LMFDB - Embedded newform 560.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.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:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.6053.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 20x - 31 $ x^3 - 20*x - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.88477$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.56288 q^{3} -5.00000 q^{5} -7.00000 q^{7} +46.3230 q^{9} -50.3535 q^{11} -62.9573 q^{13} +42.8144 q^{15} +37.5324 q^{17} -156.341 q^{19} +59.9402 q^{21} -17.3009 q^{23} +25.0000 q^{25} -165.460 q^{27} -91.8017 q^{29} -307.464 q^{31} +431.171 q^{33} +35.0000 q^{35} -287.010 q^{37} +539.096 q^{39} -424.591 q^{41} -8.88215 q^{43} -231.615 q^{45} +137.338 q^{47} +49.0000 q^{49} -321.385 q^{51} +567.325 q^{53} +251.767 q^{55} +1338.73 q^{57} +507.568 q^{59} +222.341 q^{61} -324.261 q^{63} +314.786 q^{65} +943.665 q^{67} +148.146 q^{69} -567.712 q^{71} +478.069 q^{73} -214.072 q^{75} +352.474 q^{77} +697.133 q^{79} +166.098 q^{81} -427.235 q^{83} -187.662 q^{85} +786.087 q^{87} -199.150 q^{89} +440.701 q^{91} +2632.77 q^{93} +781.706 q^{95} -1634.66 q^{97} -2332.52 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 15 q^{5} - 21 q^{7} + 41 q^{9} - 10 q^{11} - 64 q^{13} + 10 q^{15} + 132 q^{17} - 160 q^{19} + 14 q^{21} - 40 q^{23} + 75 q^{25} - 182 q^{27} + 348 q^{29} - 396 q^{31} + 822 q^{33} + 105 q^{35}+ \cdots - 744 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 15 * q^5 - 21 * q^7 + 41 * q^9 - 10 * q^11 - 64 * q^13 + 10 * q^15 + 132 * q^17 - 160 * q^19 + 14 * q^21 - 40 * q^23 + 75 * q^25 - 182 * q^27 + 348 * q^29 - 396 * q^31 + 822 * q^33 + 105 * q^35 - 18 * q^37 + 98 * q^39 + 478 * q^41 + 220 * q^43 - 205 * q^45 + 270 * q^47 + 147 * q^49 + 398 * q^51 + 718 * q^53 + 50 * q^55 + 1584 * q^57 - 480 * q^59 + 358 * q^61 - 287 * q^63 + 320 * q^65 + 2172 * q^67 + 1152 * q^69 - 368 * q^71 + 510 * q^73 - 50 * q^75 + 70 * q^77 + 150 * q^79 + 35 * q^81 - 1056 * q^83 - 660 * q^85 + 2266 * q^87 + 422 * q^89 + 448 * q^91 + 332 * q^93 + 800 * q^95 - 1172 * q^97 - 744 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−8.56288	−1.64793	−0.823964	−	0.566642i	$-0.808242\pi$
$3$	−8.56288	−1.64793	−0.823964	+	0.566642i	$0.808242\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	46.3230	1.71567
$10$	0	0
$11$	−50.3535	−1.38020	−0.690098	−	0.723716i	$-0.742432\pi$
$11$	−50.3535	−1.38020	−0.690098	+	0.723716i	$0.742432\pi$
$12$	0	0
$13$	−62.9573	−1.34317	−0.671585	−	0.740927i	$-0.734386\pi$
$13$	−62.9573	−1.34317	−0.671585	+	0.740927i	$0.734386\pi$
$14$	0	0
$15$	42.8144	0.736976
$16$	0	0
$17$	37.5324	0.535467	0.267733	−	0.963493i	$-0.413725\pi$
$17$	37.5324	0.535467	0.267733	+	0.963493i	$0.413725\pi$
$18$	0	0
$19$	−156.341	−1.88774	−0.943872	−	0.330310i	$-0.892847\pi$
$19$	−156.341	−1.88774	−0.943872	+	0.330310i	$0.892847\pi$
$20$	0	0
$21$	59.9402	0.622858
$22$	0	0
$23$	−17.3009	−0.156848	−0.0784238	−	0.996920i	$-0.524989\pi$
$23$	−17.3009	−0.156848	−0.0784238	+	0.996920i	$0.524989\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−165.460	−1.17937
$28$	0	0
$29$	−91.8017	−0.587832	−0.293916	−	0.955831i	$-0.594959\pi$
$29$	−91.8017	−0.587832	−0.293916	+	0.955831i	$0.594959\pi$
$30$	0	0
$31$	−307.464	−1.78136	−0.890679	−	0.454633i	$-0.849770\pi$
$31$	−307.464	−1.78136	−0.890679	+	0.454633i	$0.849770\pi$
$32$	0	0
$33$	431.171	2.27446
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−287.010	−1.27525	−0.637625	−	0.770347i	$-0.720083\pi$
$37$	−287.010	−1.27525	−0.637625	+	0.770347i	$0.720083\pi$
$38$	0	0
$39$	539.096	2.21345
$40$	0	0
$41$	−424.591	−1.61731	−0.808657	−	0.588280i	$-0.799805\pi$
$41$	−424.591	−1.61731	−0.808657	+	0.588280i	$0.799805\pi$
$42$	0	0
$43$	−8.88215	−0.0315003	−0.0157502	−	0.999876i	$-0.505014\pi$
$43$	−8.88215	−0.0315003	−0.0157502	+	0.999876i	$0.505014\pi$
$44$	0	0
$45$	−231.615	−0.767269
$46$	0	0
$47$	137.338	0.426231	0.213115	−	0.977027i	$-0.431639\pi$
$47$	137.338	0.426231	0.213115	+	0.977027i	$0.431639\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−321.385	−0.882411
$52$	0	0
$53$	567.325	1.47034	0.735171	−	0.677882i	$-0.237102\pi$
$53$	567.325	1.47034	0.735171	+	0.677882i	$0.237102\pi$
$54$	0	0
$55$	251.767	0.617242
$56$	0	0
$57$	1338.73	3.11087
$58$	0	0
$59$	507.568	1.12000	0.559998	−	0.828494i	$-0.310802\pi$
$59$	507.568	1.12000	0.559998	+	0.828494i	$0.310802\pi$
$60$	0	0
$61$	222.341	0.466687	0.233343	−	0.972394i	$-0.425033\pi$
$61$	222.341	0.466687	0.233343	+	0.972394i	$0.425033\pi$
$62$	0	0
$63$	−324.261	−0.648461
$64$	0	0
$65$	314.786	0.600684
$66$	0	0
$67$	943.665	1.72070	0.860351	−	0.509701i	$-0.170244\pi$
$67$	943.665	1.72070	0.860351	+	0.509701i	$0.170244\pi$
