LMFDB - Embedded newform 1472.4.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2822449.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 27x^{2} - 24x - 5 $ x^4 - x^3 - 27x^2 - 24x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.09653$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.15452 q^{3} -1.44721 q^{5} -23.2480 q^{7} +10.8781 q^{9} +14.1993 q^{11} +16.7274 q^{13} -8.90687 q^{15} +35.5362 q^{17} +77.4222 q^{19} -143.080 q^{21} -23.0000 q^{23} -122.906 q^{25} -99.2226 q^{27} -87.5927 q^{29} +19.1830 q^{31} +87.3901 q^{33} +33.6447 q^{35} -94.8859 q^{37} +102.949 q^{39} +248.138 q^{41} -123.286 q^{43} -15.7429 q^{45} +84.9619 q^{47} +197.468 q^{49} +218.708 q^{51} -372.171 q^{53} -20.5494 q^{55} +476.496 q^{57} -542.472 q^{59} +238.437 q^{61} -252.893 q^{63} -24.2081 q^{65} -1000.91 q^{67} -141.554 q^{69} +15.5404 q^{71} +18.5309 q^{73} -756.424 q^{75} -330.106 q^{77} +754.689 q^{79} -904.376 q^{81} -923.951 q^{83} -51.4283 q^{85} -539.091 q^{87} -1310.65 q^{89} -388.879 q^{91} +118.062 q^{93} -112.046 q^{95} -605.897 q^{97} +154.462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} + 2 q^{5} + 32 q^{7} + 79 q^{9} + 56 q^{11} - 139 q^{13} - 230 q^{15} - 6 q^{17} - 108 q^{21} - 92 q^{23} + 160 q^{25} + 119 q^{27} - 489 q^{29} + 127 q^{31} + 438 q^{33} - 408 q^{35} - 826 q^{37}+ \cdots - 1054 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 + 2 * q^5 + 32 * q^7 + 79 * q^9 + 56 * q^11 - 139 * q^13 - 230 * q^15 - 6 * q^17 - 108 * q^21 - 92 * q^23 + 160 * q^25 + 119 * q^27 - 489 * q^29 + 127 * q^31 + 438 * q^33 - 408 * q^35 - 826 * q^37 - 323 * q^39 + 91 * q^41 - 762 * q^43 - 1240 * q^45 - 397 * q^47 + 748 * q^49 - 1132 * q^51 + 430 * q^53 + 380 * q^55 - 918 * q^57 + 340 * q^59 - 714 * q^61 + 2348 * q^63 - 194 * q^65 - 1832 * q^67 - 115 * q^69 + 1047 * q^71 - 27 * q^73 + 1539 * q^75 - 568 * q^77 + 794 * q^79 - 116 * q^81 - 502 * q^83 + 3752 * q^85 + 2191 * q^87 + 6 * q^89 - 2532 * q^91 + 217 * q^93 - 172 * q^95 - 1970 * q^97 - 1054 * 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
$3$	6.15452	1.18444	0.592219	−	0.805777i	$-0.298252\pi$
$3$	6.15452	1.18444	0.592219	+	0.805777i	$0.298252\pi$
$4$	0	0
$5$	−1.44721	−0.129442	−0.0647212	−	0.997903i	$-0.520616\pi$
$5$	−1.44721	−0.129442	−0.0647212	+	0.997903i	$0.520616\pi$
$6$	0	0
$7$	−23.2480	−1.25527	−0.627636	−	0.778507i	$-0.715977\pi$
$7$	−23.2480	−1.25527	−0.627636	+	0.778507i	$0.715977\pi$
$8$	0	0
$9$	10.8781	0.402892
$10$	0	0
$11$	14.1993	0.389206	0.194603	−	0.980882i	$-0.437658\pi$
$11$	14.1993	0.389206	0.194603	+	0.980882i	$0.437658\pi$
$12$	0	0
$13$	16.7274	0.356874	0.178437	−	0.983951i	$-0.442896\pi$
$13$	16.7274	0.356874	0.178437	+	0.983951i	$0.442896\pi$
$14$	0	0
$15$	−8.90687	−0.153316
$16$	0	0
$17$	35.5362	0.506988	0.253494	−	0.967337i	$-0.418420\pi$
$17$	35.5362	0.506988	0.253494	+	0.967337i	$0.418420\pi$
$18$	0	0
$19$	77.4222	0.934835	0.467418	−	0.884037i	$-0.345184\pi$
$19$	77.4222	0.934835	0.467418	+	0.884037i	$0.345184\pi$
$20$	0	0
$21$	−143.080	−1.48679
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−122.906	−0.983245
$26$	0	0
$27$	−99.2226	−0.707237
$28$	0	0
$29$	−87.5927	−0.560881	−0.280441	−	0.959871i	$-0.590481\pi$
$29$	−87.5927	−0.560881	−0.280441	+	0.959871i	$0.590481\pi$
$30$	0	0
$31$	19.1830	0.111141	0.0555705	−	0.998455i	$-0.482302\pi$
$31$	19.1830	0.111141	0.0555705	+	0.998455i	$0.482302\pi$
$32$	0	0
$33$	87.3901	0.460990
$34$	0	0
$35$	33.6447	0.162485
$36$	0	0
$37$	−94.8859	−0.421599	−0.210799	−	0.977529i	$-0.567607\pi$
$37$	−94.8859	−0.421599	−0.210799	+	0.977529i	$0.567607\pi$
$38$	0	0
$39$	102.949	0.422695
$40$	0	0
$41$	248.138	0.945187	0.472593	−	0.881281i	$-0.343318\pi$
$41$	248.138	0.945187	0.472593	+	0.881281i	$0.343318\pi$
$42$	0	0
$43$	−123.286	−0.437229	−0.218615	−	0.975811i	$-0.570154\pi$
$43$	−123.286	−0.437229	−0.218615	+	0.975811i	$0.570154\pi$
$44$	0	0
$45$	−15.7429	−0.0521512
$46$	0	0
$47$	84.9619	0.263680	0.131840	−	0.991271i	$-0.457911\pi$
$47$	84.9619	0.263680	0.131840	+	0.991271i	$0.457911\pi$
$48$	0	0
$49$	197.468	0.575707
$50$	0	0
$51$	218.708	0.600496
$52$	0	0
$53$	−372.171	−0.964559	−0.482280	−	0.876017i	$-0.660191\pi$
$53$	−372.171	−0.964559	−0.482280	+	0.876017i	$0.660191\pi$
$54$	0	0
$55$	−20.5494	−0.0503797
$56$	0	0
$57$	476.496	1.10725
$58$	0	0
$59$	−542.472	−1.19701	−0.598507	−	0.801117i	$-0.704239\pi$
$59$	−542.472	−1.19701	−0.598507	+	0.801117i	$0.704239\pi$
$60$	0	0
$61$	238.437	0.500471	0.250235	−	0.968185i	$-0.419492\pi$
$61$	238.437	0.500471	0.250235	+	0.968185i	$0.419492\pi$
$62$	0	0
$63$	−252.893	−0.505739
