514	−	0.803345i	$-0.296949\pi$
$727$	23346.6	1.19103	0.595514	+	0.803345i	$0.296949\pi$
$728$	0	0
$729$	17034.7	0.865451
$730$	13079.3	0.663132
$731$	−18543.7	−0.938254
$732$	−11817.8	−0.596717
$733$	−9077.04	−0.457392	−0.228696	−	0.973498i	$-0.573446\pi$
$733$	−9077.04	−0.457392	−0.228696	+	0.973498i	$0.573446\pi$
$734$	3044.63	0.153105
$735$	0	0
$736$	−736.000	−0.0368605
$737$	15410.6	0.770224
$738$	−1802.33	−0.0898978
$739$	9440.48	0.469924	0.234962	−	0.972005i	$-0.424503\pi$
$739$	9440.48	0.469924	0.234962	+	0.972005i	$0.424503\pi$
$740$	4977.20	0.247251
$741$	2571.70	0.127495
$742$	0	0
$743$	10012.1	0.494360	0.247180	−	0.968970i	$-0.420496\pi$
$743$	10012.1	0.494360	0.247180	+	0.968970i	$0.420496\pi$
$744$	13510.4	0.665744
$745$	−5950.34	−0.292622
$746$	21811.9	1.07050
$747$	11.3127	0.000554098 0
$748$	−17340.9	−0.847654
$749$	0	0
$750$	14371.7	0.699709
$751$	−35032.6	−1.70221	−0.851103	−	0.524998i	$-0.824066\pi$
$751$	−35032.6	−1.70221	−0.851103	+	0.524998i	$0.824066\pi$
$752$	3956.50	0.191860
$753$	484.344	0.0234402
$754$	4620.25	0.223156
$755$	−15063.2	−0.726099
$756$	0	0
$757$	20473.5	0.982990	0.491495	−	0.870880i	$-0.336451\pi$
$757$	20473.5	0.982990	0.491495	+	0.870880i	$0.336451\pi$
$758$	5222.44	0.250248
$759$	−4168.64	−0.199357
$760$	674.446	0.0321904
$761$	−11785.2	−0.561384	−0.280692	−	0.959798i	$-0.590564\pi$
$761$	−11785.2	−0.561384	−0.280692	+	0.959798i	$0.590564\pi$
$762$	−19977.3	−0.949737
$763$	0	0
$764$	81.2315	0.00384667
$765$	2439.40	0.115290
$766$	−10465.9	−0.493664
$767$	−28270.3	−1.33087
$768$	1402.22	0.0658833
$769$	39423.6	1.84870	0.924352	−	0.381541i	$-0.124607\pi$
$769$	39423.6	1.84870	0.924352	+	0.381541i	$0.124607\pi$
$770$	0	0
$771$	5391.53	0.251843
$772$	12852.9	0.599203
$773$	1520.27	0.0707377	0.0353689	−	0.999374i	$-0.488739\pi$
$773$	1520.27	0.0707377	0.0353689	+	0.999374i	$0.488739\pi$
$774$	849.872	0.0394677
$775$	26681.5	1.23668
$776$	9814.75	0.454032
$777$	0	0
$778$	16335.6	0.752777
$779$	−4080.37	−0.187669
$780$	−4692.97	−0.215430
$781$	7082.42	0.324493
$782$	−6026.70	−0.275594
$783$	−8791.95	−0.401275
$784$	0	0
$785$	17892.2	0.813505
$786$	22456.2	1.01907
$787$	−30954.4	−1.40204	−0.701020	−	0.713141i	$-0.747272\pi$
$787$	−30954.4	−1.40204	−0.701020	+	0.713141i	$0.747272\pi$
$788$	−9657.68	−0.436600
$789$	−6362.41	−0.287082
$790$	−8926.50	−0.402013
$791$	0	0
$792$	794.745	0.0356566
$793$	18629.3	0.834232
$794$	−5214.56	−0.233070
$795$	6476.39	0.288923
$796$	13041.2	0.580694
$797$	−8988.94	−0.399504	−0.199752	−	0.979846i	$-0.564014\pi$
$797$	−8988.94	−0.399504	−0.199752	+	0.979846i	$0.564014\pi$
$798$	0	0
$799$	32397.6	1.43447
$800$	2769.24	0.122384
$801$	−3208.66	−0.141539
$802$	260.014	0.0114482
$803$	34892.5	1.53341
$804$	−10203.9	−0.447591
$805$	0	0
$806$	−21297.5	−0.930734
$807$	22716.6	0.990906
$808$	14274.7	0.621511
$809$	−15850.2	−0.688832	−0.344416	−	0.938817i	$-0.611923\pi$