$68$	0	0
$69$	148.146	0.258474
$70$	0	0
$71$	−567.712	−0.948944	−0.474472	−	0.880271i	$-0.657361\pi$
$71$	−567.712	−0.948944	−0.474472	+	0.880271i	$0.657361\pi$
$72$	0	0
$73$	478.069	0.766490	0.383245	−	0.923647i	$-0.374806\pi$
$73$	478.069	0.766490	0.383245	+	0.923647i	$0.374806\pi$
$74$	0	0
$75$	−214.072	−0.329586
$76$	0	0
$77$	352.474	0.521665
$78$	0	0
$79$	697.133	0.992830	0.496415	−	0.868085i	$-0.334649\pi$
$79$	697.133	0.992830	0.496415	+	0.868085i	$0.334649\pi$
$80$	0	0
$81$	166.098	0.227843
$82$	0	0
$83$	−427.235	−0.565001	−0.282501	−	0.959267i	$-0.591164\pi$
$83$	−427.235	−0.565001	−0.282501	+	0.959267i	$0.591164\pi$
$84$	0	0
$85$	−187.662	−0.239468
$86$	0	0
$87$	786.087	0.968705
$88$	0	0
$89$	−199.150	−0.237189	−0.118595	−	0.992943i	$-0.537839\pi$
$89$	−199.150	−0.237189	−0.118595	+	0.992943i	$0.537839\pi$
$90$	0	0
$91$	440.701	0.507671
$92$	0	0
$93$	2632.77	2.93555
$94$	0	0
$95$	781.706	0.844225
$96$	0	0
$97$	−1634.66	−1.71108	−0.855538	−	0.517740i	$-0.826774\pi$
$97$	−1634.66	−1.71108	−0.855538	+	0.517740i	$0.826774\pi$
$98$	0	0
$99$	−2332.52	−2.36795
$100$	0	0
$101$	552.725	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$101$	552.725	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$102$	0	0
$103$	866.615	0.829031	0.414515	−	0.910042i	$-0.363951\pi$
$103$	866.615	0.829031	0.414515	+	0.910042i	$0.363951\pi$
$104$	0	0
$105$	−299.701	−0.278551
$106$	0	0
$107$	−524.040	−0.473466	−0.236733	−	0.971575i	$-0.576077\pi$
$107$	−524.040	−0.473466	−0.236733	+	0.971575i	$0.576077\pi$
$108$	0	0
$109$	−943.667	−0.829238	−0.414619	−	0.909995i	$-0.636085\pi$
$109$	−943.667	−0.829238	−0.414619	+	0.909995i	$0.636085\pi$
$110$	0	0
$111$	2457.64	2.10152
$112$	0	0
$113$	−438.555	−0.365096	−0.182548	−	0.983197i	$-0.558434\pi$
$113$	−438.555	−0.365096	−0.182548	+	0.983197i	$0.558434\pi$
$114$	0	0
$115$	86.5047	0.0701444
$116$	0	0
$117$	−2916.37	−2.30443
$118$	0	0
$119$	−262.727	−0.202387
$120$	0	0
$121$	1204.47	0.904939
$122$	0	0
$123$	3635.72	2.66522
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1650.14	1.15296	0.576482	−	0.817110i	$-0.304425\pi$
$127$	1650.14	1.15296	0.576482	+	0.817110i	$0.304425\pi$
$128$	0	0
$129$	76.0568	0.0519103
$130$	0	0
$131$	−1798.09	−1.19924	−0.599619	−	0.800286i	$-0.704681\pi$
$131$	−1798.09	−1.19924	−0.599619	+	0.800286i	$0.704681\pi$
$132$	0	0
$133$	1094.39	0.713501
$134$	0	0
$135$	827.302	0.527428
$136$	0	0
$137$	−443.140	−0.276350	−0.138175	−	0.990408i	$-0.544124\pi$
$137$	−443.140	−0.276350	−0.138175	+	0.990408i	$0.544124\pi$
$138$	0	0
$139$	1282.10	0.782347	0.391174	−	0.920317i	$-0.372069\pi$
$139$	1282.10	0.782347	0.391174	+	0.920317i	$0.372069\pi$
$140$	0	0
$141$	−1176.01	−0.702398
$142$	0	0
$143$	3170.12	1.85384
$144$	0	0
$145$	459.008	0.262887
$146$	0	0
$147$	−419.581	−0.235418
$148$	0	0
$149$	−1777.65	−0.977387	−0.488693	−	0.872456i	$-0.662526\pi$
$149$	−1777.65	−0.977387	−0.488693	+	0.872456i	$0.662526\pi$
$150$	0	0
$151$	1835.91	0.989433	0.494717	−	0.869054i	$-0.335272\pi$
$151$	1835.91	0.989433	0.494717	+	0.869054i	$0.335272\pi$
$152$	0	0
$153$	1738.61	0.918682
$154$	0	0
$155$	1537.32	0.796647
$156$	0	0
$157$	−1813.06	−0.921641	−0.460820	−	0.887493i	$-0.652445\pi$
$157$	−1813.06	−0.921641	−0.460820	+	0.887493i	$0.652445\pi$
$158$	0	0
$159$	−4857.94	−2.42302
$160$	0	0
$161$	121.107	0.0592828
$162$	0	0
$163$	−785.406	−0.377409	−0.188705	−	0.982034i	$-0.560429\pi$
$163$	−785.406	−0.377409	−0.188705	+	0.982034i	$0.560429\pi$
$164$	0	0
$165$	−2155.86	−1.01717
$166$	0	0
$167$	−2313.08	−1.07181	−0.535904	−	0.844279i	$-0.680029\pi$
$167$	−2313.08	−1.07181	−0.535904	+	0.844279i	$0.680029\pi$
$168$	0	0
$169$	1766.62	0.804106
$170$	0	0
$171$	−7242.19	−3.23874
$172$	0	0
$173$	−3527.06	−1.55004	−0.775021	−	0.631936i	$-0.782261\pi$
$173$	−3527.06	−1.55004	−0.775021	+	0.631936i	$0.782261\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−4346.25	−1.84567
$178$	0	0
$179$	−1776.08	−0.741621	−0.370811	−	0.928709i	$-0.620920\pi$
$179$	−1776.08	−0.741621	−0.370811	+	0.928709i	$0.620920\pi$
$180$	0	0
$181$	−700.812	−0.287795	−0.143898	−	0.989593i	$-0.545964\pi$
$181$	−700.812	−0.287795	−0.143898	+	0.989593i	$0.545964\pi$
$182$	0	0
$183$	−1903.88	−0.769066
$184$	0	0
$185$	1435.05	0.570309
$186$	0	0
$187$	−1889.89	−0.739049
$188$	0	0
$189$	1158.22	0.445758
$190$	0	0
$191$	3472.93	1.31567	0.657834	−	0.753163i	$-0.271473\pi$
$191$	3472.93	1.31567	0.657834	+	0.753163i	$0.271473\pi$
$192$	0	0
$193$	1023.95	0.381896	0.190948	−	0.981600i	$-0.438844\pi$
$193$	1023.95	0.381896	0.190948	+	0.981600i	$0.438844\pi$
$194$	0	0
$195$	−2695.48	−0.989884
$196$	0	0
$197$	1781.28	0.644219	0.322110	−	0.946702i	$-0.395608\pi$
