LMFDB - Embedded newform 2254.4.a.z.1.2

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			1.2
Root			$-7.18301$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} -7.18301 q^{3} +4.00000 q^{4} +20.3352 q^{5} +14.3660 q^{6} -8.00000 q^{8} +24.5957 q^{9} -40.6703 q^{10} +1.47400 q^{11} -28.7320 q^{12} -32.4909 q^{13} -146.068 q^{15} +16.0000 q^{16} -31.8290 q^{17} -49.1913 q^{18} +51.2910 q^{19} +81.3406 q^{20} -2.94800 q^{22} +23.0000 q^{23} +57.4641 q^{24} +288.519 q^{25} +64.9819 q^{26} +17.2705 q^{27} -39.6714 q^{29} +292.135 q^{30} -35.8933 q^{31} -32.0000 q^{32} -10.5878 q^{33} +63.6579 q^{34} +98.3826 q^{36} +76.6668 q^{37} -102.582 q^{38} +233.383 q^{39} -162.681 q^{40} -162.203 q^{41} -336.129 q^{43} +5.89600 q^{44} +500.156 q^{45} -46.0000 q^{46} +215.290 q^{47} -114.928 q^{48} -577.037 q^{50} +228.628 q^{51} -129.964 q^{52} -304.769 q^{53} -34.5409 q^{54} +29.9740 q^{55} -368.424 q^{57} +79.3427 q^{58} +97.3925 q^{59} -584.271 q^{60} -923.709 q^{61} +71.7865 q^{62} +64.0000 q^{64} -660.708 q^{65} +21.1755 q^{66} +604.397 q^{67} -127.316 q^{68} -165.209 q^{69} -769.020 q^{71} -196.765 q^{72} +400.707 q^{73} -153.334 q^{74} -2072.43 q^{75} +205.164 q^{76} -466.765 q^{78} +423.609 q^{79} +325.363 q^{80} -788.137 q^{81} +324.405 q^{82} -437.830 q^{83} -647.247 q^{85} +672.258 q^{86} +284.960 q^{87} -11.7920 q^{88} -651.246 q^{89} -1000.31 q^{90} +92.0000 q^{92} +257.822 q^{93} -430.581 q^{94} +1043.01 q^{95} +229.856 q^{96} +610.711 q^{97} +36.2540 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.18301	−1.38237	−0.691186	−	0.722677i	$-0.742911\pi$
$3$	−7.18301	−1.38237	−0.691186	+	0.722677i	$0.742911\pi$
$4$	4.00000	0.500000
$5$	20.3352	1.81883	0.909416	−	0.415888i	$-0.136529\pi$
$5$	20.3352	1.81883	0.909416	+	0.415888i	$0.136529\pi$
$6$	14.3660	0.977484
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	24.5957	0.910950
$10$	−40.6703	−1.28611
$11$	1.47400	0.0404025	0.0202013	−	0.999796i	$-0.493569\pi$
$11$	1.47400	0.0404025	0.0202013	+	0.999796i	$0.493569\pi$
$12$	−28.7320	−0.691186
$13$	−32.4909	−0.693182	−0.346591	−	0.938016i	$-0.612661\pi$
$13$	−32.4909	−0.693182	−0.346591	+	0.938016i	$0.612661\pi$
$14$	0	0
$15$	−146.068	−2.51430
$16$	16.0000	0.250000
$17$	−31.8290	−0.454097	−0.227049	−	0.973883i	$-0.572908\pi$
$17$	−31.8290	−0.454097	−0.227049	+	0.973883i	$0.572908\pi$
$18$	−49.1913	−0.644139
$19$	51.2910	0.619314	0.309657	−	0.950848i	$-0.399786\pi$
$19$	51.2910	0.619314	0.309657	+	0.950848i	$0.399786\pi$
$20$	81.3406	0.909416
$21$	0	0
$22$	−2.94800	−0.0285689
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	57.4641	0.488742
$25$	288.519	2.30815
$26$	64.9819	0.490154
$27$	17.2705	0.123100
$28$	0	0
$29$	−39.6714	−0.254027	−0.127014	−	0.991901i	$-0.540539\pi$
$29$	−39.6714	−0.254027	−0.127014	+	0.991901i	$0.540539\pi$
$30$	292.135	1.77788
$31$	−35.8933	−0.207955	−0.103978	−	0.994580i	$-0.533157\pi$
$31$	−35.8933	−0.207955	−0.103978	+	0.994580i	$0.533157\pi$
$32$	−32.0000	−0.176777
$33$	−10.5878	−0.0558513
$34$	63.6579	0.321095
$35$	0	0
$36$	98.3826	0.455475
$37$	76.6668	0.340647	0.170324	−	0.985388i	$-0.445519\pi$
$37$	76.6668	0.340647	0.170324	+	0.985388i	$0.445519\pi$
$38$	−102.582	−0.437921
$39$	233.383	0.958234
$40$	−162.681	−0.643054
$41$	−162.203	−0.617848	−0.308924	−	0.951087i	$-0.599969\pi$
$41$	−162.203	−0.617848	−0.308924	+	0.951087i	$0.599969\pi$
$42$	0	0
$43$	−336.129	−1.19207	−0.596037	−	0.802957i	$-0.703259\pi$
$43$	−336.129	−1.19207	−0.596037	+	0.802957i	$0.703259\pi$
$44$	5.89600	0.0202013
$45$	500.156	1.65686
$46$	−46.0000	−0.147442
$47$	215.290	0.668156	0.334078	−	0.942545i	$-0.391575\pi$
$47$	215.290	0.668156	0.334078	+	0.942545i	$0.391575\pi$
$48$	−114.928	−0.345593
$49$	0	0
$50$	−577.037	−1.63211
$51$	228.628	0.627731
$52$	−129.964	−0.346591
$53$	−304.769	−0.789871	−0.394936	−	0.918709i	$-0.629233\pi$
$53$	−304.769	−0.789871	−0.394936	+	0.918709i	$0.629233\pi$
$54$	−34.5409	−0.0870449
$55$	29.9740	0.0734854
$56$	0	0
$57$	−368.424	−0.856122
$58$	79.3427	0.179624
$59$	97.3925	0.214905	0.107453	−	0.994210i	$-0.465731\pi$
$59$	97.3925	0.214905	0.107453	+	0.994210i	$0.465731\pi$
$60$	−584.271	−1.25715
$61$	−923.709	−1.93883	−0.969417	−	0.245420i	$-0.921074\pi$
$61$	−923.709	−1.93883	−0.969417	+	0.245420i	$0.921074\pi$
$62$	71.7865	0.147047
$63$	0	0
$64$	64.0000	0.125000
$65$	−660.708	−1.26078
$66$	21.1755	0.0394928
$67$	604.397	1.10207	0.551036	−	0.834482i	$-0.314233\pi$
$67$	604.397	1.10207	0.551036	+	0.834482i	$0.314233\pi$
$68$	−127.316	−0.227049
$69$	−165.209	−0.288244
$70$	0	0
$71$	−769.020	−1.28543	−0.642717	−	0.766104i	$-0.722193\pi$
$71$	−769.020	−1.28543	−0.642717	+	0.766104i	$0.722193\pi$
$72$	−196.765	−0.322069
$73$	400.707	0.642455	0.321227	−	0.947002i	$-0.395905\pi$
$73$	400.707	0.642455	0.321227	+	0.947002i	$0.395905\pi$
$74$	−153.334	−0.240874
$75$	−2072.43	−3.19072
$76$	205.164	0.309657
$77$	0	0
$78$	−466.765	−0.677574
$79$	423.609	0.603288	0.301644	−	0.953421i	$-0.402465\pi$
$79$	423.609	0.603288	0.301644	+	0.953421i	$0.402465\pi$
$80$	325.363	0.454708
$81$	−788.137	−1.08112
$82$	324.405	0.436885
$83$	−437.830	−0.579013	−0.289507	−	0.957176i	$-0.593491\pi$