$64$	0	0
$65$	−24.2081	−0.0461946
$66$	0	0
$67$	−1000.91	−1.82509	−0.912545	−	0.408977i	$-0.865886\pi$
$67$	−1000.91	−1.82509	−0.912545	+	0.408977i	$0.865886\pi$
$68$	0	0
$69$	−141.554	−0.246972
$70$	0	0
$71$	15.5404	0.0259761	0.0129880	−	0.999916i	$-0.495866\pi$
$71$	15.5404	0.0259761	0.0129880	+	0.999916i	$0.495866\pi$
$72$	0	0
$73$	18.5309	0.0297107	0.0148554	−	0.999890i	$-0.495271\pi$
$73$	18.5309	0.0297107	0.0148554	+	0.999890i	$0.495271\pi$
$74$	0	0
$75$	−756.424	−1.16459
$76$	0	0
$77$	−330.106	−0.488559
$78$	0	0
$79$	754.689	1.07480	0.537400	−	0.843328i	$-0.319407\pi$
$79$	754.689	1.07480	0.537400	+	0.843328i	$0.319407\pi$
$80$	0	0
$81$	−904.376	−1.24057
$82$	0	0
$83$	−923.951	−1.22189	−0.610945	−	0.791673i	$-0.709210\pi$
$83$	−923.951	−1.22189	−0.610945	+	0.791673i	$0.709210\pi$
$84$	0	0
$85$	−51.4283	−0.0656257
$86$	0	0
$87$	−539.091	−0.664329
$88$	0	0
$89$	−1310.65	−1.56099	−0.780497	−	0.625159i	$-0.785034\pi$
$89$	−1310.65	−1.56099	−0.780497	+	0.625159i	$0.785034\pi$
$90$	0	0
$91$	−388.879	−0.447974
$92$	0	0
$93$	118.062	0.131639
$94$	0	0
$95$	−112.046	−0.121007
$96$	0	0
$97$	−605.897	−0.634222	−0.317111	−	0.948388i	$-0.602713\pi$
$97$	−605.897	−0.634222	−0.317111	+	0.948388i	$0.602713\pi$
$98$	0	0
$99$	154.462	0.156808
$100$	0	0
$101$	−401.451	−0.395503	−0.197752	−	0.980252i	$-0.563364\pi$
$101$	−401.451	−0.395503	−0.197752	+	0.980252i	$0.563364\pi$
$102$	0	0
$103$	−1007.34	−0.963657	−0.481828	−	0.876266i	$-0.660027\pi$
$103$	−1007.34	−0.963657	−0.481828	+	0.876266i	$0.660027\pi$
$104$	0	0
$105$	207.067	0.192454
$106$	0	0
$107$	218.166	0.197111	0.0985556	−	0.995132i	$-0.468578\pi$
$107$	218.166	0.197111	0.0985556	+	0.995132i	$0.468578\pi$
$108$	0	0
$109$	657.112	0.577430	0.288715	−	0.957415i	$-0.406772\pi$
$109$	657.112	0.577430	0.288715	+	0.957415i	$0.406772\pi$
$110$	0	0
$111$	−583.977	−0.499357
$112$	0	0
$113$	−677.125	−0.563704	−0.281852	−	0.959458i	$-0.590949\pi$
$113$	−677.125	−0.563704	−0.281852	+	0.959458i	$0.590949\pi$
$114$	0	0
$115$	33.2858	0.0269906
$116$	0	0
$117$	181.962	0.143781
$118$	0	0
$119$	−826.145	−0.636408
$120$	0	0
$121$	−1129.38	−0.848519
$122$	0	0
$123$	1527.17	1.11951
$124$	0	0
$125$	358.771	0.256716
$126$	0	0
$127$	−1316.92	−0.920138	−0.460069	−	0.887883i	$-0.652175\pi$
$127$	−1316.92	−0.920138	−0.460069	+	0.887883i	$0.652175\pi$
$128$	0	0
$129$	−758.763	−0.517871
$130$	0	0
$131$	−78.7994	−0.0525552	−0.0262776	−	0.999655i	$-0.508365\pi$
$131$	−78.7994	−0.0525552	−0.0262776	+	0.999655i	$0.508365\pi$
$132$	0	0
$133$	−1799.91	−1.17347
$134$	0	0
$135$	143.596	0.0915464
$136$	0	0
$137$	1415.41	0.882677	0.441338	−	0.897341i	$-0.354504\pi$
$137$	1415.41	0.882677	0.441338	+	0.897341i	$0.354504\pi$
$138$	0	0
$139$	1377.84	0.840771	0.420386	−	0.907346i	$-0.361895\pi$
$139$	1377.84	0.840771	0.420386	+	0.907346i	$0.361895\pi$
$140$	0	0
$141$	522.900	0.312313
$142$	0	0
$143$	237.519	0.138897
$144$	0	0
$145$	126.765	0.0726018
$146$	0	0
$147$	1215.32	0.681889
$148$	0	0
$149$	−2366.57	−1.30119	−0.650593	−	0.759427i	$-0.725480\pi$
$149$	−2366.57	−1.30119	−0.650593	+	0.759427i	$0.725480\pi$
$150$	0	0
$151$	2080.82	1.12142	0.560711	−	0.828011i	$-0.310528\pi$
$151$	2080.82	1.12142	0.560711	+	0.828011i	$0.310528\pi$
$152$	0	0
$153$	386.566	0.204261
$154$	0	0
$155$	−27.7618	−0.0143863
$156$	0	0
$157$	−2289.83	−1.16400	−0.582000	−	0.813189i	$-0.697730\pi$
$157$	−2289.83	−1.16400	−0.582000	+	0.813189i	$0.697730\pi$
$158$	0	0
$159$	−2290.53	−1.14246
$160$	0	0
$161$	534.703	0.261742
$162$	0	0
$163$	−2258.91	−1.08547	−0.542735	−	0.839904i	$-0.682611\pi$
$163$	−2258.91	−1.08547	−0.542735	+	0.839904i	$0.682611\pi$
$164$	0	0
$165$	−126.472	−0.0596716
$166$	0	0
$167$	−1765.94	−0.818277	−0.409138	−	0.912472i	$-0.634171\pi$
$167$	−1765.94	−0.818277	−0.409138	+	0.912472i	$0.634171\pi$
$168$	0	0
$169$	−1917.19	−0.872641
$170$	0	0
$171$	842.205	0.376637
$172$	0	0
$173$	−206.784	−0.0908755	−0.0454378	−	0.998967i	$-0.514468\pi$
$173$	−206.784	−0.0908755	−0.0454378	+	0.998967i	$0.514468\pi$
$174$	0	0
$175$	2857.30	1.23424
$176$	0	0
$177$	−3338.66	−1.41779
$178$	0	0
$179$	4581.89	1.91322	0.956611	−	0.291368i	$-0.0941104\pi$
$179$	4581.89	1.91322	0.956611	+	0.291368i	$0.0941104\pi$
$180$	0	0
$181$	−169.992	−0.0698088	−0.0349044	−	0.999391i	$-0.511113\pi$
$181$	−169.992	−0.0698088	−0.0349044	+	0.999391i	$0.511113\pi$
$182$	0	0
$183$	1467.46	0.592776
$184$	0	0
$185$	137.320	0.0545727
$186$	0	0
$187$	504.591	0.197323
$188$	0	0
$189$	2306.72	0.887775
$190$	0	0
$191$	−423.842	−0.160566	−0.0802831	−	0.996772i	$-0.525582\pi$
$191$	−423.842	−0.160566	−0.0802831	+	0.996772i	$0.525582\pi$