$197$	1781.28	0.644219	0.322110	+	0.946702i	$0.395608\pi$
$198$	0	0
$199$	−3764.21	−1.34089	−0.670446	−	0.741958i	$-0.733897\pi$
$199$	−3764.21	−1.34089	−0.670446	+	0.741958i	$0.733897\pi$
$200$	0	0
$201$	−8080.50	−2.83559
$202$	0	0
$203$	642.612	0.222180
$204$	0	0
$205$	2122.95	0.723285
$206$	0	0
$207$	−801.431	−0.269098
$208$	0	0
$209$	7872.33	2.60546
$210$	0	0
$211$	3052.77	0.996025	0.498012	−	0.867170i	$-0.334063\pi$
$211$	3052.77	0.996025	0.498012	+	0.867170i	$0.334063\pi$
$212$	0	0
$213$	4861.25	1.56379
$214$	0	0
$215$	44.4107	0.0140874
$216$	0	0
$217$	2152.24	0.673290
$218$	0	0
$219$	−4093.65	−1.26312
$220$	0	0
$221$	−2362.94	−0.719223
$222$	0	0
$223$	−34.2588	−0.0102876	−0.00514382	−	0.999987i	$-0.501637\pi$
$223$	−34.2588	−0.0102876	−0.00514382	+	0.999987i	$0.501637\pi$
$224$	0	0
$225$	1158.07	0.343133
$226$	0	0
$227$	5176.61	1.51359	0.756793	−	0.653655i	$-0.226765\pi$
$227$	5176.61	1.51359	0.756793	+	0.653655i	$0.226765\pi$
$228$	0	0
$229$	−1900.70	−0.548479	−0.274239	−	0.961661i	$-0.588426\pi$
$229$	−1900.70	−0.548479	−0.274239	+	0.961661i	$0.588426\pi$
$230$	0	0
$231$	−3018.20	−0.859666
$232$	0	0
$233$	2386.78	0.671087	0.335544	−	0.942025i	$-0.391080\pi$
$233$	2386.78	0.671087	0.335544	+	0.942025i	$0.391080\pi$
$234$	0	0
$235$	−686.692	−0.190616
$236$	0	0
$237$	−5969.47	−1.63611
$238$	0	0
$239$	−4000.94	−1.08284	−0.541421	−	0.840752i	$-0.682113\pi$
$239$	−4000.94	−1.08284	−0.541421	+	0.840752i	$0.682113\pi$
$240$	0	0
$241$	5578.43	1.49103	0.745516	−	0.666488i	$-0.232203\pi$
$241$	5578.43	1.49103	0.745516	+	0.666488i	$0.232203\pi$
$242$	0	0
$243$	3045.16	0.803896
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	9842.83	2.53556
$248$	0	0
$249$	3658.36	0.931081
$250$	0	0
$251$	−6094.25	−1.53253	−0.766267	−	0.642523i	$-0.777888\pi$
$251$	−6094.25	−1.53253	−0.766267	+	0.642523i	$0.777888\pi$
$252$	0	0
$253$	871.163	0.216480
$254$	0	0
$255$	1606.93	0.394626
$256$	0	0
$257$	−935.737	−0.227119	−0.113560	−	0.993531i	$-0.536225\pi$
$257$	−935.737	−0.227119	−0.113560	+	0.993531i	$0.536225\pi$
$258$	0	0
$259$	2009.07	0.481999
$260$	0	0
$261$	−4252.53	−1.00852
$262$	0	0
$263$	−2449.50	−0.574305	−0.287153	−	0.957885i	$-0.592709\pi$
$263$	−2449.50	−0.574305	−0.287153	+	0.957885i	$0.592709\pi$
$264$	0	0
$265$	−2836.62	−0.657557
$266$	0	0
$267$	1705.30	0.390871
$268$	0	0
$269$	3040.59	0.689174	0.344587	−	0.938754i	$-0.388019\pi$
$269$	3040.59	0.689174	0.344587	+	0.938754i	$0.388019\pi$
$270$	0	0
$271$	−6226.83	−1.39577	−0.697884	−	0.716211i	$-0.745875\pi$
$271$	−6226.83	−1.39577	−0.697884	+	0.716211i	$0.745875\pi$
$272$	0	0
$273$	−3773.67	−0.836604
$274$	0	0
$275$	−1258.84	−0.276039
$276$	0	0
$277$	−2251.87	−0.488455	−0.244227	−	0.969718i	$-0.578534\pi$
$277$	−2251.87	−0.488455	−0.244227	+	0.969718i	$0.578534\pi$
$278$	0	0
$279$	−14242.6	−3.05621
$280$	0	0
$281$	4923.65	1.04527	0.522634	−	0.852557i	$-0.324950\pi$
$281$	4923.65	1.04527	0.522634	+	0.852557i	$0.324950\pi$
$282$	0	0
$283$	7678.49	1.61286	0.806429	−	0.591331i	$-0.201397\pi$
$283$	7678.49	1.61286	0.806429	+	0.591331i	$0.201397\pi$
$284$	0	0
$285$	−6693.66	−1.39122
$286$	0	0
$287$	2972.13	0.611288
$288$	0	0
$289$	−3504.32	−0.713275
$290$	0	0
$291$	13997.4	2.81973
$292$	0	0
$293$	6707.41	1.33738	0.668688	−	0.743544i	$-0.266856\pi$
$293$	6707.41	1.33738	0.668688	+	0.743544i	$0.266856\pi$
$294$	0	0
$295$	−2537.84	−0.500877
$296$	0	0
$297$	8331.51	1.62775
$298$	0	0
$299$	1089.22	0.210673
$300$	0	0
$301$	62.1750	0.0119060
$302$	0	0
$303$	−4732.92	−0.897357
$304$	0	0
$305$	−1111.71	−0.208709
$306$	0	0
$307$	−9563.80	−1.77796	−0.888982	−	0.457942i	$-0.848587\pi$
$307$	−9563.80	−1.77796	−0.888982	+	0.457942i	$0.848587\pi$
$308$	0	0
$309$	−7420.73	−1.36618
$310$	0	0
$311$	−8143.75	−1.48486	−0.742428	−	0.669926i	$-0.766326\pi$
$311$	−8143.75	−1.48486	−0.742428	+	0.669926i	$0.766326\pi$
$312$	0	0
$313$	−3482.85	−0.628954	−0.314477	−	0.949265i	$-0.601829\pi$
$313$	−3482.85	−0.628954	−0.314477	+	0.949265i	$0.601829\pi$
$314$	0	0
$315$	1621.30	0.290000
$316$	0	0
$317$	68.3992	0.0121189	0.00605943	−	0.999982i	$-0.498071\pi$
$317$	68.3992	0.0121189	0.00605943	+	0.999982i	$0.498071\pi$
$318$	0	0
$319$	4622.53	0.811324
$320$	0	0
$321$	4487.29	0.780237
$322$	0	0
$323$	−5867.86	−1.01082
$324$	0	0
$325$	−1573.93	−0.268634
$326$	0	0
$327$	8080.51	1.36652
$328$	0	0
$329$	−961.368	−0.161100
$330$	0	0
$331$	9215.53	1.53031	0.765153	−	0.643848i	$-0.222663\pi$
$331$	9215.53	1.53031	0.765153	+	0.643848i	$0.222663\pi$
$332$	0	0
$333$	−13295.2	−2.18790
$334$	0	0
$335$	−4718.33	−0.769522
$336$	0	0
$337$	−4931.89	−0.797203	−0.398601	−	0.917124i	$-0.630504\pi$
$337$	−4931.89	−0.797203	−0.398601	+	0.917124i	$0.630504\pi$
$338$	0	0