$809$	−15850.2	−0.688832	−0.344416	+	0.938817i	$0.611923\pi$
$810$	9935.75	0.430996
$811$	−1135.12	−0.0491485	−0.0245742	−	0.999698i	$-0.507823\pi$
$811$	−1135.12	−0.0491485	−0.0245742	+	0.999698i	$0.507823\pi$
$812$	0	0
$813$	20159.7	0.869658
$814$	13278.0	0.571737
$815$	567.228	0.0243793
$816$	11482.0	0.492587
$817$	1924.06	0.0823923
$818$	−17714.7	−0.757189
$819$	0	0
$820$	7446.08	0.317108
$821$	38631.2	1.64219	0.821094	−	0.570793i	$-0.193364\pi$
$821$	38631.2	1.64219	0.821094	+	0.570793i	$0.193364\pi$
$822$	13021.1	0.552511
$823$	2640.09	0.111820	0.0559100	−	0.998436i	$-0.482194\pi$
$823$	2640.09	0.111820	0.0559100	+	0.998436i	$0.482194\pi$
$824$	8672.10	0.366634
$825$	15684.7	0.661905
$826$	0	0
$827$	10061.4	0.423058	0.211529	−	0.977372i	$-0.432156\pi$
$827$	10061.4	0.423058	0.211529	+	0.977372i	$0.432156\pi$
$828$	276.208	0.0115929
$829$	22225.7	0.931160	0.465580	−	0.885006i	$-0.345846\pi$
$829$	22225.7	0.931160	0.465580	+	0.885006i	$0.345846\pi$
$830$	−46.7371	−0.00195454
$831$	−27160.3	−1.13379
$832$	−2210.44	−0.0921071
$833$	0	0
$834$	−19350.6	−0.803425
$835$	−12137.0	−0.503017
$836$	1799.26	0.0744363
$837$	40527.3	1.67363
$838$	−6835.17	−0.281763
$839$	7883.29	0.324388	0.162194	−	0.986759i	$-0.448143\pi$
$839$	7883.29	0.324388	0.162194	+	0.986759i	$0.448143\pi$
$840$	0	0
$841$	−19915.2	−0.816565
$842$	−18383.0	−0.752401
$843$	−15004.0	−0.613007
$844$	12871.7	0.524957
$845$	−6227.29	−0.253521
$846$	−1484.80	−0.0603411
$847$	0	0
$848$	3050.45	0.123529
$849$	7098.42	0.286946
$850$	22675.7	0.915025
$851$	4614.67	0.185886
$852$	−4689.53	−0.188569
$853$	−37123.6	−1.49014	−0.745069	−	0.666987i	$-0.767584\pi$
$853$	−37123.6	−1.49014	−0.745069	+	0.666987i	$0.767584\pi$
$854$	0	0
$855$	−253.108	−0.0101241
$856$	12517.3	0.499803
$857$	−6403.50	−0.255238	−0.127619	−	0.991823i	$-0.540734\pi$
$857$	−6403.50	−0.255238	−0.127619	+	0.991823i	$0.540734\pi$
$858$	−12519.7	−0.498154
$859$	−12039.5	−0.478209	−0.239105	−	0.970994i	$-0.576854\pi$
$859$	−12039.5	−0.478209	−0.239105	+	0.970994i	$0.576854\pi$
$860$	−3511.13	−0.139219
$861$	0	0
$862$	6416.02	0.253515
$863$	−24935.7	−0.983570	−0.491785	−	0.870717i	$-0.663655\pi$
$863$	−24935.7	−0.983570	−0.491785	+	0.870717i	$0.663655\pi$
$864$	4206.27	0.165625
$865$	−8860.89	−0.348300
$866$	−17261.4	−0.677330
$867$	67109.3	2.62878
$868$	0	0
$869$	−23813.8	−0.929605
$870$	−4544.20	−0.177084
$871$	16085.2	0.625748
$872$	12069.3	0.468712
$873$	−3683.31	−0.142796
$874$	625.320	0.0242011
$875$	0	0
$876$	−23103.6	−0.891093
$877$	−16043.3	−0.617724	−0.308862	−	0.951107i	$-0.599948\pi$
$877$	−16043.3	−0.617724	−0.308862	+	0.951107i	$0.599948\pi$
$878$	−35032.2	−1.34656
$879$	49385.1	1.89502
$880$	−3283.39	−0.125776
$881$	8585.77	0.328333	0.164167	−	0.986433i	$-0.447506\pi$
$881$	8585.77	0.328333	0.164167	+	0.986433i	$0.447506\pi$
$882$	0	0
$883$	−10383.8	−0.395745	−0.197873	−	0.980228i	$-0.563403\pi$