$192$	0	0
$193$	291.603	0.108757	0.0543784	−	0.998520i	$-0.482682\pi$
$193$	291.603	0.108757	0.0543784	+	0.998520i	$0.482682\pi$
$194$	0	0
$195$	−148.989	−0.0547146
$196$	0	0
$197$	−4361.99	−1.57756	−0.788778	−	0.614678i	$-0.789286\pi$
$197$	−4361.99	−1.57756	−0.788778	+	0.614678i	$0.789286\pi$
$198$	0	0
$199$	4418.17	1.57385	0.786924	−	0.617050i	$-0.211672\pi$
$199$	4418.17	1.57385	0.786924	+	0.617050i	$0.211672\pi$
$200$	0	0
$201$	−6160.14	−2.16170
$202$	0	0
$203$	2036.35	0.704058
$204$	0	0
$205$	−359.108	−0.122347
$206$	0	0
$207$	−250.196	−0.0840087
$208$	0	0
$209$	1099.34	0.363843
$210$	0	0
$211$	4709.41	1.53654	0.768268	−	0.640129i	$-0.221119\pi$
$211$	4709.41	1.53654	0.768268	+	0.640129i	$0.221119\pi$
$212$	0	0
$213$	95.6434	0.0307670
$214$	0	0
$215$	178.420	0.0565960
$216$	0	0
$217$	−445.966	−0.139512
$218$	0	0
$219$	114.049	0.0351905
$220$	0	0
$221$	594.430	0.180931
$222$	0	0
$223$	4936.13	1.48228	0.741138	−	0.671353i	$-0.234286\pi$
$223$	4936.13	1.48228	0.741138	+	0.671353i	$0.234286\pi$
$224$	0	0
$225$	−1336.98	−0.396141
$226$	0	0
$227$	5010.95	1.46515	0.732574	−	0.680687i	$-0.238319\pi$
$227$	5010.95	1.46515	0.732574	+	0.680687i	$0.238319\pi$
$228$	0	0
$229$	−4251.82	−1.22694	−0.613468	−	0.789720i	$-0.710226\pi$
$229$	−4251.82	−1.22694	−0.613468	+	0.789720i	$0.710226\pi$
$230$	0	0
$231$	−2031.64	−0.578667
$232$	0	0
$233$	−175.891	−0.0494550	−0.0247275	−	0.999694i	$-0.507872\pi$
$233$	−175.891	−0.0494550	−0.0247275	+	0.999694i	$0.507872\pi$
$234$	0	0
$235$	−122.958	−0.0341314
$236$	0	0
$237$	4644.75	1.27303
$238$	0	0
$239$	−4979.92	−1.34780	−0.673900	−	0.738822i	$-0.735382\pi$
$239$	−4979.92	−1.34780	−0.673900	+	0.738822i	$0.735382\pi$
$240$	0	0
$241$	2153.46	0.575588	0.287794	−	0.957692i	$-0.407078\pi$
$241$	2153.46	0.575588	0.287794	+	0.957692i	$0.407078\pi$
$242$	0	0
$243$	−2886.98	−0.762140
$244$	0	0
$245$	−285.777	−0.0745209
$246$	0	0
$247$	1295.08	0.333618
$248$	0	0
$249$	−5686.47	−1.44725
$250$	0	0
$251$	−3030.37	−0.762053	−0.381027	−	0.924564i	$-0.624429\pi$
$251$	−3030.37	−0.762053	−0.381027	+	0.924564i	$0.624429\pi$
$252$	0	0
$253$	−326.585	−0.0811550
$254$	0	0
$255$	−316.517	−0.0777296
$256$	0	0
$257$	5813.04	1.41092	0.705462	−	0.708747i	$-0.250739\pi$
$257$	5813.04	1.41092	0.705462	+	0.708747i	$0.250739\pi$
$258$	0	0
$259$	2205.90	0.529221
$260$	0	0
$261$	−952.840	−0.225974
$262$	0	0
$263$	3238.30	0.759247	0.379623	−	0.925141i	$-0.376054\pi$
$263$	3238.30	0.759247	0.379623	+	0.925141i	$0.376054\pi$
$264$	0	0
$265$	538.609	0.124855
$266$	0	0
$267$	−8066.41	−1.84890
$268$	0	0
$269$	1884.90	0.427227	0.213614	−	0.976918i	$-0.431477\pi$
$269$	1884.90	0.427227	0.213614	+	0.976918i	$0.431477\pi$
$270$	0	0
$271$	8743.27	1.95984	0.979919	−	0.199397i	$-0.0638984\pi$
$271$	8743.27	1.95984	0.979919	+	0.199397i	$0.0638984\pi$
$272$	0	0
$273$	−2393.36	−0.530597
$274$	0	0
$275$	−1745.18	−0.382684
$276$	0	0
$277$	−3108.71	−0.674312	−0.337156	−	0.941449i	$-0.609465\pi$
$277$	−3108.71	−0.674312	−0.337156	+	0.941449i	$0.609465\pi$
$278$	0	0
$279$	208.674	0.0447778
$280$	0	0
$281$	−6270.05	−1.33110	−0.665551	−	0.746352i	$-0.731803\pi$
$281$	−6270.05	−1.33110	−0.665551	+	0.746352i	$0.731803\pi$
$282$	0	0
$283$	−2274.38	−0.477730	−0.238865	−	0.971053i	$-0.576775\pi$
$283$	−2274.38	−0.477730	−0.238865	+	0.971053i	$0.576775\pi$
$284$	0	0
$285$	−689.590	−0.143326
$286$	0	0
$287$	−5768.70	−1.18647
$288$	0	0
$289$	−3650.18	−0.742963
$290$	0	0
$291$	−3729.01	−0.751196
$292$	0	0
$293$	5526.41	1.10190	0.550950	−	0.834538i	$-0.314266\pi$
$293$	5526.41	1.10190	0.550950	+	0.834538i	$0.314266\pi$
$294$	0	0
$295$	785.071	0.154944
$296$	0	0
$297$	−1408.90	−0.275261
$298$	0	0
$299$	−384.731	−0.0744133
$300$	0	0
$301$	2866.14	0.548842
$302$	0	0
$303$	−2470.73	−0.468449
$304$	0	0
$305$	−345.068	−0.0647821
$306$	0	0
$307$	−3569.23	−0.663539	−0.331770	−	0.943360i	$-0.607646\pi$
$307$	−3569.23	−0.663539	−0.331770	+	0.943360i	$0.607646\pi$
$308$	0	0
$309$	−6199.72	−1.14139
$310$	0	0
$311$	−3991.96	−0.727856	−0.363928	−	0.931427i	$-0.618565\pi$
$311$	−3991.96	−0.727856	−0.363928	+	0.931427i	$0.618565\pi$
$312$	0	0
$313$	−1750.31	−0.316081	−0.158040	−	0.987433i	$-0.550518\pi$
$313$	−1750.31	−0.316081	−0.158040	+	0.987433i	$0.550518\pi$
$314$	0	0
$315$	365.989	0.0654640
$316$	0	0
$317$	−8794.63	−1.55822	−0.779110	−	0.626888i	$-0.784328\pi$
$317$	−8794.63	−1.55822	−0.779110	+	0.626888i	$0.784328\pi$
$318$	0	0
$319$	−1243.76	−0.218298
$320$	0	0
$321$	1342.71	0.233466
$322$	0	0
$323$	2751.29	0.473951
$324$	0	0
$325$	−2055.90	−0.350894
$326$	0	0
$327$	4044.20	0.683930