$339$	3755.30	0.601652
$340$	0	0
$341$	15481.9	2.45862
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−740.730	−0.115593
$346$	0	0
$347$	−13.7153	−0.00212183	−0.00106091	−	0.999999i	$-0.500338\pi$
$347$	−13.7153	−0.00212183	−0.00106091	+	0.999999i	$0.500338\pi$
$348$	0	0
$349$	−5025.11	−0.770739	−0.385370	−	0.922762i	$-0.625926\pi$
$349$	−5025.11	−0.770739	−0.385370	+	0.922762i	$0.625926\pi$
$350$	0	0
$351$	10416.9	1.58409
$352$	0	0
$353$	1896.14	0.285896	0.142948	−	0.989730i	$-0.454342\pi$
$353$	1896.14	0.285896	0.142948	+	0.989730i	$0.454342\pi$
$354$	0	0
$355$	2838.56	0.424381
$356$	0	0
$357$	2249.70	0.333520
$358$	0	0
$359$	−7546.30	−1.10941	−0.554706	−	0.832047i	$-0.687169\pi$
$359$	−7546.30	−1.10941	−0.554706	+	0.832047i	$0.687169\pi$
$360$	0	0
$361$	17583.6	2.56358
$362$	0	0
$363$	−10313.8	−1.49127
$364$	0	0
$365$	−2390.35	−0.342785
$366$	0	0
$367$	−11910.9	−1.69413	−0.847065	−	0.531490i	$-0.821632\pi$
$367$	−11910.9	−1.69413	−0.847065	+	0.531490i	$0.821632\pi$
$368$	0	0
$369$	−19668.3	−2.77477
$370$	0	0
$371$	−3971.27	−0.555737
$372$	0	0
$373$	7194.68	0.998731	0.499365	−	0.866392i	$-0.333567\pi$
$373$	7194.68	0.998731	0.499365	+	0.866392i	$0.333567\pi$
$374$	0	0
$375$	1070.36	0.147395
$376$	0	0
$377$	5779.58	0.789559
$378$	0	0
$379$	4505.98	0.610704	0.305352	−	0.952240i	$-0.401226\pi$
$379$	4505.98	0.610704	0.305352	+	0.952240i	$0.401226\pi$
$380$	0	0
$381$	−14130.0	−1.90000
$382$	0	0
$383$	−3908.84	−0.521495	−0.260748	−	0.965407i	$-0.583969\pi$
$383$	−3908.84	−0.521495	−0.260748	+	0.965407i	$0.583969\pi$
$384$	0	0
$385$	−1762.37	−0.233296
$386$	0	0
$387$	−411.448	−0.0540441
$388$	0	0
$389$	−3693.58	−0.481419	−0.240709	−	0.970597i	$-0.577380\pi$
$389$	−3693.58	−0.481419	−0.240709	+	0.970597i	$0.577380\pi$
$390$	0	0
$391$	−649.345	−0.0839867
$392$	0	0
$393$	15396.9	1.97626
$394$	0	0
$395$	−3485.66	−0.444007
$396$	0	0
$397$	4763.16	0.602157	0.301079	−	0.953599i	$-0.402653\pi$
$397$	4763.16	0.602157	0.301079	+	0.953599i	$0.402653\pi$
$398$	0	0
$399$	−9371.13	−1.17580
$400$	0	0
$401$	−9889.66	−1.23159	−0.615793	−	0.787908i	$-0.711164\pi$
$401$	−9889.66	−1.23159	−0.615793	+	0.787908i	$0.711164\pi$
$402$	0	0
$403$	19357.1	2.39267
$404$	0	0
$405$	−830.488	−0.101894
$406$	0	0
$407$	14452.0	1.76009
$408$	0	0
$409$	12451.6	1.50536	0.752678	−	0.658389i	$-0.228762\pi$
$409$	12451.6	1.50536	0.752678	+	0.658389i	$0.228762\pi$
$410$	0	0
$411$	3794.55	0.455405
$412$	0	0
$413$	−3552.98	−0.423318
$414$	0	0
$415$	2136.17	0.252676
$416$	0	0
$417$	−10978.5	−1.28925
$418$	0	0
$419$	12910.7	1.50532	0.752662	−	0.658407i	$-0.228769\pi$
$419$	12910.7	1.50532	0.752662	+	0.658407i	$0.228769\pi$
$420$	0	0
$421$	3422.51	0.396207	0.198103	−	0.980181i	$-0.436522\pi$
$421$	3422.51	0.396207	0.198103	+	0.980181i	$0.436522\pi$
$422$	0	0
$423$	6361.92	0.731270
$424$	0	0
$425$	938.309	0.107093
$426$	0	0
$427$	−1556.39	−0.176391
$428$	0	0
$429$	−27145.4	−3.05499
$430$	0	0
$431$	−4930.33	−0.551010	−0.275505	−	0.961300i	$-0.588845\pi$
$431$	−4930.33	−0.551010	−0.275505	+	0.961300i	$0.588845\pi$
$432$	0	0
$433$	5890.88	0.653805	0.326902	−	0.945058i	$-0.393995\pi$
$433$	5890.88	0.653805	0.326902	+	0.945058i	$0.393995\pi$
$434$	0	0
$435$	−3930.43	−0.433218
$436$	0	0
$437$	2704.85	0.296088
$438$	0	0
$439$	2632.66	0.286218	0.143109	−	0.989707i	$-0.454290\pi$
$439$	2632.66	0.286218	0.143109	+	0.989707i	$0.454290\pi$
$440$	0	0
$441$	2269.83	0.245095
$442$	0	0
$443$	−8701.76	−0.933258	−0.466629	−	0.884453i	$-0.654532\pi$
$443$	−8701.76	−0.933258	−0.466629	+	0.884453i	$0.654532\pi$
$444$	0	0
$445$	995.749	0.106074
$446$	0	0
$447$	15221.8	1.61066
$448$	0	0
$449$	−9393.15	−0.987283	−0.493641	−	0.869666i	$-0.664334\pi$
$449$	−9393.15	−0.987283	−0.493641	+	0.869666i	$0.664334\pi$
$450$	0	0
$451$	21379.6	2.23221
$452$	0	0
$453$	−15720.7	−1.63051
$454$	0	0
$455$	−2203.51	−0.227037
$456$	0	0
$457$	−8489.35	−0.868961	−0.434480	−	0.900681i	$-0.643068\pi$
$457$	−8489.35	−0.868961	−0.434480	+	0.900681i	$0.643068\pi$
$458$	0	0
$459$	−6210.12	−0.631511
$460$	0	0
$461$	−17481.3	−1.76612	−0.883062	−	0.469256i	$-0.844522\pi$
$461$	−17481.3	−1.76612	−0.883062	+	0.469256i	$0.844522\pi$
$462$	0	0
$463$	−9318.16	−0.935317	−0.467658	−	0.883909i	$-0.654902\pi$
$463$	−9318.16	−0.935317	−0.467658	+	0.883909i	$0.654902\pi$
$464$	0	0
$465$	−13163.9	−1.31282
$466$	0	0
$467$	−14620.7	−1.44875	−0.724373	−	0.689408i	$-0.757871\pi$
$467$	−14620.7	−1.44875	−0.724373	+	0.689408i	$0.757871\pi$
$468$	0	0
$469$	−6605.66	−0.650365
$470$	0	0
$471$	15525.0	1.51880
$472$	0	0
$473$	447.247	0.0434766
$474$	0	0
$475$	−3908.53	−0.377549
$476$	0	0
$477$	26280.2	2.52261
$478$	0	0
$479$	−6775.20	−0.646277	−0.323139	−	0.946352i	$-0.604738\pi$