$83$	−437.830	−0.579013	−0.289507	+	0.957176i	$0.593491\pi$
$84$	0	0
$85$	−647.247	−0.825927
$86$	672.258	0.842923
$87$	284.960	0.351160
$88$	−11.7920	−0.0142844
$89$	−651.246	−0.775640	−0.387820	−	0.921735i	$-0.626772\pi$
$89$	−651.246	−0.775640	−0.387820	+	0.921735i	$0.626772\pi$
$90$	−1000.31	−1.17158
$91$	0	0
$92$	92.0000	0.104257
$93$	257.822	0.287472
$94$	−430.581	−0.472458
$95$	1043.01	1.12643
$96$	229.856	0.244371
$97$	610.711	0.639261	0.319631	−	0.947542i	$-0.396441\pi$
$97$	610.711	0.639261	0.319631	+	0.947542i	$0.396441\pi$
$98$	0	0
$99$	36.2540	0.0368047
$100$	1154.07	1.15407
$101$	−532.486	−0.524598	−0.262299	−	0.964987i	$-0.584481\pi$
$101$	−532.486	−0.524598	−0.262299	+	0.964987i	$0.584481\pi$
$102$	−457.256	−0.443873
$103$	−1788.73	−1.71115	−0.855575	−	0.517679i	$-0.826796\pi$
$103$	−1788.73	−1.71115	−0.855575	+	0.517679i	$0.826796\pi$
$104$	259.927	0.245077
$105$	0	0
$106$	609.537	0.558523
$107$	1093.46	0.987935	0.493968	−	0.869480i	$-0.335546\pi$
$107$	1093.46	0.987935	0.493968	+	0.869480i	$0.335546\pi$
$108$	69.0819	0.0615501
$109$	−565.511	−0.496937	−0.248468	−	0.968640i	$-0.579927\pi$
$109$	−565.511	−0.496937	−0.248468	+	0.968640i	$0.579927\pi$
$110$	−59.9480	−0.0519620
$111$	−550.698	−0.470901
$112$	0	0
$113$	−93.8808	−0.0781554	−0.0390777	−	0.999236i	$-0.512442\pi$
$113$	−93.8808	−0.0781554	−0.0390777	+	0.999236i	$0.512442\pi$
$114$	736.848	0.605370
$115$	467.709	0.379253
$116$	−158.685	−0.127014
$117$	−799.136	−0.631454
$118$	−194.785	−0.151961
$119$	0	0
$120$	1168.54	0.888940
$121$	−1328.83	−0.998368
$122$	1847.42	1.37096
$123$	1165.10	0.854096
$124$	−143.573	−0.103978
$125$	3325.18	2.37930
$126$	0	0
$127$	1865.89	1.30371	0.651854	−	0.758344i	$-0.273991\pi$
$127$	1865.89	1.30371	0.651854	+	0.758344i	$0.273991\pi$
$128$	−128.000	−0.0883883
$129$	2414.42	1.64789
$130$	1321.42	0.891507
$131$	2776.35	1.85168	0.925842	−	0.377910i	$-0.123357\pi$
$131$	2776.35	1.85168	0.925842	+	0.377910i	$0.123357\pi$
$132$	−42.3510	−0.0279256
$133$	0	0
$134$	−1208.79	−0.779282
$135$	351.198	0.223898
$136$	254.632	0.160548
$137$	13.7411	0.00856918	0.00428459	−	0.999991i	$-0.498636\pi$
$137$	13.7411	0.00856918	0.00428459	+	0.999991i	$0.498636\pi$
$138$	330.419	0.203820
$139$	2592.52	1.58197	0.790987	−	0.611833i	$-0.209567\pi$
$139$	2592.52	1.58197	0.790987	+	0.611833i	$0.209567\pi$
$140$	0	0
$141$	−1546.43	−0.923640
$142$	1538.04	0.908939
$143$	−47.8916	−0.0280063
$144$	393.530	0.227738
$145$	−806.724	−0.462033
$146$	−801.414	−0.454284
$147$	0	0
$148$	306.667	0.170324
$149$	−1494.45	−0.821678	−0.410839	−	0.911708i	$-0.634764\pi$
$149$	−1494.45	−0.821678	−0.410839	+	0.911708i	$0.634764\pi$
$150$	4144.87	2.25618
$151$	−416.293	−0.224354	−0.112177	−	0.993688i	$-0.535782\pi$
$151$	−416.293	−0.224354	−0.112177	+	0.993688i	$0.535782\pi$
$152$	−410.328	−0.218961
$153$	−782.854	−0.413660
$154$	0	0
$155$	−729.895	−0.378236
$156$	933.531	0.479117
$157$	218.157	0.110897	0.0554484	−	0.998462i	$-0.482341\pi$
$157$	218.157	0.110897	0.0554484	+	0.998462i	$0.482341\pi$
$158$	−847.218	−0.426589
$159$	2189.16	1.09190
$160$	−650.725	−0.321527
$161$	0	0
$162$	1576.27	0.764467
$163$	−2570.65	−1.23527	−0.617633	−	0.786466i	$-0.711908\pi$
$163$	−2570.65	−1.23527	−0.617633	+	0.786466i	$0.711908\pi$
$164$	−648.810	−0.308924
$165$	−215.304	−0.101584
$166$	875.660	0.409424
$167$	3712.89	1.72043	0.860216	−	0.509931i	$-0.170329\pi$
$167$	3712.89	1.72043	0.860216	+	0.509931i	$0.170329\pi$
$168$	0	0
$169$	−1141.34	−0.519499
$170$	1294.49	0.584019
$171$	1261.54	0.564164
$172$	−1344.52	−0.596037
$173$	1474.27	0.647902	0.323951	−	0.946074i	$-0.394989\pi$
$173$	1474.27	0.647902	0.323951	+	0.946074i	$0.394989\pi$
$174$	−569.920	−0.248308
$175$	0	0
$176$	23.5840	0.0101006
$177$	−699.571	−0.297079
$178$	1302.49	0.548460
$179$	1913.64	0.799064	0.399532	−	0.916719i	$-0.369173\pi$
$179$	1913.64	0.799064	0.399532	+	0.916719i	$0.369173\pi$
$180$	2000.63	0.828432
$181$	916.510	0.376374	0.188187	−	0.982133i	$-0.439739\pi$
$181$	916.510	0.376374	0.188187	+	0.982133i	$0.439739\pi$
$182$	0	0
$183$	6635.01	2.68019
$184$	−184.000	−0.0737210
$185$	1559.03	0.619580
$186$	−515.643	−0.203273
$187$	−46.9159	−0.0183467
$188$	861.162	0.334078
$189$	0	0
$190$	−2086.02	−0.796505
$191$	−4751.71	−1.80011	−0.900057	−	0.435773i	$-0.856475\pi$
$191$	−4751.71	−1.80011	−0.900057	+	0.435773i	$0.856475\pi$
$192$	−459.713	−0.172796
$193$	−4068.24	−1.51730	−0.758649	−	0.651500i	$-0.774140\pi$
$193$	−4068.24	−1.51730	−0.758649	+	0.651500i	$0.774140\pi$
$194$	−1221.42	−0.452026
$195$	4745.87	1.74287
$196$	0	0
$197$	−3976.49	−1.43814	−0.719068	−	0.694939i	$-0.755431\pi$
$197$	−3976.49	−1.43814	−0.719068	+	0.694939i	$0.755431\pi$
$198$	−72.5080	−0.0260248
$199$	−1007.05	−0.358733	−0.179366	−	0.983782i	$-0.557405\pi$
$199$	−1007.05	−0.358733	−0.179366	+	0.983782i	$0.557405\pi$
$200$	−2308.15	−0.816054
$201$	−4341.39	−1.52347
$202$	1064.97	0.370947
$203$	0	0
$204$	914.511	0.313866
$205$	−3298.42	−1.12376
$206$	3577.45	1.20997
$207$	565.700	0.189946
$208$	−519.855	−0.173295
$209$	75.6030	0.0250218
$210$	0	0
$211$	−1559.98	−0.508975	−0.254488	−	0.967076i	$-0.581907\pi$
$211$	−1559.98	−0.508975	−0.254488	+	0.967076i	$0.581907\pi$