$883$	−10383.8	−0.395745	−0.197873	+	0.980228i	$0.563403\pi$
$884$	−18100.0	−0.688654
$885$	27804.9	1.05610
$886$	11455.4	0.434368
$887$	−136.647	−0.00517267	−0.00258633	−	0.999997i	$-0.500823\pi$
$887$	−136.647	−0.00517267	−0.00258633	+	0.999997i	$0.500823\pi$
$888$	−8791.84	−0.332247
$889$	0	0
$890$	13256.2	0.499268
$891$	26506.2	0.996624
$892$	−20250.4	−0.760128
$893$	−3361.52	−0.125967
$894$	10510.8	0.393215
$895$	21470.2	0.801867
$896$	0	0
$897$	−4351.14	−0.161962
$898$	21260.0	0.790039
$899$	−20622.3	−0.765064
$900$	−1039.25	−0.0384906
$901$	24978.4	0.923587
$902$	19864.4	0.733272
$903$	0	0
$904$	−16304.4	−0.599861
$905$	−21837.2	−0.802093
$906$	26607.9	0.975706
$907$	27468.1	1.00558	0.502790	−	0.864408i	$-0.332307\pi$
$907$	27468.1	1.00558	0.502790	+	0.864408i	$0.332307\pi$
$908$	−18777.6	−0.686295
$909$	−5357.03	−0.195469
$910$	0	0
$911$	−11545.9	−0.419905	−0.209953	−	0.977712i	$-0.567331\pi$
$911$	−11545.9	−0.419905	−0.209953	+	0.977712i	$0.567331\pi$
$912$	−1191.36	−0.0432563
$913$	−124.683	−0.00451963
$914$	23540.0	0.851898
$915$	−18322.6	−0.661997
$916$	−11086.7	−0.399907
$917$	0	0
$918$	34442.8	1.23833
$919$	−11651.4	−0.418221	−0.209111	−	0.977892i	$-0.567057\pi$
$919$	−11651.4	−0.418221	−0.209111	+	0.977892i	$0.567057\pi$
$920$	−1141.12	−0.0408930
$921$	7564.06	0.270624
$922$	14682.7	0.524455
$923$	7392.49	0.263626
$924$	0	0
$925$	−17362.9	−0.617178
$926$	1580.17	0.0560775
$927$	−3254.49	−0.115309
$928$	−2140.36	−0.0757122
$929$	−861.210	−0.0304148	−0.0152074	−	0.999884i	$-0.504841\pi$
$929$	−861.210	−0.0304148	−0.0152074	+	0.999884i	$0.504841\pi$
$930$	20946.9	0.738575
$931$	0	0
$932$	−19645.4	−0.690458
$933$	−1633.57	−0.0573213
$934$	22394.7	0.784557
$935$	−26885.8	−0.940386
$936$	829.538	0.0289683
$937$	8059.54	0.280997	0.140498	−	0.990081i	$-0.455130\pi$
$937$	8059.54	0.280997	0.140498	+	0.990081i	$0.455130\pi$
$938$	0	0
$939$	−44921.4	−1.56119
$940$	6134.27	0.212849
$941$	−36315.8	−1.25809	−0.629045	−	0.777369i	$-0.716554\pi$
$941$	−36315.8	−1.25809	−0.629045	+	0.777369i	$0.716554\pi$
$942$	−31605.3	−1.09316
$943$	6903.72	0.238405
$944$	13096.4	0.451538
$945$	0	0
$946$	−9366.87	−0.321927
$947$	38394.7	1.31749	0.658743	−	0.752368i	$-0.271088\pi$
$947$	38394.7	1.31749	0.658743	+	0.752368i	$0.271088\pi$
$948$	15768.0	0.540211
$949$	36420.0	1.24578
$950$	−2352.80	−0.0803524
$951$	−18523.6	−0.631619
$952$	0	0
$953$	12564.4	0.427072	0.213536	−	0.976935i	$-0.431502\pi$
$953$	12564.4	0.427072	0.213536	+	0.976935i	$0.431502\pi$
$954$	−1144.78	−0.0388507
$955$	125.944	0.00426749
$956$	11367.5	0.384571
$957$	−12122.8	−0.409483
$958$	6386.07	0.215370
$959$	0	0
$960$	2174.05	0.0730908
$961$	65269.4	2.19091
$962$	13859.3	0.464492
$963$	−4697.51	−0.157191
$964$	2178.12	0.0727725
$965$	19927.5	0.664755
$966$	0	0
$967$	14676.0	0.488053	0.244026	−	0.969769i	$-0.421532\pi$
$967$	14676.0	0.488053	0.244026	+	0.969769i	$0.421532\pi$
$968$	1888.70	0.0627121
$969$	−9755.35	−0.323413