$328$	0	0
$329$	−1975.19	−0.330990
$330$	0	0
$331$	1026.21	0.170410	0.0852052	−	0.996363i	$-0.472845\pi$
$331$	1026.21	0.170410	0.0852052	+	0.996363i	$0.472845\pi$
$332$	0	0
$333$	−1032.18	−0.169859
$334$	0	0
$335$	1448.53	0.236244
$336$	0	0
$337$	9712.53	1.56996	0.784978	−	0.619523i	$-0.212674\pi$
$337$	9712.53	1.56996	0.784978	+	0.619523i	$0.212674\pi$
$338$	0	0
$339$	−4167.38	−0.667672
$340$	0	0
$341$	272.386	0.0432567
$342$	0	0
$343$	3383.33	0.532603
$344$	0	0
$345$	204.858	0.0319687
$346$	0	0
$347$	−7413.55	−1.14692	−0.573458	−	0.819235i	$-0.694399\pi$
$347$	−7413.55	−1.14692	−0.573458	+	0.819235i	$0.694399\pi$
$348$	0	0
$349$	−10049.2	−1.54132	−0.770659	−	0.637248i	$-0.780073\pi$
$349$	−10049.2	−1.54132	−0.770659	+	0.637248i	$0.780073\pi$
$350$	0	0
$351$	−1659.74	−0.252394
$352$	0	0
$353$	8742.65	1.31820	0.659099	−	0.752056i	$-0.270938\pi$
$353$	8742.65	1.31820	0.659099	+	0.752056i	$0.270938\pi$
$354$	0	0
$355$	−22.4902	−0.00336240
$356$	0	0
$357$	−5084.52	−0.753785
$358$	0	0
$359$	−2347.49	−0.345114	−0.172557	−	0.985000i	$-0.555203\pi$
$359$	−2347.49	−0.345114	−0.172557	+	0.985000i	$0.555203\pi$
$360$	0	0
$361$	−864.801	−0.126083
$362$	0	0
$363$	−6950.78	−1.00502
$364$	0	0
$365$	−26.8181	−0.00384582
$366$	0	0
$367$	−2505.57	−0.356376	−0.178188	−	0.983996i	$-0.557023\pi$
$367$	−2505.57	−0.356376	−0.178188	+	0.983996i	$0.557023\pi$
$368$	0	0
$369$	2699.27	0.380808
$370$	0	0
$371$	8652.22	1.21078
$372$	0	0
$373$	6462.76	0.897129	0.448564	−	0.893751i	$-0.351935\pi$
$373$	6462.76	0.897129	0.448564	+	0.893751i	$0.351935\pi$
$374$	0	0
$375$	2208.06	0.304064
$376$	0	0
$377$	−1465.20	−0.200164
$378$	0	0
$379$	6909.79	0.936497	0.468248	−	0.883597i	$-0.344885\pi$
$379$	6909.79	0.936497	0.468248	+	0.883597i	$0.344885\pi$
$380$	0	0
$381$	−8104.99	−1.08985
$382$	0	0
$383$	4482.58	0.598039	0.299019	−	0.954247i	$-0.403340\pi$
$383$	4482.58	0.598039	0.299019	+	0.954247i	$0.403340\pi$
$384$	0	0
$385$	477.732	0.0632402
$386$	0	0
$387$	−1341.11	−0.176156
$388$	0	0
$389$	−12178.5	−1.58733	−0.793667	−	0.608352i	$-0.791831\pi$
$389$	−12178.5	−1.58733	−0.793667	+	0.608352i	$0.791831\pi$
$390$	0	0
$391$	−817.333	−0.105714
$392$	0	0
$393$	−484.972	−0.0622484
$394$	0	0
$395$	−1092.19	−0.139125
$396$	0	0
$397$	−5727.55	−0.724075	−0.362037	−	0.932164i	$-0.617919\pi$
$397$	−5727.55	−0.724075	−0.362037	+	0.932164i	$0.617919\pi$
$398$	0	0
$399$	−11077.6	−1.38990
$400$	0	0
$401$	2982.55	0.371425	0.185712	−	0.982604i	$-0.440541\pi$
$401$	2982.55	0.371425	0.185712	+	0.982604i	$0.440541\pi$
$402$	0	0
$403$	320.883	0.0396633
$404$	0	0
$405$	1308.82	0.160582
$406$	0	0
$407$	−1347.32	−0.164089
$408$	0	0
$409$	7957.98	0.962095	0.481047	−	0.876695i	$-0.340257\pi$
$409$	7957.98	0.962095	0.481047	+	0.876695i	$0.340257\pi$
$410$	0	0
$411$	8711.17	1.04548
$412$	0	0
$413$	12611.4	1.50258
$414$	0	0
$415$	1337.15	0.158164
$416$	0	0
$417$	8479.96	0.995841
$418$	0	0
$419$	2621.06	0.305602	0.152801	−	0.988257i	$-0.451171\pi$
$419$	2621.06	0.305602	0.152801	+	0.988257i	$0.451171\pi$
$420$	0	0
$421$	2.56163	0.000296547 0	0.000148274	−	1.00000i	$-0.499953\pi$
$421$	2.56163	0.000296547 0	0.000148274		1.00000i	$0.499953\pi$
$422$	0	0
$423$	924.223	0.106235
$424$	0	0
$425$	−4367.60	−0.498493
$426$	0	0
$427$	−5543.17	−0.628227
$428$	0	0
$429$	1461.81	0.164515
$430$	0	0
$431$	−8453.37	−0.944744	−0.472372	−	0.881399i	$-0.656602\pi$
$431$	−8453.37	−0.944744	−0.472372	+	0.881399i	$0.656602\pi$
$432$	0	0
$433$	−11383.7	−1.26343	−0.631717	−	0.775199i	$-0.717650\pi$
$433$	−11383.7	−1.26343	−0.631717	+	0.775199i	$0.717650\pi$
$434$	0	0
$435$	780.177	0.0859922
$436$	0	0
$437$	−1780.71	−0.194927
$438$	0	0
$439$	17958.4	1.95241	0.976207	−	0.216839i	$-0.0695748\pi$
$439$	17958.4	1.95241	0.976207	+	0.216839i	$0.0695748\pi$
$440$	0	0
$441$	2148.07	0.231948
$442$	0	0
$443$	−135.469	−0.0145289	−0.00726446	−	0.999974i	$-0.502312\pi$
$443$	−135.469	−0.0145289	−0.00726446	+	0.999974i	$0.502312\pi$
$444$	0	0
$445$	1896.78	0.202059
$446$	0	0
$447$	−14565.1	−1.54117
$448$	0	0
$449$	−807.866	−0.0849121	−0.0424561	−	0.999098i	$-0.513518\pi$
$449$	−807.866	−0.0849121	−0.0424561	+	0.999098i	$0.513518\pi$
$450$	0	0
$451$	3523.40	0.367872
$452$	0	0
$453$	12806.5	1.32826
$454$	0	0
$455$	562.789	0.0579867
$456$	0	0
$457$	−7610.73	−0.779026	−0.389513	−	0.921021i	$-0.627357\pi$
$457$	−7610.73	−0.779026	−0.389513	+	0.921021i	$0.627357\pi$
$458$	0	0
$459$	−3526.00	−0.358561
$460$	0	0
$461$	18801.7	1.89953	0.949765	−	0.312964i	$-0.101322\pi$
$461$	18801.7	1.89953	0.949765	+	0.312964i	$0.101322\pi$
$462$	0	0