$212$	−1219.07	−0.394936
$213$	5523.88	1.77695
$214$	−2186.93	−0.698576
$215$	−6835.23	−2.16818
$216$	−138.164	−0.0435225
$217$	0	0
$218$	1131.02	0.351387
$219$	−2878.28	−0.888111
$220$	119.896	0.0367427
$221$	1034.15	0.314772
$222$	1101.40	0.332977
$223$	−315.399	−0.0947116	−0.0473558	−	0.998878i	$-0.515079\pi$
$223$	−315.399	−0.0947116	−0.0473558	+	0.998878i	$0.515079\pi$
$224$	0	0
$225$	7096.30	2.10261
$226$	187.762	0.0552642
$227$	1858.86	0.543512	0.271756	−	0.962366i	$-0.412396\pi$
$227$	1858.86	0.543512	0.271756	+	0.962366i	$0.412396\pi$
$228$	−1473.70	−0.428061
$229$	373.471	0.107771	0.0538857	−	0.998547i	$-0.482839\pi$
$229$	373.471	0.107771	0.0538857	+	0.998547i	$0.482839\pi$
$230$	−935.417	−0.268172
$231$	0	0
$232$	317.371	0.0898122
$233$	−3174.43	−0.892548	−0.446274	−	0.894896i	$-0.647249\pi$
$233$	−3174.43	−0.892548	−0.446274	+	0.894896i	$0.647249\pi$
$234$	1598.27	0.446505
$235$	4377.97	1.21526
$236$	389.570	0.107453
$237$	−3042.79	−0.833968
$238$	0	0
$239$	−2124.46	−0.574978	−0.287489	−	0.957784i	$-0.592820\pi$
$239$	−2124.46	−0.574978	−0.287489	+	0.957784i	$0.592820\pi$
$240$	−2337.08	−0.628575
$241$	5603.04	1.49761	0.748804	−	0.662791i	$-0.230628\pi$
$241$	5603.04	1.49761	0.748804	+	0.662791i	$0.230628\pi$
$242$	2657.65	0.705953
$243$	5194.89	1.37141
$244$	−3694.84	−0.969417
$245$	0	0
$246$	−2330.21	−0.603937
$247$	−1666.49	−0.429297
$248$	287.146	0.0735233
$249$	3144.94	0.800411
$250$	−6650.36	−1.68242
$251$	−1370.33	−0.344599	−0.172299	−	0.985045i	$-0.555120\pi$
$251$	−1370.33	−0.344599	−0.172299	+	0.985045i	$0.555120\pi$
$252$	0	0
$253$	33.9020	0.00842451
$254$	−3731.78	−0.921861
$255$	4649.18	1.14174
$256$	256.000	0.0625000
$257$	7322.81	1.77737	0.888686	−	0.458516i	$-0.151619\pi$
$257$	7322.81	1.77737	0.888686	+	0.458516i	$0.151619\pi$
$258$	−4828.83	−1.16523
$259$	0	0
$260$	−2642.83	−0.630391
$261$	−975.743	−0.231406
$262$	−5552.70	−1.30934
$263$	4355.02	1.02107	0.510537	−	0.859856i	$-0.329447\pi$
$263$	4355.02	1.02107	0.510537	+	0.859856i	$0.329447\pi$
$264$	84.7021	0.0197464
$265$	−6197.52	−1.43664
$266$	0	0
$267$	4677.91	1.07222
$268$	2417.59	0.551036
$269$	−3763.78	−0.853091	−0.426545	−	0.904466i	$-0.640270\pi$
$269$	−3763.78	−0.853091	−0.426545	+	0.904466i	$0.640270\pi$
$270$	−702.395	−0.158320
$271$	−558.317	−0.125149	−0.0625744	−	0.998040i	$-0.519931\pi$
$271$	−558.317	−0.125149	−0.0625744	+	0.998040i	$0.519931\pi$
$272$	−509.263	−0.113524
$273$	0	0
$274$	−27.4821	−0.00605933
$275$	425.276	0.0932550
$276$	−660.837	−0.144122
$277$	6137.45	1.33128	0.665638	−	0.746275i	$-0.268159\pi$
$277$	6137.45	1.33128	0.665638	+	0.746275i	$0.268159\pi$
$278$	−5185.03	−1.11862
$279$	−882.818	−0.189437
$280$	0	0
$281$	957.943	0.203367	0.101683	−	0.994817i	$-0.467577\pi$
$281$	957.943	0.203367	0.101683	+	0.994817i	$0.467577\pi$
$282$	3092.87	0.653112
$283$	−7076.50	−1.48641	−0.743206	−	0.669063i	$-0.766696\pi$
$283$	−7076.50	−1.48641	−0.743206	+	0.669063i	$0.766696\pi$
$284$	−3076.08	−0.642717
$285$	−7491.96	−1.55714
$286$	95.7832	0.0198034
$287$	0	0
$288$	−787.061	−0.161035
$289$	−3899.92	−0.793795
$290$	1613.45	0.326706
$291$	−4386.75	−0.883696
$292$	1602.83	0.321227
$293$	2120.04	0.422710	0.211355	−	0.977409i	$-0.432212\pi$
$293$	2120.04	0.422710	0.211355	+	0.977409i	$0.432212\pi$
$294$	0	0
$295$	1980.49	0.390877
$296$	−613.334	−0.120437
$297$	25.4567	0.00497355
$298$	2988.90	0.581014
$299$	−747.291	−0.144538
$300$	−8289.73	−1.59536
$301$	0	0
$302$	832.587	0.158642
$303$	3824.86	0.725189
$304$	820.656	0.154829
$305$	−18783.8	−3.52641
$306$	1565.71	0.292502
$307$	110.678	0.0205756	0.0102878	−	0.999947i	$-0.496725\pi$
$307$	110.678	0.0205756	0.0102878	+	0.999947i	$0.496725\pi$
$308$	0	0
$309$	12848.4	2.36544
$310$	1459.79	0.267453
$311$	−2581.75	−0.470732	−0.235366	−	0.971907i	$-0.575629\pi$
$311$	−2581.75	−0.470732	−0.235366	+	0.971907i	$0.575629\pi$
$312$	−1867.06	−0.338787
$313$	−1404.56	−0.253643	−0.126821	−	0.991926i	$-0.540478\pi$
$313$	−1404.56	−0.253643	−0.126821	+	0.991926i	$0.540478\pi$
$314$	−436.314	−0.0784159
$315$	0	0
$316$	1694.44	0.301644
$317$	78.7013	0.0139442	0.00697209	−	0.999976i	$-0.497781\pi$
$317$	78.7013	0.0139442	0.00697209	+	0.999976i	$0.497781\pi$
$318$	−4378.31	−0.772087
$319$	−58.4756	−0.0102633
$320$	1301.45	0.227354
$321$	−7854.36	−1.36569
$322$	0	0
$323$	−1632.54	−0.281229
$324$	−3152.55	−0.540560
$325$	−9374.24	−1.59997
$326$	5141.29	0.873465
$327$	4062.07	0.686951
$328$	1297.62	0.218442
$329$	0	0
$330$	430.607	0.0718308
$331$	−1466.58	−0.243537	−0.121768	−	0.992559i	$-0.538857\pi$
$331$	−1466.58	−0.243537	−0.121768	+	0.992559i	$0.538857\pi$
$332$	−1751.32	−0.289507
$333$	1885.67	0.310312
$334$	−7425.78	−1.21653
$335$	12290.5	2.00448
$336$	0	0
$337$	−902.848	−0.145939	−0.0729693	−	0.997334i	$-0.523248\pi$
$337$	−902.848	−0.145939	−0.0729693	+	0.997334i	$0.523248\pi$
$338$	2282.68	0.367341
$339$	674.346	0.108040
$340$	−2588.99	−0.412963
$341$	−52.9066	−0.00840192
$342$	−2523.07	−0.398924
$343$	0	0
$344$	2689.03	0.421462
$345$	−3359.56	−0.524268
$346$	−2948.55	−0.458136
$347$	−8618.21	−1.33328	−0.666642	−	0.745378i	$-0.732269\pi$