$970$	15217.1	0.503703
$971$	−11481.6	−0.379468	−0.189734	−	0.981836i	$-0.560763\pi$
$971$	−11481.6	−0.379468	−0.189734	+	0.981836i	$0.560763\pi$
$972$	−3354.57	−0.110697
$973$	0	0
$974$	−465.018	−0.0152979
$975$	16371.4	0.537747
$976$	−8630.15	−0.283037
$977$	−38428.8	−1.25839	−0.629195	−	0.777248i	$-0.716615\pi$
$977$	−38428.8	−1.25839	−0.629195	+	0.777248i	$0.716615\pi$
$978$	−1001.96	−0.0327600
$979$	35364.3	1.15449
$980$	0	0
$981$	−4529.38	−0.147413
$982$	−20372.2	−0.662021
$983$	57386.4	1.86200	0.930998	−	0.365024i	$-0.118939\pi$
$983$	57386.4	1.86200	0.930998	+	0.365024i	$0.118939\pi$
$984$	−13152.9	−0.426118
$985$	−14973.6	−0.484363
$986$	−17526.3	−0.566075
$987$	0	0
$988$	1878.03	0.0604738
$989$	−3255.39	−0.104667
$990$	1232.20	0.0395574
$991$	−36085.3	−1.15670	−0.578349	−	0.815790i	$-0.696303\pi$
$991$	−36085.3	−1.15670	−0.578349	+	0.815790i	$0.696303\pi$
$992$	9866.20	0.315778
$993$	−29563.8	−0.944793
$994$	0	0
$995$	20219.5	0.644221
$996$	82.5574	0.00262644
$997$	−31194.6	−0.990917	−0.495459	−	0.868632i	$-0.665000\pi$
$997$	−31194.6	−0.990917	−0.495459	+	0.868632i	$0.665000\pi$
$998$	20641.5	0.654705
$999$	−26373.0	−0.835242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.z.1.12	yes	14
7.6	odd	2	inner	2254.4.a.z.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.z.1.3	✓	14	7.6	odd	2	inner
2254.4.a.z.1.12	yes	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +5.47743 q^{3} +4.00000 q^{4} +6.20172 q^{5} -10.9549 q^{6} -8.00000 q^{8} +3.00226 q^{9} -12.4034 q^{10} -33.0895 q^{11} +21.9097 q^{12} -34.5381 q^{13} +33.9695 q^{15} +16.0000 q^{16} +131.015 q^{17} -6.00452 q^{18} -13.5939 q^{19} +24.8069 q^{20} +66.1789 q^{22} +23.0000 q^{23} -43.8195 q^{24} -86.5386 q^{25} +69.0761 q^{26} -131.446 q^{27} +66.8864 q^{29} -67.9390 q^{30} -308.319 q^{31} -32.0000 q^{32} -181.245 q^{33} -262.030 q^{34} +12.0090 q^{36} +200.638 q^{37} +27.1878 q^{38} -189.180 q^{39} -49.6138 q^{40} +300.162 q^{41} -141.539 q^{43} -132.358 q^{44} +18.6192 q^{45} -46.0000 q^{46} +247.281 q^{47} +87.6389 q^{48} +173.077 q^{50} +717.626 q^{51} -138.152 q^{52} +190.653 q^{53} +262.892 q^{54} -205.212 q^{55} -74.4597 q^{57} -133.773 q^{58} +818.525 q^{59} +135.878 q^{60} -539.384 q^{61} +616.637 q^{62} +64.0000 q^{64} -214.196 q^{65} +362.490 q^{66} -465.724 q^{67} +524.061 q^{68} +125.981 q^{69} -214.039 q^{71} -24.0181 q^{72} -1054.49 q^{73} -401.276 q^{74} -474.009 q^{75} -54.3757 q^{76} +378.360 q^{78} +719.679 q^{79} +99.2276 q^{80} -801.047 q^{81} -600.324 q^{82} +3.76807 q^{83} +812.520 q^{85} +283.077 q^{86} +366.366 q^{87} +264.716 q^{88} -1068.75 q^{89} -37.2384 q^{90} +92.0000 q^{92} -1688.79 q^{93} -494.562 q^{94} -84.3057 q^{95} -175.278 q^{96} -1226.84 q^{97} -99.3431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) \) 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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