$463$	−11011.6	−1.10530	−0.552651	−	0.833413i	$-0.686384\pi$
$463$	−11011.6	−1.10530	−0.552651	+	0.833413i	$0.686384\pi$
$464$	0	0
$465$	−170.861	−0.0170397
$466$	0	0
$467$	−8572.93	−0.849481	−0.424741	−	0.905315i	$-0.639635\pi$
$467$	−8572.93	−0.849481	−0.424741	+	0.905315i	$0.639635\pi$
$468$	0	0
$469$	23269.2	2.29098
$470$	0	0
$471$	−14092.8	−1.37869
$472$	0	0
$473$	−1750.57	−0.170172
$474$	0	0
$475$	−9515.62	−0.919172
$476$	0	0
$477$	−4048.51	−0.388613
$478$	0	0
$479$	8679.82	0.827957	0.413978	−	0.910287i	$-0.364139\pi$
$479$	8679.82	0.827957	0.413978	+	0.910287i	$0.364139\pi$
$480$	0	0
$481$	−1587.20	−0.150458
$482$	0	0
$483$	3290.84	0.310017
$484$	0	0
$485$	876.860	0.0820952
$486$	0	0
$487$	2238.51	0.208289	0.104144	−	0.994562i	$-0.466790\pi$
$487$	2238.51	0.208289	0.104144	+	0.994562i	$0.466790\pi$
$488$	0	0
$489$	−13902.5	−1.28567
$490$	0	0
$491$	13646.4	1.25428	0.627142	−	0.778905i	$-0.284225\pi$
$491$	13646.4	1.25428	0.627142	+	0.778905i	$0.284225\pi$
$492$	0	0
$493$	−3112.71	−0.284360
$494$	0	0
$495$	−223.538	−0.0202976
$496$	0	0
$497$	−361.282	−0.0326070
$498$	0	0
$499$	11961.9	1.07313	0.536563	−	0.843860i	$-0.319722\pi$
$499$	11961.9	1.07313	0.536563	+	0.843860i	$0.319722\pi$
$500$	0	0
$501$	−10868.5	−0.969197
$502$	0	0
$503$	−3138.63	−0.278219	−0.139110	−	0.990277i	$-0.544424\pi$
$503$	−3138.63	−0.278219	−0.139110	+	0.990277i	$0.544424\pi$
$504$	0	0
$505$	580.983	0.0511949
$506$	0	0
$507$	−11799.4	−1.03359
$508$	0	0
$509$	3777.32	0.328933	0.164466	−	0.986383i	$-0.447410\pi$
$509$	3777.32	0.328933	0.164466	+	0.986383i	$0.447410\pi$
$510$	0	0
$511$	−430.807	−0.0372950
$512$	0	0
$513$	−7682.04	−0.661150
$514$	0	0
$515$	1457.84	0.124738
$516$	0	0
$517$	1206.40	0.102626
$518$	0	0
$519$	−1272.65	−0.107636
$520$	0	0
$521$	8252.84	0.693980	0.346990	−	0.937869i	$-0.387204\pi$
$521$	8252.84	0.693980	0.346990	+	0.937869i	$0.387204\pi$
$522$	0	0
$523$	2581.34	0.215821	0.107910	−	0.994161i	$-0.465584\pi$
$523$	2581.34	0.215821	0.107910	+	0.994161i	$0.465584\pi$
$524$	0	0
$525$	17585.3	1.46188
$526$	0	0
$527$	681.691	0.0563471
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−5901.06	−0.482267
$532$	0	0
$533$	4150.72	0.337312
$534$	0	0
$535$	−315.732	−0.0255145
$536$	0	0
$537$	28199.3	2.26609
$538$	0	0
$539$	2803.91	0.224069
$540$	0	0
$541$	−13690.3	−1.08797	−0.543987	−	0.839094i	$-0.683086\pi$
$541$	−13690.3	−1.08797	−0.543987	+	0.839094i	$0.683086\pi$
$542$	0	0
$543$	−1046.22	−0.0826842
$544$	0	0
$545$	−950.978	−0.0747439
$546$	0	0
$547$	−9978.29	−0.779965	−0.389983	−	0.920822i	$-0.627519\pi$
$547$	−9978.29	−0.779965	−0.389983	+	0.920822i	$0.627519\pi$
$548$	0	0
$549$	2593.73	0.201635
$550$	0	0
$551$	−6781.62	−0.524332
$552$	0	0
$553$	−17545.0	−1.34917
$554$	0	0
$555$	845.137	0.0646380
$556$	0	0
$557$	6824.80	0.519167	0.259583	−	0.965721i	$-0.416415\pi$
$557$	6824.80	0.519167	0.259583	+	0.965721i	$0.416415\pi$
$558$	0	0
$559$	−2062.25	−0.156036
$560$	0	0
$561$	3105.51	0.233716
$562$	0	0
$563$	20979.3	1.57046	0.785232	−	0.619202i	$-0.212544\pi$
$563$	20979.3	1.57046	0.785232	+	0.619202i	$0.212544\pi$
$564$	0	0
$565$	979.942	0.0729672
$566$	0	0
$567$	21024.9	1.55725
$568$	0	0
$569$	−6367.00	−0.469101	−0.234550	−	0.972104i	$-0.575362\pi$
$569$	−6367.00	−0.469101	−0.234550	+	0.972104i	$0.575362\pi$
$570$	0	0
$571$	−26735.8	−1.95947	−0.979735	−	0.200296i	$-0.935810\pi$
$571$	−26735.8	−1.95947	−0.979735	+	0.200296i	$0.935810\pi$
$572$	0	0
$573$	−2608.54	−0.190181
$574$	0	0
$575$	2826.83	0.205021
$576$	0	0
$577$	13525.1	0.975835	0.487917	−	0.872890i	$-0.337757\pi$
$577$	13525.1	0.975835	0.487917	+	0.872890i	$0.337757\pi$
$578$	0	0
$579$	1794.68	0.128816
$580$	0	0
$581$	21480.0	1.53380
$582$	0	0
$583$	−5284.58	−0.375412
$584$	0	0
$585$	−263.338	−0.0186114
$586$	0	0
$587$	−1447.65	−0.101791	−0.0508953	−	0.998704i	$-0.516207\pi$
$587$	−1447.65	−0.101791	−0.0508953	+	0.998704i	$0.516207\pi$
$588$	0	0
$589$	1485.19	0.103898
$590$	0	0
$591$	−26845.9	−1.86852
$592$	0	0
$593$	−25979.8	−1.79909	−0.899547	−	0.436825i	$-0.856103\pi$
$593$	−25979.8	−1.79909	−0.899547	+	0.436825i	$0.856103\pi$
$594$	0	0
$595$	1195.60	0.0823781
$596$	0	0
$597$	27191.7	1.86412
$598$	0	0
$599$	12211.5	0.832966	0.416483	−	0.909143i	$-0.363263\pi$
$599$	12211.5	0.832966	0.416483	+	0.909143i	$0.363263\pi$
$600$	0	0
$601$	25772.1	1.74920	0.874598	−	0.484849i	$-0.161125\pi$
$601$	25772.1	1.74920	0.874598	+	0.484849i	$0.161125\pi$
$602$	0	0
$603$	−10888.0	−0.735313
$604$	0	0
$605$	1634.45	0.109834
$606$	0	0
$607$	14255.6	0.953243	0.476621	−	0.879109i	$-0.341861\pi$
$607$	14255.6	0.953243	0.476621	+	0.879109i	$0.341861\pi$