$347$	−8618.21	−1.33328	−0.666642	+	0.745378i	$0.732269\pi$
$348$	1139.84	0.175580
$349$	−11350.6	−1.74093	−0.870466	−	0.492228i	$-0.836182\pi$
$349$	−11350.6	−1.74093	−0.870466	+	0.492228i	$0.836182\pi$
$350$	0	0
$351$	−561.134	−0.0853307
$352$	−47.1680	−0.00714222
$353$	9747.51	1.46971	0.734855	−	0.678224i	$-0.237250\pi$
$353$	9747.51	1.46971	0.734855	+	0.678224i	$0.237250\pi$
$354$	1399.14	0.210067
$355$	−15638.1	−2.33799
$356$	−2604.98	−0.387820
$357$	0	0
$358$	−3827.29	−0.565023
$359$	−5328.86	−0.783417	−0.391708	−	0.920089i	$-0.628116\pi$
$359$	−5328.86	−0.783417	−0.391708	+	0.920089i	$0.628116\pi$
$360$	−4001.25	−0.585790
$361$	−4228.23	−0.616450
$362$	−1833.02	−0.266136
$363$	9544.98	1.38011
$364$	0	0
$365$	8148.44	1.16852
$366$	−13270.0	−1.89518
$367$	5593.55	0.795588	0.397794	−	0.917475i	$-0.369776\pi$
$367$	5593.55	0.795588	0.397794	+	0.917475i	$0.369776\pi$
$368$	368.000	0.0521286
$369$	−3989.48	−0.562829
$370$	−3118.06	−0.438109
$371$	0	0
$372$	1031.29	0.143736
$373$	1179.34	0.163710	0.0818550	−	0.996644i	$-0.473916\pi$
$373$	1179.34	0.163710	0.0818550	+	0.996644i	$0.473916\pi$
$374$	93.8318	0.0129731
$375$	−23884.8	−3.28908
$376$	−1722.32	−0.236229
$377$	1288.96	0.176087
$378$	0	0
$379$	−2763.33	−0.374519	−0.187259	−	0.982311i	$-0.559960\pi$
$379$	−2763.33	−0.374519	−0.187259	+	0.982311i	$0.559960\pi$
$380$	4172.04	0.563214
$381$	−13402.7	−1.80221
$382$	9503.42	1.27287
$383$	−2985.88	−0.398359	−0.199179	−	0.979963i	$-0.563828\pi$
$383$	−2985.88	−0.398359	−0.199179	+	0.979963i	$0.563828\pi$
$384$	919.425	0.122186
$385$	0	0
$386$	8136.48	1.07289
$387$	−8267.31	−1.08592
$388$	2442.84	0.319631
$389$	−8966.43	−1.16868	−0.584339	−	0.811509i	$-0.698646\pi$
$389$	−8966.43	−1.16868	−0.584339	+	0.811509i	$0.698646\pi$
$390$	−9491.75	−1.23239
$391$	−732.066	−0.0946859
$392$	0	0
$393$	−19942.5	−2.55972
$394$	7952.97	1.01692
$395$	8614.16	1.09728
$396$	145.016	0.0184023
$397$	4745.30	0.599899	0.299949	−	0.953955i	$-0.403030\pi$
$397$	4745.30	0.599899	0.299949	+	0.953955i	$0.403030\pi$
$398$	2014.10	0.253662
$399$	0	0
$400$	4616.30	0.577037
$401$	7696.33	0.958445	0.479222	−	0.877693i	$-0.340919\pi$
$401$	7696.33	0.958445	0.479222	+	0.877693i	$0.340919\pi$
$402$	8682.78	1.07726
$403$	1166.21	0.144151
$404$	−2129.95	−0.262299
$405$	−16026.9	−1.96638
$406$	0	0
$407$	113.007	0.0137630
$408$	−1829.02	−0.221937
$409$	−2170.05	−0.262353	−0.131176	−	0.991359i	$-0.541875\pi$
$409$	−2170.05	−0.262353	−0.131176	+	0.991359i	$0.541875\pi$
$410$	6596.83	0.794620
$411$	−98.7022	−0.0118458
$412$	−7154.90	−0.855575
$413$	0	0
$414$	−1131.40	−0.134312
$415$	−8903.34	−1.05313
$416$	1039.71	0.122538
$417$	−18622.1	−2.18688
$418$	−151.206	−0.0176931
$419$	−12771.5	−1.48909	−0.744547	−	0.667570i	$-0.767334\pi$
$419$	−12771.5	−1.48909	−0.744547	+	0.667570i	$0.767334\pi$
$420$	0	0
$421$	−16190.5	−1.87429	−0.937147	−	0.348934i	$-0.886544\pi$
$421$	−16190.5	−1.87429	−0.937147	+	0.348934i	$0.886544\pi$
$422$	3119.97	0.359900
$423$	5295.21	0.608657
$424$	2438.15	0.279262
$425$	−9183.25	−1.04812
$426$	−11047.8	−1.25649
$427$	0	0
$428$	4373.85	0.493968
$429$	344.006	0.0387151
$430$	13670.5	1.53314
$431$	6321.36	0.706471	0.353236	−	0.935534i	$-0.385081\pi$
$431$	6321.36	0.706471	0.353236	+	0.935534i	$0.385081\pi$
$432$	276.327	0.0307750
$433$	−13821.8	−1.53402	−0.767011	−	0.641634i	$-0.778257\pi$
$433$	−13821.8	−1.53402	−0.767011	+	0.641634i	$0.778257\pi$
$434$	0	0
$435$	5794.70	0.638701
$436$	−2262.04	−0.248468
$437$	1179.69	0.129136
$438$	5756.57	0.627989
$439$	−5015.04	−0.545227	−0.272613	−	0.962124i	$-0.587888\pi$
$439$	−5015.04	−0.545227	−0.272613	+	0.962124i	$0.587888\pi$
$440$	−239.792	−0.0259810
$441$	0	0
$442$	−2068.31	−0.222577
$443$	−10170.5	−1.09078	−0.545391	−	0.838182i	$-0.683619\pi$
$443$	−10170.5	−1.09078	−0.545391	+	0.838182i	$0.683619\pi$
$444$	−2202.79	−0.235450
$445$	−13243.2	−1.41076
$446$	630.798	0.0669712
$447$	10734.6	1.13586
$448$	0	0
$449$	6325.09	0.664809	0.332405	−	0.943137i	$-0.392140\pi$
$449$	6325.09	0.664809	0.332405	+	0.943137i	$0.392140\pi$
$450$	−14192.6	−1.48677
$451$	−239.087	−0.0249626
$452$	−375.523	−0.0390777
$453$	2990.24	0.310141
$454$	−3717.73	−0.384321
$455$	0	0
$456$	2947.39	0.302685
$457$	−9126.72	−0.934201	−0.467101	−	0.884204i	$-0.654701\pi$
$457$	−9126.72	−0.934201	−0.467101	+	0.884204i	$0.654701\pi$
$458$	−746.942	−0.0762059
$459$	−549.701	−0.0558994
$460$	1870.83	0.189626
$461$	−14594.2	−1.47445	−0.737223	−	0.675650i	$-0.763863\pi$
$461$	−14594.2	−1.47445	−0.737223	+	0.675650i	$0.763863\pi$
$462$	0	0
$463$	−14629.3	−1.46843	−0.734213	−	0.678919i	$-0.762449\pi$
$463$	−14629.3	−1.46843	−0.734213	+	0.678919i	$0.762449\pi$
$464$	−634.742	−0.0635068
$465$	5242.84	0.522862
$466$	6348.86	0.631127
$467$	13861.5	1.37352	0.686759	−	0.726885i	$-0.259033\pi$
$467$	13861.5	1.37352	0.686759	+	0.726885i	$0.259033\pi$
$468$	−3196.54	−0.315727
$469$	0	0
$470$	−8755.93	−0.859321
$471$	−1567.02	−0.153301
$472$	−779.140	−0.0759806
$473$	−495.454	−0.0481628
$474$	6085.58	0.589704
$475$	14798.4	1.42947
$476$	0	0
$477$	−7495.98	−0.719533
$478$	4248.91	0.406571
$479$	15515.2	1.47997	0.739987	−	0.672621i	$-0.234832\pi$