$479$	−6775.20	−0.646277	−0.323139	+	0.946352i	$0.604738\pi$
$480$	0	0
$481$	18069.4	1.71288
$482$	0	0
$483$	−1037.02	−0.0976938
$484$	0	0
$485$	8173.29	0.765217
$486$	0	0
$487$	12188.5	1.13411	0.567057	−	0.823678i	$-0.308082\pi$
$487$	12188.5	1.13411	0.567057	+	0.823678i	$0.308082\pi$
$488$	0	0
$489$	6725.34	0.621944
$490$	0	0
$491$	−6590.32	−0.605737	−0.302869	−	0.953032i	$-0.597944\pi$
$491$	−6590.32	−0.605737	−0.302869	+	0.953032i	$0.597944\pi$
$492$	0	0
$493$	−3445.53	−0.314765
$494$	0	0
$495$	11662.6	1.05898
$496$	0	0
$497$	3973.98	0.358667
$498$	0	0
$499$	8661.16	0.777008	0.388504	−	0.921447i	$-0.372992\pi$
$499$	8661.16	0.777008	0.388504	+	0.921447i	$0.372992\pi$
$500$	0	0
$501$	19806.7	1.76626
$502$	0	0
$503$	2756.60	0.244355	0.122178	−	0.992508i	$-0.461012\pi$
$503$	2756.60	0.244355	0.122178	+	0.992508i	$0.461012\pi$
$504$	0	0
$505$	−2763.63	−0.243524
$506$	0	0
$507$	−15127.4	−1.32511
$508$	0	0
$509$	5847.93	0.509244	0.254622	−	0.967041i	$-0.418049\pi$
$509$	5847.93	0.509244	0.254622	+	0.967041i	$0.418049\pi$
$510$	0	0
$511$	−3346.49	−0.289706
$512$	0	0
$513$	25868.3	2.22634
$514$	0	0
$515$	−4333.08	−0.370754
$516$	0	0
$517$	−6915.46	−0.588282
$518$	0	0
$519$	30201.8	2.55436
$520$	0	0
$521$	22727.6	1.91116	0.955580	−	0.294732i	$-0.0952302\pi$
$521$	22727.6	1.91116	0.955580	+	0.294732i	$0.0952302\pi$
$522$	0	0
$523$	−339.387	−0.0283754	−0.0141877	−	0.999899i	$-0.504516\pi$
$523$	−339.387	−0.0283754	−0.0141877	+	0.999899i	$0.504516\pi$
$524$	0	0
$525$	1498.50	0.124572
$526$	0	0
$527$	−11539.8	−0.953858
$528$	0	0
$529$	−11867.7	−0.975399
$530$	0	0
$531$	23512.1	1.92154
$532$	0	0
$533$	26731.1	2.17233
$534$	0	0
$535$	2620.20	0.211740
$536$	0	0
$537$	15208.3	1.22214
$538$	0	0
$539$	−2467.32	−0.197171
$540$	0	0
$541$	−1807.30	−0.143626	−0.0718132	−	0.997418i	$-0.522879\pi$
$541$	−1807.30	−0.143626	−0.0718132	+	0.997418i	$0.522879\pi$
$542$	0	0
$543$	6000.97	0.474266
$544$	0	0
$545$	4718.34	0.370846
$546$	0	0
$547$	7047.70	0.550892	0.275446	−	0.961317i	$-0.411174\pi$
$547$	7047.70	0.550892	0.275446	+	0.961317i	$0.411174\pi$
$548$	0	0
$549$	10299.5	0.800678
$550$	0	0
$551$	14352.4	1.10968
$552$	0	0
$553$	−4879.93	−0.375254
$554$	0	0
$555$	−12288.2	−0.939828
$556$	0	0
$557$	−17815.6	−1.35524	−0.677622	−	0.735410i	$-0.736990\pi$
$557$	−17815.6	−1.35524	−0.677622	+	0.735410i	$0.736990\pi$
$558$	0	0
$559$	559.196	0.0423103
$560$	0	0
$561$	16182.9	1.21790
$562$	0	0
$563$	13002.1	0.973307	0.486654	−	0.873595i	$-0.338217\pi$
$563$	13002.1	0.973307	0.486654	+	0.873595i	$0.338217\pi$
$564$	0	0
$565$	2192.78	0.163276
$566$	0	0
$567$	−1162.68	−0.0861166
$568$	0	0
$569$	4414.05	0.325213	0.162607	−	0.986691i	$-0.448010\pi$
$569$	4414.05	0.325213	0.162607	+	0.986691i	$0.448010\pi$
$570$	0	0
$571$	−7957.50	−0.583206	−0.291603	−	0.956539i	$-0.594189\pi$
$571$	−7957.50	−0.583206	−0.291603	+	0.956539i	$0.594189\pi$
$572$	0	0
$573$	−29738.3	−2.16813
$574$	0	0
$575$	−432.524	−0.0313695
$576$	0	0
$577$	17205.7	1.24139	0.620697	−	0.784051i	$-0.286850\pi$
$577$	17205.7	1.24139	0.620697	+	0.784051i	$0.286850\pi$
$578$	0	0
$579$	−8768.00	−0.629336
$580$	0	0
$581$	2990.64	0.213550
$582$	0	0
$583$	−28566.8	−2.02936
$584$	0	0
$585$	14581.8	1.03057
$586$	0	0
$587$	2045.50	0.143828	0.0719138	−	0.997411i	$-0.477089\pi$
$587$	2045.50	0.143828	0.0719138	+	0.997411i	$0.477089\pi$
$588$	0	0
$589$	48069.2	3.36275
$590$	0	0
$591$	−15252.9	−1.06163
$592$	0	0
$593$	10844.0	0.750943	0.375471	−	0.926834i	$-0.377481\pi$
$593$	10844.0	0.750943	0.375471	+	0.926834i	$0.377481\pi$
$594$	0	0
$595$	1313.63	0.0905104
$596$	0	0
$597$	32232.5	2.20969
$598$	0	0
$599$	−3607.63	−0.246083	−0.123042	−	0.992402i	$-0.539265\pi$
$599$	−3607.63	−0.246083	−0.123042	+	0.992402i	$0.539265\pi$
$600$	0	0
$601$	1287.61	0.0873920	0.0436960	−	0.999045i	$-0.486087\pi$
$601$	1287.61	0.0873920	0.0436960	+	0.999045i	$0.486087\pi$
$602$	0	0
$603$	43713.4	2.95215
$604$	0	0
$605$	−6022.37	−0.404701
$606$	0	0
$607$	−7342.44	−0.490973	−0.245486	−	0.969400i	$-0.578948\pi$
$607$	−7342.44	−0.490973	−0.245486	+	0.969400i	$0.578948\pi$
$608$	0	0
$609$	−5502.61	−0.366136
$610$	0	0
$611$	−8646.45	−0.572501
$612$	0	0
$613$	−23046.3	−1.51848	−0.759242	−	0.650809i	$-0.774430\pi$
$613$	−23046.3	−1.51848	−0.759242	+	0.650809i	$0.774430\pi$
$614$	0	0
$615$	−18178.6	−1.19192
$616$	0	0
$617$	17708.3	1.15544	0.577721	−	0.816234i	$-0.303942\pi$
$617$	17708.3	1.15544	0.577721	+	0.816234i	$0.303942\pi$
$618$	0	0
$619$	677.370	0.0439835	0.0219918	−	0.999758i	$-0.492999\pi$
$619$	677.370	0.0439835	0.0219918	+	0.999758i	$0.492999\pi$
$620$	0	0
$621$	2862.62	0.184981
$622$	0	0
$623$	1394.05	0.0896491
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−67409.8	−4.29360