$608$	0	0
$609$	12532.8	0.833913
$610$	0	0
$611$	1421.20	0.0941006
$612$	0	0
$613$	−20679.7	−1.36256	−0.681278	−	0.732025i	$-0.738575\pi$
$613$	−20679.7	−1.36256	−0.681278	+	0.732025i	$0.738575\pi$
$614$	0	0
$615$	−2210.13	−0.144913
$616$	0	0
$617$	−13850.4	−0.903722	−0.451861	−	0.892088i	$-0.649240\pi$
$617$	−13850.4	−0.903722	−0.451861	+	0.892088i	$0.649240\pi$
$618$	0	0
$619$	7948.31	0.516106	0.258053	−	0.966131i	$-0.416919\pi$
$619$	7948.31	0.516106	0.258053	+	0.966131i	$0.416919\pi$
$620$	0	0
$621$	2282.12	0.147469
$622$	0	0
$623$	30469.9	1.95947
$624$	0	0
$625$	14844.0	0.950015
$626$	0	0
$627$	6765.93	0.430949
$628$	0	0
$629$	−3371.89	−0.213746
$630$	0	0
$631$	−14165.9	−0.893718	−0.446859	−	0.894604i	$-0.647457\pi$
$631$	−14165.9	−0.893718	−0.446859	+	0.894604i	$0.647457\pi$
$632$	0	0
$633$	28984.1	1.81993
$634$	0	0
$635$	1905.85	0.119105
$636$	0	0
$637$	3303.13	0.205455
$638$	0	0
$639$	169.049	0.0104655
$640$	0	0
$641$	8355.41	0.514850	0.257425	−	0.966298i	$-0.417126\pi$
$641$	8355.41	0.514850	0.257425	+	0.966298i	$0.417126\pi$
$642$	0	0
$643$	1810.08	0.111015	0.0555073	−	0.998458i	$-0.482322\pi$
$643$	1810.08	0.111015	0.0555073	+	0.998458i	$0.482322\pi$
$644$	0	0
$645$	1098.09	0.0670344
$646$	0	0
$647$	−3129.33	−0.190149	−0.0950746	−	0.995470i	$-0.530309\pi$
$647$	−3129.33	−0.190149	−0.0950746	+	0.995470i	$0.530309\pi$
$648$	0	0
$649$	−7702.75	−0.465885
$650$	0	0
$651$	−2744.70	−0.165243
$652$	0	0
$653$	27462.4	1.64577	0.822885	−	0.568208i	$-0.192363\pi$
$653$	27462.4	1.64577	0.822885	+	0.568208i	$0.192363\pi$
$654$	0	0
$655$	114.039	0.00680287
$656$	0	0
$657$	201.581	0.0119702
$658$	0	0
$659$	−12765.1	−0.754561	−0.377281	−	0.926099i	$-0.623141\pi$
$659$	−12765.1	−0.754561	−0.377281	+	0.926099i	$0.623141\pi$
$660$	0	0
$661$	−17897.1	−1.05313	−0.526564	−	0.850135i	$-0.676520\pi$
$661$	−17897.1	−1.05313	−0.526564	+	0.850135i	$0.676520\pi$
$662$	0	0
$663$	3658.43	0.214301
$664$	0	0
$665$	2604.84	0.151897
$666$	0	0
$667$	2014.63	0.116952
$668$	0	0
$669$	30379.5	1.75566
$670$	0	0
$671$	3385.65	0.194786
$672$	0	0
$673$	−9158.54	−0.524570	−0.262285	−	0.964990i	$-0.584476\pi$
$673$	−9158.54	−0.524570	−0.262285	+	0.964990i	$0.584476\pi$
$674$	0	0
$675$	12195.0	0.695387
$676$	0	0
$677$	−22782.7	−1.29337	−0.646684	−	0.762758i	$-0.723845\pi$
$677$	−22782.7	−1.29337	−0.646684	+	0.762758i	$0.723845\pi$
$678$	0	0
$679$	14085.9	0.796121
$680$	0	0
$681$	30840.0	1.73538
$682$	0	0
$683$	11678.9	0.654289	0.327144	−	0.944974i	$-0.393914\pi$
$683$	11678.9	0.654289	0.327144	+	0.944974i	$0.393914\pi$
$684$	0	0
$685$	−2048.40	−0.114256
$686$	0	0
$687$	−26167.9	−1.45323
$688$	0	0
$689$	−6225.47	−0.344226
$690$	0	0
$691$	18799.9	1.03500	0.517498	−	0.855684i	$-0.326863\pi$
$691$	18799.9	1.03500	0.517498	+	0.855684i	$0.326863\pi$
$692$	0	0
$693$	−3590.91	−0.196836
$694$	0	0
$695$	−1994.03	−0.108831
$696$	0	0
$697$	8817.89	0.479199
$698$	0	0
$699$	−1082.53	−0.0585764
$700$	0	0
$701$	−14703.5	−0.792214	−0.396107	−	0.918204i	$-0.629639\pi$
$701$	−14703.5	−0.792214	−0.396107	+	0.918204i	$0.629639\pi$
$702$	0	0
$703$	−7346.28	−0.394125
$704$	0	0
$705$	−756.745	−0.0404265
$706$	0	0
$707$	9332.91	0.496464
$708$	0	0
$709$	20507.7	1.08629	0.543146	−	0.839638i	$-0.317233\pi$
$709$	20507.7	1.08629	0.543146	+	0.839638i	$0.317233\pi$
$710$	0	0
$711$	8209.57	0.433028
$712$	0	0
$713$	−441.209	−0.0231745
$714$	0	0
$715$	−343.739	−0.0179792
$716$	0	0
$717$	−30649.0	−1.59639
$718$	0	0
$719$	−13728.9	−0.712104	−0.356052	−	0.934466i	$-0.615877\pi$
$719$	−13728.9	−0.712104	−0.356052	+	0.934466i	$0.615877\pi$
$720$	0	0
$721$	23418.7	1.20965
$722$	0	0
$723$	13253.5	0.681748
$724$	0	0
$725$	10765.6	0.551483
$726$	0	0
$727$	24721.1	1.26115	0.630573	−	0.776130i	$-0.282820\pi$
$727$	24721.1	1.26115	0.630573	+	0.776130i	$0.282820\pi$
$728$	0	0
$729$	6650.15	0.337863
$730$	0	0
$731$	−4381.10	−0.221670
$732$	0	0
$733$	−20639.1	−1.04001	−0.520003	−	0.854165i	$-0.674069\pi$
$733$	−20639.1	−1.04001	−0.520003	+	0.854165i	$0.674069\pi$
$734$	0	0
$735$	−1758.82	−0.0882653
$736$	0	0
$737$	−14212.3	−0.710335
$738$	0	0
$739$	10399.6	0.517665	0.258833	−	0.965922i	$-0.416662\pi$
$739$	10399.6	0.517665	0.258833	+	0.965922i	$0.416662\pi$
$740$	0	0
$741$	7970.57	0.395150
$742$	0	0
$743$	40046.7	1.97735	0.988675	−	0.150074i	$-0.0479513\pi$
$743$	40046.7	1.97735	0.988675	+	0.150074i	$0.0479513\pi$
$744$	0	0
$745$	3424.92	0.168429
$746$	0	0
$747$	−10050.8	−0.492289
$748$	0	0
$749$	−5071.91	−0.247428
$750$	0	0
$751$	−26088.7	−1.26763	−0.633815	−	0.773485i	$-0.718512\pi$
$751$	−26088.7	−1.26763	−0.633815	+	0.773485i	$0.718512\pi$