$479$	15515.2	1.47997	0.739987	+	0.672621i	$0.234832\pi$
$480$	4674.17	0.444470
$481$	−2490.98	−0.236130
$482$	−11206.1	−1.05897
$483$	0	0
$484$	−5315.31	−0.499184
$485$	12418.9	1.16271
$486$	−10389.8	−0.969733
$487$	14994.6	1.39521	0.697607	−	0.716480i	$-0.254248\pi$
$487$	14994.6	1.39521	0.697607	+	0.716480i	$0.254248\pi$
$488$	7389.67	0.685481
$489$	18465.0	1.70760
$490$	0	0
$491$	−8488.36	−0.780192	−0.390096	−	0.920774i	$-0.627558\pi$
$491$	−8488.36	−0.780192	−0.390096	+	0.920774i	$0.627558\pi$
$492$	4660.41	0.427048
$493$	1262.70	0.115353
$494$	3332.99	0.303559
$495$	737.231	0.0669415
$496$	−574.292	−0.0519889
$497$	0	0
$498$	−6289.87	−0.565976
$499$	1808.90	0.162280	0.0811399	−	0.996703i	$-0.474144\pi$
$499$	1808.90	0.162280	0.0811399	+	0.996703i	$0.474144\pi$
$500$	13300.7	1.18965
$501$	−26669.7	−2.37827
$502$	2740.65	0.243668
$503$	12894.4	1.14301	0.571505	−	0.820599i	$-0.306360\pi$
$503$	12894.4	1.14301	0.571505	+	0.820599i	$0.306360\pi$
$504$	0	0
$505$	−10828.2	−0.954155
$506$	−67.8040	−0.00595703
$507$	8198.25	0.718141
$508$	7463.56	0.651854
$509$	−5045.96	−0.439407	−0.219703	−	0.975567i	$-0.570509\pi$
$509$	−5045.96	−0.439407	−0.219703	+	0.975567i	$0.570509\pi$
$510$	−9298.36	−0.807330
$511$	0	0
$512$	−512.000	−0.0441942
$513$	885.820	0.0762376
$514$	−14645.6	−1.25679
$515$	−36374.0	−3.11229
$516$	9657.67	0.823944
$517$	317.338	0.0269952
$518$	0	0
$519$	−10589.7	−0.895641
$520$	5285.67	0.445753
$521$	−12167.5	−1.02316	−0.511580	−	0.859235i	$-0.670940\pi$
$521$	−12167.5	−1.02316	−0.511580	+	0.859235i	$0.670940\pi$
$522$	1951.49	0.163629
$523$	4974.77	0.415930	0.207965	−	0.978136i	$-0.433316\pi$
$523$	4974.77	0.415930	0.207965	+	0.978136i	$0.433316\pi$
$524$	11105.4	0.925842
$525$	0	0
$526$	−8710.05	−0.722008
$527$	1142.44	0.0944320
$528$	−169.404	−0.0139628
$529$	529.000	0.0434783
$530$	12395.0	1.01586
$531$	2395.43	0.195768
$532$	0	0
$533$	5270.11	0.428281
$534$	−9355.82	−0.758176
$535$	22235.7	1.79689
$536$	−4835.17	−0.389641
$537$	−13745.7	−1.10460
$538$	7527.55	0.603226
$539$	0	0
$540$	1404.79	0.111949
$541$	20242.3	1.60866	0.804328	−	0.594186i	$-0.202526\pi$
$541$	20242.3	1.60866	0.804328	+	0.594186i	$0.202526\pi$
$542$	1116.63	0.0884936
$543$	−6583.30	−0.520288
$544$	1018.53	0.0802739
$545$	−11499.7	−0.903844
$546$	0	0
$547$	−10362.8	−0.810024	−0.405012	−	0.914311i	$-0.632733\pi$
$547$	−10362.8	−0.810024	−0.405012	+	0.914311i	$0.632733\pi$
$548$	54.9642	0.00428459
$549$	−22719.2	−1.76618
$550$	−850.553	−0.0659413
$551$	−2034.79	−0.157323
$552$	1321.67	0.101910
$553$	0	0
$554$	−12274.9	−0.941354
$555$	−11198.5	−0.856489
$556$	10370.1	0.790987
$557$	20976.3	1.59568	0.797840	−	0.602869i	$-0.205976\pi$
$557$	20976.3	1.59568	0.797840	+	0.602869i	$0.205976\pi$
$558$	1765.64	0.133952
$559$	10921.1	0.826324
$560$	0	0
$561$	336.997	0.0253619
$562$	−1915.89	−0.143802
$563$	−15834.4	−1.18533	−0.592665	−	0.805449i	$-0.701924\pi$
$563$	−15834.4	−1.18533	−0.592665	+	0.805449i	$0.701924\pi$
$564$	−6185.73	−0.461820
$565$	−1909.08	−0.142152
$566$	14153.0	1.05105
$567$	0	0
$568$	6152.16	0.454470
$569$	−5730.50	−0.422206	−0.211103	−	0.977464i	$-0.567706\pi$
$569$	−5730.50	−0.422206	−0.211103	+	0.977464i	$0.567706\pi$
$570$	14983.9	1.10107
$571$	27207.3	1.99403	0.997013	−	0.0772378i	$-0.0246101\pi$
$571$	27207.3	1.99403	0.997013	+	0.0772378i	$0.0246101\pi$
$572$	−191.566	−0.0140031
$573$	34131.6	2.48842
$574$	0	0
$575$	6635.93	0.481282
$576$	1574.12	0.113869
$577$	−21743.7	−1.56880	−0.784402	−	0.620253i	$-0.787030\pi$
$577$	−21743.7	−1.56880	−0.784402	+	0.620253i	$0.787030\pi$
$578$	7799.83	0.561298
$579$	29222.2	2.09747
$580$	−3226.89	−0.231016
$581$	0	0
$582$	8773.49	0.624867
$583$	−449.229	−0.0319128
$584$	−3205.66	−0.227142
$585$	−16250.5	−1.14851
$586$	−4240.08	−0.298901
$587$	−22903.0	−1.61040	−0.805201	−	0.593002i	$-0.797943\pi$
$587$	−22903.0	−1.61040	−0.805201	+	0.593002i	$0.797943\pi$
$588$	0	0
$589$	−1841.00	−0.128790
$590$	−3960.98	−0.276392
$591$	28563.1	1.98804
$592$	1226.67	0.0851618
$593$	20655.2	1.43037	0.715183	−	0.698937i	$-0.246343\pi$
$593$	20655.2	1.43037	0.715183	+	0.698937i	$0.246343\pi$
$594$	−50.9133	−0.00351683
$595$	0	0
$596$	−5977.80	−0.410839
$597$	7233.65	0.495902
$598$	1494.58	0.102204
$599$	−8692.83	−0.592954	−0.296477	−	0.955040i	$-0.595812\pi$
$599$	−8692.83	−0.592954	−0.296477	+	0.955040i	$0.595812\pi$
$600$	16579.5	1.12809
$601$	−1689.69	−0.114682	−0.0573408	−	0.998355i	$-0.518262\pi$
$601$	−1689.69	−0.114682	−0.0573408	+	0.998355i	$0.518262\pi$
$602$	0	0
$603$	14865.5	1.00393
$604$	−1665.17	−0.112177
$605$	−27021.9	−1.81586
$606$	−7649.71	−0.512786
$607$	−6775.64	−0.453072	−0.226536	−	0.974003i	$-0.572740\pi$
$607$	−6775.64	−0.453072	−0.226536	+	0.974003i	$0.572740\pi$
$608$	−1641.31	−0.109480
$609$	0	0
$610$	37567.5	2.49355
$611$	−6994.99	−0.463154
$612$	−3131.42	−0.206830
$613$	13609.0	0.896677	0.448339	−	0.893864i	$-0.352016\pi$
$613$	13609.0	0.896677	0.448339	+	0.893864i	$0.352016\pi$
$614$	−221.355	−0.0145491
$615$	23692.6	1.55346
$616$	0	0
$617$	−27132.6	−1.77037	−0.885184	−	0.465242i	$-0.845967\pi$
$617$	−27132.6	−1.77037	−0.885184	+	0.465242i	$0.845967\pi$
$618$	−25696.9	−1.67262