$628$	0	0
$629$	−10772.2	−0.682854
$630$	0	0
$631$	−12453.9	−0.785706	−0.392853	−	0.919601i	$-0.628512\pi$
$631$	−12453.9	−0.785706	−0.392853	+	0.919601i	$0.628512\pi$
$632$	0	0
$633$	−26140.5	−1.64138
$634$	0	0
$635$	−8250.71	−0.515621
$636$	0	0
$637$	−3084.91	−0.191881
$638$	0	0
$639$	−26298.1	−1.62807
$640$	0	0
$641$	−23183.4	−1.42853	−0.714265	−	0.699876i	$-0.753239\pi$
$641$	−23183.4	−1.42853	−0.714265	+	0.699876i	$0.753239\pi$
$642$	0	0
$643$	20218.9	1.24006	0.620028	−	0.784579i	$-0.287121\pi$
$643$	20218.9	1.24006	0.620028	+	0.784579i	$0.287121\pi$
$644$	0	0
$645$	−380.284	−0.0232150
$646$	0	0
$647$	25370.2	1.54158	0.770792	−	0.637087i	$-0.219861\pi$
$647$	25370.2	1.54158	0.770792	+	0.637087i	$0.219861\pi$
$648$	0	0
$649$	−25557.8	−1.54581
$650$	0	0
$651$	−18429.4	−1.10953
$652$	0	0
$653$	875.999	0.0524969	0.0262485	−	0.999655i	$-0.491644\pi$
$653$	875.999	0.0524969	0.0262485	+	0.999655i	$0.491644\pi$
$654$	0	0
$655$	8990.46	0.536315
$656$	0	0
$657$	22145.6	1.31504
$658$	0	0
$659$	7407.23	0.437853	0.218926	−	0.975741i	$-0.429745\pi$
$659$	7407.23	0.437853	0.218926	+	0.975741i	$0.429745\pi$
$660$	0	0
$661$	−11390.1	−0.670232	−0.335116	−	0.942177i	$-0.608776\pi$
$661$	−11390.1	−0.670232	−0.335116	+	0.942177i	$0.608776\pi$
$662$	0	0
$663$	20233.5	1.18523
$664$	0	0
$665$	−5471.95	−0.319087
$666$	0	0
$667$	1588.26	0.0922001
$668$	0	0
$669$	293.354	0.0169533
$670$	0	0
$671$	−11195.7	−0.644119
$672$	0	0
$673$	−30992.3	−1.77514	−0.887568	−	0.460676i	$-0.847607\pi$
$673$	−30992.3	−1.77514	−0.887568	+	0.460676i	$0.847607\pi$
$674$	0	0
$675$	−4136.51	−0.235873
$676$	0	0
$677$	−14659.2	−0.832201	−0.416101	−	0.909319i	$-0.636604\pi$
$677$	−14659.2	−0.832201	−0.416101	+	0.909319i	$0.636604\pi$
$678$	0	0
$679$	11442.6	0.646726
$680$	0	0
$681$	−44326.8	−2.49428
$682$	0	0
$683$	7693.91	0.431039	0.215519	−	0.976500i	$-0.430856\pi$
$683$	7693.91	0.431039	0.215519	+	0.976500i	$0.430856\pi$
$684$	0	0
$685$	2215.70	0.123588
$686$	0	0
$687$	16275.5	0.903853
$688$	0	0
$689$	−35717.2	−1.97492
$690$	0	0
$691$	1483.17	0.0816534	0.0408267	−	0.999166i	$-0.487001\pi$
$691$	1483.17	0.0816534	0.0408267	+	0.999166i	$0.487001\pi$
$692$	0	0
$693$	16327.7	0.895002
$694$	0	0
$695$	−6410.50	−0.349876
$696$	0	0
$697$	−15935.9	−0.866018
$698$	0	0
$699$	−20437.7	−1.10590
$700$	0	0
$701$	−6316.56	−0.340332	−0.170166	−	0.985415i	$-0.554430\pi$
$701$	−6316.56	−0.340332	−0.170166	+	0.985415i	$0.554430\pi$
$702$	0	0
$703$	44871.6	2.40735
$704$	0	0
$705$	5880.06	0.314122
$706$	0	0
$707$	−3869.08	−0.205815
$708$	0	0
$709$	15141.3	0.802035	0.401017	−	0.916071i	$-0.368657\pi$
$709$	15141.3	0.802035	0.401017	+	0.916071i	$0.368657\pi$
$710$	0	0
$711$	32293.3	1.70336
$712$	0	0
$713$	5319.41	0.279402
$714$	0	0
$715$	−15850.6	−0.829061
$716$	0	0
$717$	34259.6	1.78444
$718$	0	0
$719$	−27594.2	−1.43128	−0.715639	−	0.698471i	$-0.753864\pi$
$719$	−27594.2	−1.43128	−0.715639	+	0.698471i	$0.753864\pi$
$720$	0	0
$721$	−6066.31	−0.313344
$722$	0	0
$723$	−47767.5	−2.45711
$724$	0	0
$725$	−2295.04	−0.117566
$726$	0	0
$727$	37646.1	1.92052	0.960260	−	0.279107i	$-0.0900384\pi$
$727$	37646.1	1.92052	0.960260	+	0.279107i	$0.0900384\pi$
$728$	0	0
$729$	−30559.9	−1.55261
$730$	0	0
$731$	−333.368	−0.0168674
$732$	0	0
$733$	16306.3	0.821676	0.410838	−	0.911708i	$-0.365236\pi$
$733$	16306.3	0.821676	0.410838	+	0.911708i	$0.365236\pi$
$734$	0	0
$735$	2097.91	0.105282
$736$	0	0
$737$	−47516.8	−2.37491
$738$	0	0
$739$	−31419.3	−1.56397	−0.781987	−	0.623295i	$-0.785794\pi$
$739$	−31419.3	−1.56397	−0.781987	+	0.623295i	$0.785794\pi$
$740$	0	0
$741$	−84283.0	−4.17842
$742$	0	0
$743$	−4475.71	−0.220993	−0.110497	−	0.993877i	$-0.535244\pi$
$743$	−4475.71	−0.220993	−0.110497	+	0.993877i	$0.535244\pi$
$744$	0	0
$745$	8888.24	0.437101
$746$	0	0
$747$	−19790.8	−0.969353
$748$	0	0
$749$	3668.28	0.178953
$750$	0	0
$751$	−13207.9	−0.641763	−0.320882	−	0.947119i	$-0.603979\pi$
$751$	−13207.9	−0.641763	−0.320882	+	0.947119i	$0.603979\pi$
$752$	0	0
$753$	52184.4	2.52550
$754$	0	0
$755$	−9179.56	−0.442488
$756$	0	0
$757$	−6040.06	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$757$	−6040.06	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$758$	0	0
$759$	−7459.66	−0.356744
$760$	0	0
$761$	22751.2	1.08375	0.541873	−	0.840460i	$-0.317715\pi$
$761$	22751.2	1.08375	0.541873	+	0.840460i	$0.317715\pi$
$762$	0	0
$763$	6605.67	0.313422
$764$	0	0
$765$	−8693.05	−0.410847
$766$	0	0
$767$	−31955.1	−1.50434
$768$	0	0
$769$	16581.3	0.777550	0.388775	−	0.921333i	$-0.372898\pi$
$769$	16581.3	0.777550	0.388775	+	0.921333i	$0.372898\pi$
$770$	0	0
$771$	8012.60	0.374276
$772$	0	0
$773$	−22529.0	−1.04827	−0.524134	−	0.851636i	$-0.675611\pi$