$752$	0	0
$753$	−18650.5	−0.902604
$754$	0	0
$755$	−3011.38	−0.145160
$756$	0	0
$757$	−39040.1	−1.87442	−0.937211	−	0.348764i	$-0.886601\pi$
$757$	−39040.1	−1.87442	−0.937211	+	0.348764i	$0.886601\pi$
$758$	0	0
$759$	−2009.97	−0.0961230
$760$	0	0
$761$	19459.6	0.926950	0.463475	−	0.886110i	$-0.346602\pi$
$761$	19459.6	0.926950	0.463475	+	0.886110i	$0.346602\pi$
$762$	0	0
$763$	−15276.5	−0.724832
$764$	0	0
$765$	−559.442	−0.0264401
$766$	0	0
$767$	−9074.18	−0.427183
$768$	0	0
$769$	30321.5	1.42188	0.710938	−	0.703255i	$-0.248271\pi$
$769$	30321.5	1.42188	0.710938	+	0.703255i	$0.248271\pi$
$770$	0	0
$771$	35776.5	1.67115
$772$	0	0
$773$	5553.94	0.258423	0.129212	−	0.991617i	$-0.458755\pi$
$773$	5553.94	0.258423	0.129212	+	0.991617i	$0.458755\pi$
$774$	0	0
$775$	−2357.70	−0.109279
$776$	0	0
$777$	13576.3	0.626829
$778$	0	0
$779$	19211.4	0.883594
$780$	0	0
$781$	220.663	0.0101100
$782$	0	0
$783$	8691.18	0.396676
$784$	0	0
$785$	3313.86	0.150671
$786$	0	0
$787$	−10531.8	−0.477023	−0.238512	−	0.971140i	$-0.576660\pi$
$787$	−10531.8	−0.477023	−0.238512	+	0.971140i	$0.576660\pi$
$788$	0	0
$789$	19930.2	0.899280
$790$	0	0
$791$	15741.8	0.707602
$792$	0	0
$793$	3988.44	0.178605
$794$	0	0
$795$	3314.88	0.147883
$796$	0	0
$797$	15836.9	0.703856	0.351928	−	0.936027i	$-0.385526\pi$
$797$	15836.9	0.703856	0.351928	+	0.936027i	$0.385526\pi$
$798$	0	0
$799$	3019.23	0.133683
$800$	0	0
$801$	−14257.3	−0.628912
$802$	0	0
$803$	263.127	0.0115636
$804$	0	0
$805$	−773.827	−0.0338805
$806$	0	0
$807$	11600.6	0.506024
$808$	0	0
$809$	34948.2	1.51881	0.759404	−	0.650620i	$-0.225491\pi$
$809$	34948.2	1.51881	0.759404	+	0.650620i	$0.225491\pi$
$810$	0	0
$811$	30170.8	1.30634	0.653169	−	0.757212i	$-0.273439\pi$
$811$	30170.8	1.30634	0.653169	+	0.757212i	$0.273439\pi$
$812$	0	0
$813$	53810.6	2.32130
$814$	0	0
$815$	3269.11	0.140506
$816$	0	0
$817$	−9545.04	−0.408738
$818$	0	0
$819$	−4230.26	−0.180485
$820$	0	0
$821$	32731.1	1.39138	0.695691	−	0.718341i	$-0.255098\pi$
$821$	32731.1	1.39138	0.695691	+	0.718341i	$0.255098\pi$
$822$	0	0
$823$	39001.6	1.65190	0.825948	−	0.563747i	$-0.190641\pi$
$823$	39001.6	1.65190	0.825948	+	0.563747i	$0.190641\pi$
$824$	0	0
$825$	−10740.7	−0.453266
$826$	0	0
$827$	−16043.5	−0.674593	−0.337297	−	0.941398i	$-0.609512\pi$
$827$	−16043.5	−0.674593	−0.337297	+	0.941398i	$0.609512\pi$
$828$	0	0
$829$	−10794.1	−0.452227	−0.226113	−	0.974101i	$-0.572602\pi$
$829$	−10794.1	−0.452227	−0.226113	+	0.974101i	$0.572602\pi$
$830$	0	0
$831$	−19132.6	−0.798680
$832$	0	0
$833$	7017.25	0.291877
$834$	0	0
$835$	2555.68	0.105920
$836$	0	0
$837$	−1903.39	−0.0786030
$838$	0	0
$839$	34642.7	1.42550	0.712752	−	0.701416i	$-0.247448\pi$
$839$	34642.7	1.42550	0.712752	+	0.701416i	$0.247448\pi$
$840$	0	0
$841$	−16716.5	−0.685412
$842$	0	0
$843$	−38589.1	−1.57661
$844$	0	0
$845$	2774.58	0.112957
$846$	0	0
$847$	26255.8	1.06512
$848$	0	0
$849$	−13997.7	−0.565842
$850$	0	0
$851$	2182.38	0.0879094
$852$	0	0
$853$	20704.0	0.831057	0.415529	−	0.909580i	$-0.363597\pi$
$853$	20704.0	0.831057	0.415529	+	0.909580i	$0.363597\pi$
$854$	0	0
$855$	−1218.85	−0.0487528
$856$	0	0
$857$	24088.7	0.960155	0.480078	−	0.877226i	$-0.340609\pi$
$857$	24088.7	0.960155	0.480078	+	0.877226i	$0.340609\pi$
$858$	0	0
$859$	6648.01	0.264060	0.132030	−	0.991246i	$-0.457851\pi$
$859$	6648.01	0.264060	0.132030	+	0.991246i	$0.457851\pi$
$860$	0	0
$861$	−35503.6	−1.40530
$862$	0	0
$863$	−12503.2	−0.493181	−0.246591	−	0.969120i	$-0.579310\pi$
$863$	−12503.2	−0.493181	−0.246591	+	0.969120i	$0.579310\pi$
$864$	0	0
$865$	299.259	0.0117631
$866$	0	0
$867$	−22465.1	−0.879993
$868$	0	0
$869$	10716.1	0.418318
$870$	0	0
$871$	−16742.7	−0.651326
$872$	0	0
$873$	−6591.00	−0.255523
$874$	0	0
$875$	−8340.70	−0.322248
$876$	0	0
$877$	−9041.20	−0.348118	−0.174059	−	0.984735i	$-0.555688\pi$
$877$	−9041.20	−0.348118	−0.174059	+	0.984735i	$0.555688\pi$
$878$	0	0
$879$	34012.4	1.30513
$880$	0	0
$881$	−33837.2	−1.29399	−0.646995	−	0.762494i	$-0.723975\pi$
$881$	−33837.2	−1.29399	−0.646995	+	0.762494i	$0.723975\pi$
$882$	0	0
$883$	−24724.4	−0.942290	−0.471145	−	0.882056i	$-0.656159\pi$
$883$	−24724.4	−0.942290	−0.471145	+	0.882056i	$0.656159\pi$
$884$	0	0
$885$	4831.73	0.183522
$886$	0	0
$887$	9427.81	0.356882	0.178441	−	0.983951i	$-0.442895\pi$
$887$	9427.81	0.356882	0.178441	+	0.983951i	$0.442895\pi$
$888$	0	0
$889$	30615.6	1.15502
$890$	0	0
$891$	−12841.5	−0.482837
$892$	0	0
$893$	6577.94	0.246498
$894$	0	0
$895$	−6630.96	−0.247652
$896$	0	0
$897$	−2367.84	−0.0881379
$898$	0	0
$899$	−1680.29	−0.0623368
$900$	0	0
$901$	−13225.6	−0.489020