$619$	−15927.0	−1.03419	−0.517093	−	0.855929i	$-0.672986\pi$
$619$	−15927.0	−1.03419	−0.517093	+	0.855929i	$0.672986\pi$
$620$	−2919.58	−0.189118
$621$	397.221	0.0256681
$622$	5163.50	0.332858
$623$	0	0
$624$	3734.12	0.239559
$625$	31553.2	2.01940
$626$	2809.11	0.179353
$627$	−543.057	−0.0345895
$628$	872.627	0.0554484
$629$	−2440.22	−0.154687
$630$	0	0
$631$	−609.908	−0.0384787	−0.0192393	−	0.999815i	$-0.506124\pi$
$631$	−609.908	−0.0384787	−0.0192393	+	0.999815i	$0.506124\pi$
$632$	−3388.87	−0.213295
$633$	11205.4	0.703593
$634$	−157.403	−0.00986003
$635$	37943.2	2.37123
$636$	8756.63	0.545948
$637$	0	0
$638$	116.951	0.00725728
$639$	−18914.5	−1.17097
$640$	−2602.90	−0.160764
$641$	4391.42	0.270594	0.135297	−	0.990805i	$-0.456801\pi$
$641$	4391.42	0.270594	0.135297	+	0.990805i	$0.456801\pi$
$642$	15708.7	0.965691
$643$	−27611.5	−1.69345	−0.846727	−	0.532028i	$-0.821430\pi$
$643$	−27611.5	−1.69345	−0.846727	+	0.532028i	$0.821430\pi$
$644$	0	0
$645$	49097.6	2.99723
$646$	3265.08	0.198859
$647$	6208.49	0.377250	0.188625	−	0.982049i	$-0.439597\pi$
$647$	6208.49	0.377250	0.188625	+	0.982049i	$0.439597\pi$
$648$	6305.09	0.382234
$649$	143.557	0.00868272
$650$	18748.5	1.13135
$651$	0	0
$652$	−10282.6	−0.617633
$653$	−7086.67	−0.424691	−0.212345	−	0.977195i	$-0.568110\pi$
$653$	−7086.67	−0.424691	−0.212345	+	0.977195i	$0.568110\pi$
$654$	−8124.14	−0.485748
$655$	56457.5	3.36790
$656$	−2595.24	−0.154462
$657$	9855.65	0.585244
$658$	0	0
$659$	−21839.3	−1.29096	−0.645478	−	0.763779i	$-0.723342\pi$
$659$	−21839.3	−1.29096	−0.645478	+	0.763779i	$0.723342\pi$
$660$	−861.215	−0.0507920
$661$	9175.62	0.539925	0.269962	−	0.962871i	$-0.412989\pi$
$661$	9175.62	0.539925	0.269962	+	0.962871i	$0.412989\pi$
$662$	2933.17	0.172207
$663$	−7428.33	−0.435132
$664$	3502.64	0.204712
$665$	0	0
$666$	−3771.34	−0.219424
$667$	−912.442	−0.0529683
$668$	14851.6	0.860216
$669$	2265.51	0.130927
$670$	−24581.0	−1.41738
$671$	−1361.55	−0.0783338
$672$	0	0
$673$	−30457.6	−1.74451	−0.872253	−	0.489055i	$-0.837342\pi$
$673$	−30457.6	−1.74451	−0.872253	+	0.489055i	$0.837342\pi$
$674$	1805.70	0.103194
$675$	4982.85	0.284133
$676$	−4565.36	−0.259750
$677$	6352.61	0.360636	0.180318	−	0.983608i	$-0.442287\pi$
$677$	6352.61	0.360636	0.180318	+	0.983608i	$0.442287\pi$
$678$	−1348.69	−0.0763956
$679$	0	0
$680$	5177.98	0.292009
$681$	−13352.2	−0.751335
$682$	105.813	0.00594106
$683$	−27906.0	−1.56339	−0.781693	−	0.623664i	$-0.785643\pi$
$683$	−27906.0	−1.56339	−0.781693	+	0.623664i	$0.785643\pi$
$684$	5046.14	0.282082
$685$	279.427	0.0155859
$686$	0	0
$687$	−2682.65	−0.148980
$688$	−5378.06	−0.298018
$689$	9902.22	0.547524
$690$	6719.11	0.370713
$691$	1272.77	0.0700703	0.0350351	−	0.999386i	$-0.488846\pi$
$691$	1272.77	0.0700703	0.0350351	+	0.999386i	$0.488846\pi$
$692$	5897.10	0.323951
$693$	0	0
$694$	17236.4	0.942775
$695$	52719.2	2.87735
$696$	−2279.68	−0.124154
$697$	5162.74	0.280563
$698$	22701.3	1.23102
$699$	22802.0	1.23383
$700$	0	0
$701$	−15411.1	−0.830342	−0.415171	−	0.909743i	$-0.636278\pi$
$701$	−15411.1	−0.830342	−0.415171	+	0.909743i	$0.636278\pi$
$702$	1122.27	0.0603380
$703$	3932.32	0.210968
$704$	94.3360	0.00505031
$705$	−31447.0	−1.67995
$706$	−19495.0	−1.03924
$707$	0	0
$708$	−2798.29	−0.148540
$709$	−23720.1	−1.25646	−0.628228	−	0.778029i	$-0.716220\pi$
$709$	−23720.1	−1.25646	−0.628228	+	0.778029i	$0.716220\pi$
$710$	31276.3	1.65321
$711$	10418.9	0.549565
$712$	5209.97	0.274230
$713$	−825.545	−0.0433617
$714$	0	0
$715$	−973.884	−0.0509387
$716$	7654.57	0.399532
$717$	15260.0	0.794832
$718$	10657.7	0.553959
$719$	2497.80	0.129558	0.0647791	−	0.997900i	$-0.479366\pi$
$719$	2497.80	0.129558	0.0647791	+	0.997900i	$0.479366\pi$
$720$	8002.50	0.414216
$721$	0	0
$722$	8456.46	0.435896
$723$	−40246.7	−2.07025
$724$	3666.04	0.188187
$725$	−11445.9	−0.586333
$726$	−19090.0	−0.975888
$727$	1811.18	0.0923977	0.0461989	−	0.998932i	$-0.485289\pi$
$727$	1811.18	0.0923977	0.0461989	+	0.998932i	$0.485289\pi$
$728$	0	0
$729$	−16035.3	−0.814676
$730$	−16296.9	−0.826267
$731$	10698.6	0.541318
$732$	26540.1	1.34009
$733$	−6029.72	−0.303837	−0.151919	−	0.988393i	$-0.548545\pi$
$733$	−6029.72	−0.303837	−0.151919	+	0.988393i	$0.548545\pi$
$734$	−11187.1	−0.562566
$735$	0	0
$736$	−736.000	−0.0368605
$737$	890.881	0.0445265
$738$	7978.96	0.397980
$739$	15068.2	0.750060	0.375030	−	0.927013i	$-0.377632\pi$
$739$	15068.2	0.750060	0.375030	+	0.927013i	$0.377632\pi$
$740$	6236.13	0.309790
$741$	11970.4	0.593448
$742$	0	0
$743$	7079.01	0.349534	0.174767	−	0.984610i	$-0.444083\pi$
$743$	7079.01	0.349534	0.174767	+	0.984610i	$0.444083\pi$
$744$	−2062.57	−0.101637
$745$	−30389.9	−1.49449
$746$	−2358.68	−0.115760
$747$	−10768.7	−0.527452
$748$	−187.664	−0.00917334
$749$	0	0
$750$	47769.6	2.32573
$751$	7405.69	0.359837	0.179918	−	0.983682i	$-0.442417\pi$
$751$	7405.69	0.359837	0.179918	+	0.983682i	$0.442417\pi$
$752$	3444.65	0.167039
$753$	9843.08	0.476363
$754$	−2577.92	−0.124512
$755$	−8465.39	−0.408062
$756$	0	0
$757$	36065.1	1.73158	0.865791	−	0.500405i	$-0.166816\pi$
$757$	36065.1	1.73158	0.865791	+	0.500405i	$0.166816\pi$
$758$	5526.65	0.264825
$759$	−243.518	−0.0116458
$760$	−8344.09	−0.398252