$773$	−22529.0	−1.04827	−0.524134	+	0.851636i	$0.675611\pi$
$774$	0	0
$775$	−7686.59	−0.356271
$776$	0	0
$777$	−17203.5	−0.794300
$778$	0	0
$779$	66381.0	3.05308
$780$	0	0
$781$	28586.3	1.30973
$782$	0	0
$783$	15189.5	0.693269
$784$	0	0
$785$	9065.28	0.412170
$786$	0	0
$787$	2748.35	0.124483	0.0622415	−	0.998061i	$-0.480175\pi$
$787$	2748.35	0.124483	0.0622415	+	0.998061i	$0.480175\pi$
$788$	0	0
$789$	20974.7	0.946414
$790$	0	0
$791$	3069.89	0.137993
$792$	0	0
$793$	−13998.0	−0.626840
$794$	0	0
$795$	24289.7	1.08361
$796$	0	0
$797$	−26825.3	−1.19222	−0.596112	−	0.802901i	$-0.703289\pi$
$797$	−26825.3	−1.19222	−0.596112	+	0.802901i	$0.703289\pi$
$798$	0	0
$799$	5154.63	0.228232
$800$	0	0
$801$	−9225.21	−0.406937
$802$	0	0
$803$	−24072.5	−1.05791
$804$	0	0
$805$	−605.533	−0.0265121
$806$	0	0
$807$	−26036.2	−1.13571
$808$	0	0
$809$	8696.46	0.377937	0.188969	−	0.981983i	$-0.439486\pi$
$809$	8696.46	0.377937	0.188969	+	0.981983i	$0.439486\pi$
$810$	0	0
$811$	−40031.6	−1.73329	−0.866646	−	0.498924i	$-0.833729\pi$
$811$	−40031.6	−1.73329	−0.866646	+	0.498924i	$0.833729\pi$
$812$	0	0
$813$	53319.6	2.30012
$814$	0	0
$815$	3927.03	0.168783
$816$	0	0
$817$	1388.65	0.0594646
$818$	0	0
$819$	20414.6	0.870993
$820$	0	0
$821$	−3291.06	−0.139901	−0.0699505	−	0.997550i	$-0.522284\pi$
$821$	−3291.06	−0.139901	−0.0699505	+	0.997550i	$0.522284\pi$
$822$	0	0
$823$	21342.1	0.903934	0.451967	−	0.892035i	$-0.350723\pi$
$823$	21342.1	0.903934	0.451967	+	0.892035i	$0.350723\pi$
$824$	0	0
$825$	10779.3	0.454892
$826$	0	0
$827$	−21039.4	−0.884657	−0.442328	−	0.896853i	$-0.645847\pi$
$827$	−21039.4	−0.884657	−0.442328	+	0.896853i	$0.645847\pi$
$828$	0	0
$829$	−22220.4	−0.930935	−0.465467	−	0.885065i	$-0.654114\pi$
$829$	−22220.4	−0.930935	−0.465467	+	0.885065i	$0.654114\pi$
$830$	0	0
$831$	19282.5	0.804938
$832$	0	0
$833$	1839.09	0.0764953
$834$	0	0
$835$	11565.4	0.479327
$836$	0	0
$837$	50873.0	2.10087
$838$	0	0
$839$	−38654.3	−1.59058	−0.795288	−	0.606232i	$-0.792680\pi$
$839$	−38654.3	−1.59058	−0.795288	+	0.606232i	$0.792680\pi$
$840$	0	0
$841$	−15961.5	−0.654453
$842$	0	0
$843$	−42160.6	−1.72253
$844$	0	0
$845$	−8833.10	−0.359607
$846$	0	0
$847$	−8431.32	−0.342035
$848$	0	0
$849$	−65750.0	−2.65787
$850$	0	0
$851$	4965.55	0.200020
$852$	0	0
$853$	25772.6	1.03451	0.517254	−	0.855832i	$-0.326954\pi$
$853$	25772.6	1.03451	0.517254	+	0.855832i	$0.326954\pi$
$854$	0	0
$855$	36211.0	1.44841
$856$	0	0
$857$	−39874.8	−1.58938	−0.794690	−	0.607016i	$-0.792366\pi$
$857$	−39874.8	−1.58938	−0.794690	+	0.607016i	$0.792366\pi$
$858$	0	0
$859$	21430.3	0.851215	0.425608	−	0.904908i	$-0.360060\pi$
$859$	21430.3	0.851215	0.425608	+	0.904908i	$0.360060\pi$
$860$	0	0
$861$	−25450.0	−1.00736
$862$	0	0
$863$	25908.7	1.02195	0.510974	−	0.859596i	$-0.329285\pi$
$863$	25908.7	1.02195	0.510974	+	0.859596i	$0.329285\pi$
$864$	0	0
$865$	17635.3	0.693200
$866$	0	0
$867$	30007.1	1.17543
$868$	0	0
$869$	−35103.1	−1.37030
$870$	0	0
$871$	−59410.6	−2.31120
$872$	0	0
$873$	−75722.2	−2.93564
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−12837.0	−0.494269	−0.247135	−	0.968981i	$-0.579489\pi$
$877$	−12837.0	−0.494269	−0.247135	+	0.968981i	$0.579489\pi$
$878$	0	0
$879$	−57434.7	−2.20390
$880$	0	0
$881$	−20254.9	−0.774580	−0.387290	−	0.921958i	$-0.626589\pi$
$881$	−20254.9	−0.774580	−0.387290	+	0.921958i	$0.626589\pi$
$882$	0	0
$883$	43532.0	1.65908	0.829541	−	0.558446i	$-0.188602\pi$
$883$	43532.0	1.65908	0.829541	+	0.558446i	$0.188602\pi$
$884$	0	0
$885$	21731.2	0.825409
$886$	0	0
$887$	−27490.1	−1.04062	−0.520309	−	0.853978i	$-0.674183\pi$
$887$	−27490.1	−1.04062	−0.520309	+	0.853978i	$0.674183\pi$
$888$	0	0
$889$	−11551.0	−0.435779
$890$	0	0
$891$	−8363.59	−0.314468
$892$	0	0
$893$	−21471.6	−0.804615
$894$	0	0
$895$	8880.38	0.331663
$896$	0	0
$897$	−9326.87	−0.347174
$898$	0	0
$899$	28225.7	1.04714
$900$	0	0
$901$	21293.1	0.787319
$902$	0	0
$903$	−532.398	−0.0196202
$904$	0	0
$905$	3504.06	0.128706
$906$	0	0
$907$	−19460.3	−0.712425	−0.356212	−	0.934405i	$-0.615932\pi$
$907$	−19460.3	−0.712425	−0.356212	+	0.934405i	$0.615932\pi$
$908$	0	0
$909$	25603.9	0.934243
$910$	0	0
$911$	24975.7	0.908322	0.454161	−	0.890920i	$-0.349939\pi$
$911$	24975.7	0.908322	0.454161	+	0.890920i	$0.349939\pi$
$912$	0	0
$913$	21512.8	0.779812
$914$	0	0
$915$	9519.41	0.343937
$916$	0	0
$917$	12586.6	0.453269
$918$	0	0
$919$	31394.8	1.12690	0.563449	−	0.826151i	$-0.309474\pi$
$919$	31394.8	1.12690	0.563449	+	0.826151i	$0.309474\pi$
$920$	0	0
$921$	81893.7	2.92996
$922$	0	0
$923$	35741.6	1.27459
$924$	0	0
$925$	−7175.26	−0.255050
$926$	0	0
$927$	40144.2	1.42234
$928$	0	0
$929$	24242.8	0.856167	0.428084	−	0.903739i	$-0.359189\pi$