$761$	−25000.1	−1.19087	−0.595437	−	0.803402i	$-0.703021\pi$
$761$	−25000.1	−1.19087	−0.595437	+	0.803402i	$0.703021\pi$
$762$	26805.4	1.27435
$763$	0	0
$764$	−19006.8	−0.900057
$765$	−15919.5	−0.752378
$766$	5971.77	0.281682
$767$	−3164.37	−0.148969
$768$	−1838.85	−0.0863982
$769$	24222.7	1.13588	0.567940	−	0.823070i	$-0.307741\pi$
$769$	24222.7	1.13588	0.567940	+	0.823070i	$0.307741\pi$
$770$	0	0
$771$	−52599.9	−2.45699
$772$	−16273.0	−0.758649
$773$	−8817.88	−0.410294	−0.205147	−	0.978731i	$-0.565767\pi$
$773$	−8817.88	−0.410294	−0.205147	+	0.978731i	$0.565767\pi$
$774$	16534.6	0.767861
$775$	−10355.9	−0.479992
$776$	−4885.69	−0.226013
$777$	0	0
$778$	17932.9	0.826380
$779$	−8319.54	−0.382642
$780$	18983.5	0.871434
$781$	−1133.53	−0.0519348
$782$	1464.13	0.0669530
$783$	−685.143	−0.0312708
$784$	0	0
$785$	4436.25	0.201703
$786$	39885.1	1.80999
$787$	33171.2	1.50245	0.751223	−	0.660048i	$-0.229464\pi$
$787$	33171.2	1.50245	0.751223	+	0.660048i	$0.229464\pi$
$788$	−15905.9	−0.719068
$789$	−31282.2	−1.41150
$790$	−17228.3	−0.775894
$791$	0	0
$792$	−290.032	−0.0130124
$793$	30012.2	1.34396
$794$	−9490.60	−0.424192
$795$	44516.8	1.98597
$796$	−4028.20	−0.179366
$797$	−27164.0	−1.20728	−0.603638	−	0.797258i	$-0.706283\pi$
$797$	−27164.0	−1.20728	−0.603638	+	0.797258i	$0.706283\pi$
$798$	0	0
$799$	−6852.47	−0.303408
$800$	−9232.60	−0.408027
$801$	−16017.8	−0.706569
$802$	−15392.7	−0.677723
$803$	590.642	0.0259568
$804$	−17365.6	−0.761736
$805$	0	0
$806$	−2332.41	−0.101930
$807$	27035.2	1.17929
$808$	4259.89	0.185473
$809$	−22955.1	−0.997600	−0.498800	−	0.866717i	$-0.666226\pi$
$809$	−22955.1	−0.997600	−0.498800	+	0.866717i	$0.666226\pi$
$810$	32053.8	1.39044
$811$	20575.6	0.890883	0.445441	−	0.895311i	$-0.353047\pi$
$811$	20575.6	0.890883	0.445441	+	0.895311i	$0.353047\pi$
$812$	0	0
$813$	4010.40	0.173002
$814$	−226.014	−0.00973191
$815$	−52274.5	−2.24674
$816$	3658.04	0.156933
$817$	−17240.4	−0.738268
$818$	4340.11	0.185511
$819$	0	0
$820$	−13193.7	−0.561881
$821$	−10861.2	−0.461703	−0.230851	−	0.972989i	$-0.574151\pi$
$821$	−10861.2	−0.461703	−0.230851	+	0.972989i	$0.574151\pi$
$822$	197.404	0.00837624
$823$	5675.36	0.240377	0.120189	−	0.992751i	$-0.461650\pi$
$823$	5675.36	0.240377	0.120189	+	0.992751i	$0.461650\pi$
$824$	14309.8	0.604983
$825$	−3054.77	−0.128913
$826$	0	0
$827$	−41862.3	−1.76021	−0.880105	−	0.474778i	$-0.842528\pi$
$827$	−41862.3	−1.76021	−0.880105	+	0.474778i	$0.842528\pi$
$828$	2262.80	0.0949731
$829$	−25890.1	−1.08468	−0.542340	−	0.840159i	$-0.682461\pi$
$829$	−25890.1	−1.08468	−0.542340	+	0.840159i	$0.682461\pi$
$830$	17806.7	0.744673
$831$	−44085.4	−1.84032
$832$	−2079.42	−0.0866477
$833$	0	0
$834$	37244.2	1.54635
$835$	75502.2	3.12918
$836$	302.412	0.0125109
$837$	−619.893	−0.0255993
$838$	25543.1	1.05295
$839$	27794.2	1.14370	0.571849	−	0.820359i	$-0.306226\pi$
$839$	27794.2	1.14370	0.571849	+	0.820359i	$0.306226\pi$
$840$	0	0
$841$	−22815.2	−0.935470
$842$	32381.1	1.32533
$843$	−6880.92	−0.281129
$844$	−6239.94	−0.254488
$845$	−23209.3	−0.944881
$846$	−10590.4	−0.430385
$847$	0	0
$848$	−4876.30	−0.197468
$849$	50830.6	2.05477
$850$	18366.5	0.741136
$851$	1763.34	0.0710298
$852$	22095.5	0.888474
$853$	5824.91	0.233812	0.116906	−	0.993143i	$-0.462702\pi$
$853$	5824.91	0.233812	0.116906	+	0.993143i	$0.462702\pi$
$854$	0	0
$855$	25653.5	1.02612
$856$	−8747.71	−0.349288
$857$	−6258.73	−0.249468	−0.124734	−	0.992190i	$-0.539808\pi$
$857$	−6258.73	−0.249468	−0.124734	+	0.992190i	$0.539808\pi$
$858$	−688.012	−0.0273757
$859$	−16699.5	−0.663305	−0.331653	−	0.943402i	$-0.607606\pi$
$859$	−16699.5	−0.663305	−0.331653	+	0.943402i	$0.607606\pi$
$860$	−27340.9	−1.08409
$861$	0	0
$862$	−12642.7	−0.499551
$863$	−30585.2	−1.20641	−0.603206	−	0.797586i	$-0.706110\pi$
$863$	−30585.2	−1.20641	−0.603206	+	0.797586i	$0.706110\pi$
$864$	−552.655	−0.0217612
$865$	29979.6	1.17842
$866$	27643.5	1.08472
$867$	28013.1	1.09732
$868$	0	0
$869$	624.400	0.0243744
$870$	−11589.4	−0.451630
$871$	−19637.4	−0.763936
$872$	4524.08	0.175694
$873$	15020.8	0.582335
$874$	−2359.39	−0.0913129
$875$	0	0
$876$	−11513.1	−0.444056
$877$	−22047.8	−0.848919	−0.424460	−	0.905447i	$-0.639536\pi$
$877$	−22047.8	−0.848919	−0.424460	+	0.905447i	$0.639536\pi$
$878$	10030.1	0.385534
$879$	−15228.3	−0.584342
$880$	479.584	0.0183713
$881$	−11579.5	−0.442820	−0.221410	−	0.975181i	$-0.571066\pi$
$881$	−11579.5	−0.442820	−0.221410	+	0.975181i	$0.571066\pi$
$882$	0	0
$883$	−35835.5	−1.36575	−0.682876	−	0.730534i	$-0.739271\pi$
$883$	−35835.5	−1.36575	−0.682876	+	0.730534i	$0.739271\pi$
$884$	4136.61	0.157386
$885$	−14225.9	−0.540337
$886$	20341.0	0.771299
$887$	−16955.9	−0.641854	−0.320927	−	0.947104i	$-0.603994\pi$
$887$	−16955.9	−0.641854	−0.320927	+	0.947104i	$0.603994\pi$
$888$	4405.59	0.166489
$889$	0	0
$890$	26486.4	0.997557
$891$	−1161.71	−0.0436800
$892$	−1261.60	−0.0473558
$893$	11042.5	0.413798
$894$	−21469.3	−0.803178
$895$	38914.2	1.45336
$896$	0	0
$897$	5367.80	0.199806
$898$	−12650.2	−0.470091
$899$	1423.93	0.0528263
$900$	28385.2	1.05130
$901$	9700.47	0.358679
$902$	478.173	0.0176512
$903$	0	0
$904$	751.046	0.0276321
$905$	18637.4	0.684560
$906$	−5980.48	−0.219303
$907$	17877.4	0.654476	0.327238	−	0.944942i	$-0.393882\pi$
$907$	17877.4	0.654476	0.327238	+	0.944942i	$0.393882\pi$
$908$	7435.46	0.271756
$909$	−13096.8	−0.477882
$910$	0	0
$911$	−17375.4	−0.631914	−0.315957	−	0.948774i	$-0.602325\pi$
$911$	−17375.4	−0.631914	−0.315957	+	0.948774i	$0.602325\pi$
$912$	−5894.78	−0.214030
$913$	−645.361	−0.0233936
$914$	18253.4	0.660580
$915$	134924.	4.87481
$916$	1493.88	0.0538857
$917$	0	0
$918$	1099.40	0.0395269
$919$	42097.2	1.51105	0.755527	−	0.655118i	$-0.227381\pi$
$919$	42097.2	1.51105	0.755527	+	0.655118i	$0.227381\pi$
$920$	−3741.67	−0.134086
$921$	−794.999	−0.0284431
$922$	29188.4	1.04259
$923$	24986.2	0.891040
$924$	0	0
$925$	22119.8	0.786264
$926$	29258.6	1.03833
$927$	−43994.9	−1.55877
$928$	1269.48	0.0449061
$929$	−35375.2	−1.24933	−0.624663	−	0.780894i	$-0.714764\pi$
$929$	−35375.2	−1.24933	−0.624663	+	0.780894i	$0.714764\pi$
$930$	−10485.7	−0.369720
$931$	0	0
$932$	−12697.7	−0.446274
$933$	18544.7	0.650726
$934$	−27723.0	−0.971224
$935$	−954.042	−0.0333695
$936$	6393.08	0.223253
$937$	−51508.1	−1.79583	−0.897917	−	0.440166i	$-0.854920\pi$
$937$	−51508.1	−1.79583	−0.897917	+	0.440166i	$0.854920\pi$
$938$	0	0
$939$	10088.9	0.350629
$940$	17511.9	0.607632
$941$	14499.3	0.502300	0.251150	−	0.967948i	$-0.419191\pi$
$941$	14499.3	0.502300	0.251150	+	0.967948i	$0.419191\pi$
$942$	3134.05	0.108400
$943$	−3730.66	−0.128830
$944$	1558.28	0.0537264
$945$	0	0
$946$	990.908	0.0340562
$947$	46723.4	1.60328	0.801641	−	0.597806i	$-0.203961\pi$
$947$	46723.4	1.60328	0.801641	+	0.597806i	$0.203961\pi$
$948$	−12171.2	−0.416984
$949$	−13019.3	−0.445338
$950$	−29596.8	−1.01079
$951$	−565.312	−0.0192760
$952$	0	0
$953$	43981.8	1.49498	0.747488	−	0.664276i	$-0.231260\pi$
$953$	43981.8	1.49498	0.747488	+	0.664276i	$0.231260\pi$
$954$	14992.0	0.508787
$955$	−96626.8	−3.27410
$956$	−8497.82	−0.287489
$957$	420.031	0.0141877
$958$	−31030.4	−1.04650
$959$	0	0
$960$	−9348.33	−0.314288
$961$	−28502.7	−0.956755
$962$	4981.95	0.166969
$963$	26894.4	0.899960
$964$	22412.2	0.748804
$965$	−82728.3	−2.75971
$966$	0	0
$967$	39862.0	1.32562	0.662810	−	0.748788i	$-0.269364\pi$
$967$	39862.0	1.32562	0.662810	+	0.748788i	$0.269364\pi$
$968$	10630.6	0.352976
$969$	11726.6	0.388763
$970$	−24837.8	−0.822159
$971$	−3719.48	−0.122929	−0.0614643	−	0.998109i	$-0.519577\pi$
$971$	−3719.48	−0.122929	−0.0614643	+	0.998109i	$0.519577\pi$
$972$	20779.6	0.685705
$973$	0	0
$974$	−29989.2	−0.986566
$975$	67335.3	2.21175
$976$	−14779.3	−0.484708
$977$	−49413.7	−1.61810	−0.809050	−	0.587740i	$-0.800018\pi$
$977$	−49413.7	−1.61810	−0.809050	+	0.587740i	$0.800018\pi$
$978$	−36929.9	−1.20745
$979$	−959.937	−0.0313378
$980$	0	0
$981$	−13909.1	−0.452684
$982$	16976.7	0.551679
$983$	−45374.9	−1.47226	−0.736131	−	0.676839i	$-0.763349\pi$
$983$	−45374.9	−1.47226	−0.736131	+	0.676839i	$0.763349\pi$
$984$	−9320.82	−0.301968
$985$	−80862.5	−2.61573
$986$	−2525.40	−0.0815670
$987$	0	0
$988$	−6665.97	−0.214649
$989$	−7730.96	−0.248565
$990$	−1474.46	−0.0473348
$991$	11620.7	0.372496	0.186248	−	0.982503i	$-0.440367\pi$
$991$	11620.7	0.372496	0.186248	+	0.982503i	$0.440367\pi$
$992$	1148.58	0.0367617
$993$	10534.5	0.336658
$994$	0	0
$995$	−20478.5	−0.652475
$996$	12579.7	0.400205
$997$	37513.6	1.19164	0.595821	−	0.803117i	$-0.296827\pi$
$997$	37513.6	1.19164	0.595821	+	0.803117i	$0.296827\pi$
$998$	−3617.80	−0.114749
$999$	1324.07	0.0419337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.z.1.2	✓	14
7.6	odd	2	inner	2254.4.a.z.1.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.z.1.2	✓	14	1.1	even	1	trivial
2254.4.a.z.1.13	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -7.18301 q^{3} +4.00000 q^{4} +20.3352 q^{5} +14.3660 q^{6} -8.00000 q^{8} +24.5957 q^{9} -40.6703 q^{10} +1.47400 q^{11} -28.7320 q^{12} -32.4909 q^{13} -146.068 q^{15} +16.0000 q^{16} -31.8290 q^{17} -49.1913 q^{18} +51.2910 q^{19} +81.3406 q^{20} -2.94800 q^{22} +23.0000 q^{23} +57.4641 q^{24} +288.519 q^{25} +64.9819 q^{26} +17.2705 q^{27} -39.6714 q^{29} +292.135 q^{30} -35.8933 q^{31} -32.0000 q^{32} -10.5878 q^{33} +63.6579 q^{34} +98.3826 q^{36} +76.6668 q^{37} -102.582 q^{38} +233.383 q^{39} -162.681 q^{40} -162.203 q^{41} -336.129 q^{43} +5.89600 q^{44} +500.156 q^{45} -46.0000 q^{46} +215.290 q^{47} -114.928 q^{48} -577.037 q^{50} +228.628 q^{51} -129.964 q^{52} -304.769 q^{53} -34.5409 q^{54} +29.9740 q^{55} -368.424 q^{57} +79.3427 q^{58} +97.3925 q^{59} -584.271 q^{60} -923.709 q^{61} +71.7865 q^{62} +64.0000 q^{64} -660.708 q^{65} +21.1755 q^{66} +604.397 q^{67} -127.316 q^{68} -165.209 q^{69} -769.020 q^{71} -196.765 q^{72} +400.707 q^{73} -153.334 q^{74} -2072.43 q^{75} +205.164 q^{76} -466.765 q^{78} +423.609 q^{79} +325.363 q^{80} -788.137 q^{81} +324.405 q^{82} -437.830 q^{83} -647.247 q^{85} +672.258 q^{86} +284.960 q^{87} -11.7920 q^{88} -651.246 q^{89} -1000.31 q^{90} +92.0000 q^{92} +257.822 q^{93} -430.581 q^{94} +1043.01 q^{95} +229.856 q^{96} +610.711 q^{97} +36.2540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) \) 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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