$929$	24242.8	0.856167	0.428084	+	0.903739i	$0.359189\pi$
$930$	0	0
$931$	−7660.72	−0.269678
$932$	0	0
$933$	69734.0	2.44693
$934$	0	0
$935$	9449.43	0.330513
$936$	0	0
$937$	40490.0	1.41169	0.705843	−	0.708368i	$-0.250568\pi$
$937$	40490.0	1.41169	0.705843	+	0.708368i	$0.250568\pi$
$938$	0	0
$939$	29823.3	1.03647
$940$	0	0
$941$	14891.5	0.515885	0.257943	−	0.966160i	$-0.416955\pi$
$941$	14891.5	0.515885	0.257943	+	0.966160i	$0.416955\pi$
$942$	0	0
$943$	7345.82	0.253672
$944$	0	0
$945$	−5791.11	−0.199349
$946$	0	0
$947$	20154.3	0.691580	0.345790	−	0.938312i	$-0.387611\pi$
$947$	20154.3	0.691580	0.345790	+	0.938312i	$0.387611\pi$
$948$	0	0
$949$	−30098.0	−1.02953
$950$	0	0
$951$	−585.694	−0.0199710
$952$	0	0
$953$	−16353.8	−0.555878	−0.277939	−	0.960599i	$-0.589651\pi$
$953$	−16353.8	−0.555878	−0.277939	+	0.960599i	$0.589651\pi$
$954$	0	0
$955$	−17364.7	−0.588384
$956$	0	0
$957$	−39582.2	−1.33700
$958$	0	0
$959$	3101.98	0.104451
$960$	0	0
$961$	64742.8	2.17323
$962$	0	0
$963$	−24275.1	−0.812309
$964$	0	0
$965$	−5119.77	−0.170789
$966$	0	0
$967$	−11441.4	−0.380488	−0.190244	−	0.981737i	$-0.560928\pi$
$967$	−11441.4	−0.380488	−0.190244	+	0.981737i	$0.560928\pi$
$968$	0	0
$969$	50245.8	1.66577
$970$	0	0
$971$	−12434.6	−0.410962	−0.205481	−	0.978661i	$-0.565876\pi$
$971$	−12434.6	−0.410962	−0.205481	+	0.978661i	$0.565876\pi$
$972$	0	0
$973$	−8974.70	−0.295700
$974$	0	0
$975$	13477.4	0.442689
$976$	0	0
$977$	−23296.9	−0.762880	−0.381440	−	0.924394i	$-0.624572\pi$
$977$	−23296.9	−0.762880	−0.381440	+	0.924394i	$0.624572\pi$
$978$	0	0
$979$	10027.9	0.327367
$980$	0	0
$981$	−43713.5	−1.42269
$982$	0	0
$983$	−16323.9	−0.529656	−0.264828	−	0.964296i	$-0.585315\pi$
$983$	−16323.9	−0.529656	−0.264828	+	0.964296i	$0.585315\pi$
$984$	0	0
$985$	−8906.42	−0.288104
$986$	0	0
$987$	8232.08	0.265481
$988$	0	0
$989$	153.670	0.00494076
$990$	0	0
$991$	42229.5	1.35365	0.676823	−	0.736146i	$-0.263356\pi$
$991$	42229.5	1.35365	0.676823	+	0.736146i	$0.263356\pi$
$992$	0	0
$993$	−78911.5	−2.52183
$994$	0	0
$995$	18821.0	0.599665
$996$	0	0
$997$	6744.97	0.214258	0.107129	−	0.994245i	$-0.465834\pi$
$997$	6744.97	0.214258	0.107129	+	0.994245i	$0.465834\pi$
$998$	0	0
$999$	47488.9	1.50399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.a.t.1.1		3
4.3	odd	2		280.4.a.h.1.3	✓	3
8.3	odd	2		2240.4.a.bu.1.1		3
8.5	even	2		2240.4.a.bx.1.3		3
20.3	even	4		1400.4.g.k.449.6		6
20.7	even	4		1400.4.g.k.449.1		6
20.19	odd	2		1400.4.a.k.1.1		3
28.27	even	2		1960.4.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.h.1.3	✓	3	4.3	odd	2
560.4.a.t.1.1		3	1.1	even	1	trivial
1400.4.a.k.1.1		3	20.19	odd	2
1400.4.g.k.449.1		6	20.7	even	4
1400.4.g.k.449.6		6	20.3	even	4
1960.4.a.p.1.1		3	28.27	even	2
2240.4.a.bu.1.1		3	8.3	odd	2
2240.4.a.bx.1.3		3	8.5	even	2

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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-8.56288 q^{3} -5.00000 q^{5} -7.00000 q^{7} +46.3230 q^{9} -50.3535 q^{11} -62.9573 q^{13} +42.8144 q^{15} +37.5324 q^{17} -156.341 q^{19} +59.9402 q^{21} -17.3009 q^{23} +25.0000 q^{25} -165.460 q^{27} -91.8017 q^{29} -307.464 q^{31} +431.171 q^{33} +35.0000 q^{35} -287.010 q^{37} +539.096 q^{39} -424.591 q^{41} -8.88215 q^{43} -231.615 q^{45} +137.338 q^{47} +49.0000 q^{49} -321.385 q^{51} +567.325 q^{53} +251.767 q^{55} +1338.73 q^{57} +507.568 q^{59} +222.341 q^{61} -324.261 q^{63} +314.786 q^{65} +943.665 q^{67} +148.146 q^{69} -567.712 q^{71} +478.069 q^{73} -214.072 q^{75} +352.474 q^{77} +697.133 q^{79} +166.098 q^{81} -427.235 q^{83} -187.662 q^{85} +786.087 q^{87} -199.150 q^{89} +440.701 q^{91} +2632.77 q^{93} +781.706 q^{95} -1634.66 q^{97} -2332.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 15 q^{5} - 21 q^{7} + 41 q^{9} - 10 q^{11} - 64 q^{13} + 10 q^{15} + 132 q^{17} - 160 q^{19} + 14 q^{21} - 40 q^{23} + 75 q^{25} - 182 q^{27} + 348 q^{29} - 396 q^{31} + 822 q^{33} + 105 q^{35}+ \cdots - 744 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 15 * q^5 - 21 * q^7 + 41 * q^9 - 10 * q^11 - 64 * q^13 + 10 * q^15 + 132 * q^17 - 160 * q^19 + 14 * q^21 - 40 * q^23 + 75 * q^25 - 182 * q^27 + 348 * q^29 - 396 * q^31 + 822 * q^33 + 105 * q^35 - 18 * q^37 + 98 * q^39 + 478 * q^41 + 220 * q^43 - 205 * q^45 + 270 * q^47 + 147 * q^49 + 398 * q^51 + 718 * q^53 + 50 * q^55 + 1584 * q^57 - 480 * q^59 + 358 * q^61 - 287 * q^63 + 320 * q^65 + 2172 * q^67 + 1152 * q^69 - 368 * q^71 + 510 * q^73 - 50 * q^75 + 70 * q^77 + 150 * q^79 + 35 * q^81 - 1056 * q^83 - 660 * q^85 + 2266 * q^87 + 422 * q^89 + 448 * q^91 + 332 * q^93 + 800 * q^95 - 1172 * q^97 - 744 * q^99

Introduction

L-functions

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