$902$	0	0
$903$	17639.7	0.650069
$904$	0	0
$905$	246.014	0.00903622
$906$	0	0
$907$	−41128.1	−1.50566	−0.752832	−	0.658213i	$-0.771313\pi$
$907$	−41128.1	−1.50566	−0.752832	+	0.658213i	$0.771313\pi$
$908$	0	0
$909$	−4367.01	−0.159345
$910$	0	0
$911$	16887.9	0.614184	0.307092	−	0.951680i	$-0.400644\pi$
$911$	16887.9	0.614184	0.307092	+	0.951680i	$0.400644\pi$
$912$	0	0
$913$	−13119.5	−0.475566
$914$	0	0
$915$	−2123.73	−0.0767303
$916$	0	0
$917$	1831.92	0.0659711
$918$	0	0
$919$	−27738.7	−0.995663	−0.497831	−	0.867274i	$-0.665870\pi$
$919$	−27738.7	−0.995663	−0.497831	+	0.867274i	$0.665870\pi$
$920$	0	0
$921$	−21966.9	−0.785921
$922$	0	0
$923$	259.951	0.00927018
$924$	0	0
$925$	11662.0	0.414535
$926$	0	0
$927$	−10958.0	−0.388249
$928$	0	0
$929$	46867.7	1.65520	0.827600	−	0.561319i	$-0.189706\pi$
$929$	46867.7	1.65520	0.827600	+	0.561319i	$0.189706\pi$
$930$	0	0
$931$	15288.4	0.538192
$932$	0	0
$933$	−24568.6	−0.862100
$934$	0	0
$935$	−730.249	−0.0255419
$936$	0	0
$937$	−17889.5	−0.623718	−0.311859	−	0.950128i	$-0.600952\pi$
$937$	−17889.5	−0.623718	−0.311859	+	0.950128i	$0.600952\pi$
$938$	0	0
$939$	−10772.3	−0.374378
$940$	0	0
$941$	−19697.0	−0.682363	−0.341182	−	0.939997i	$-0.610827\pi$
$941$	−19697.0	−0.682363	−0.341182	+	0.939997i	$0.610827\pi$
$942$	0	0
$943$	−5707.18	−0.197085
$944$	0	0
$945$	−3338.31	−0.114916
$946$	0	0
$947$	15836.0	0.543401	0.271700	−	0.962382i	$-0.412414\pi$
$947$	15836.0	0.543401	0.271700	+	0.962382i	$0.412414\pi$
$948$	0	0
$949$	309.975	0.0106030
$950$	0	0
$951$	−54126.7	−1.84561
$952$	0	0
$953$	−4023.05	−0.136746	−0.0683732	−	0.997660i	$-0.521781\pi$
$953$	−4023.05	−0.136746	−0.0683732	+	0.997660i	$0.521781\pi$
$954$	0	0
$955$	613.388	0.0207841
$956$	0	0
$957$	−7654.73	−0.258560
$958$	0	0
$959$	−32905.4	−1.10800
$960$	0	0
$961$	−29423.0	−0.987648
$962$	0	0
$963$	2373.23	0.0794145
$964$	0	0
$965$	−422.011	−0.0140777
$966$	0	0
$967$	−14863.9	−0.494304	−0.247152	−	0.968977i	$-0.579495\pi$
$967$	−14863.9	−0.494304	−0.247152	+	0.968977i	$0.579495\pi$
$968$	0	0
$969$	16932.9	0.561365
$970$	0	0
$971$	−32532.6	−1.07520	−0.537601	−	0.843199i	$-0.680669\pi$
$971$	−32532.6	−1.07520	−0.537601	+	0.843199i	$0.680669\pi$
$972$	0	0
$973$	−32032.1	−1.05540
$974$	0	0
$975$	−12653.1	−0.415612
$976$	0	0
$977$	20908.4	0.684667	0.342334	−	0.939578i	$-0.388783\pi$
$977$	20908.4	0.684667	0.342334	+	0.939578i	$0.388783\pi$
$978$	0	0
$979$	−18610.4	−0.607548
$980$	0	0
$981$	7148.11	0.232642
$982$	0	0
$983$	43012.9	1.39562	0.697812	−	0.716281i	$-0.254157\pi$
$983$	43012.9	1.39562	0.697812	+	0.716281i	$0.254157\pi$
$984$	0	0
$985$	6312.71	0.204203
$986$	0	0
$987$	−12156.4	−0.392037
$988$	0	0
$989$	2835.57	0.0911686
$990$	0	0
$991$	−41354.1	−1.32559	−0.662794	−	0.748802i	$-0.730630\pi$
$991$	−41354.1	−1.32559	−0.662794	+	0.748802i	$0.730630\pi$
$992$	0	0
$993$	6315.85	0.201840
$994$	0	0
$995$	−6394.02	−0.203723
$996$	0	0
$997$	27650.9	0.878346	0.439173	−	0.898402i	$-0.355271\pi$
$997$	27650.9	0.878346	0.439173	+	0.898402i	$0.355271\pi$
$998$	0	0
$999$	9414.83	0.298170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.be.1.3		4
4.3	odd	2		1472.4.a.z.1.2		4
8.3	odd	2		184.4.a.f.1.3	✓	4
8.5	even	2		368.4.a.m.1.2		4
24.11	even	2		1656.4.a.l.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.a.f.1.3	✓	4	8.3	odd	2
368.4.a.m.1.2		4	8.5	even	2
1472.4.a.z.1.2		4	4.3	odd	2
1472.4.a.be.1.3		4	1.1	even	1	trivial
1656.4.a.l.1.2		4	24.11	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+6.15452 q^{3} -1.44721 q^{5} -23.2480 q^{7} +10.8781 q^{9} +14.1993 q^{11} +16.7274 q^{13} -8.90687 q^{15} +35.5362 q^{17} +77.4222 q^{19} -143.080 q^{21} -23.0000 q^{23} -122.906 q^{25} -99.2226 q^{27} -87.5927 q^{29} +19.1830 q^{31} +87.3901 q^{33} +33.6447 q^{35} -94.8859 q^{37} +102.949 q^{39} +248.138 q^{41} -123.286 q^{43} -15.7429 q^{45} +84.9619 q^{47} +197.468 q^{49} +218.708 q^{51} -372.171 q^{53} -20.5494 q^{55} +476.496 q^{57} -542.472 q^{59} +238.437 q^{61} -252.893 q^{63} -24.2081 q^{65} -1000.91 q^{67} -141.554 q^{69} +15.5404 q^{71} +18.5309 q^{73} -756.424 q^{75} -330.106 q^{77} +754.689 q^{79} -904.376 q^{81} -923.951 q^{83} -51.4283 q^{85} -539.091 q^{87} -1310.65 q^{89} -388.879 q^{91} +118.062 q^{93} -112.046 q^{95} -605.897 q^{97} +154.462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} + 2 q^{5} + 32 q^{7} + 79 q^{9} + 56 q^{11} - 139 q^{13} - 230 q^{15} - 6 q^{17} - 108 q^{21} - 92 q^{23} + 160 q^{25} + 119 q^{27} - 489 q^{29} + 127 q^{31} + 438 q^{33} - 408 q^{35} - 826 q^{37}+ \cdots - 1054 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 + 2 * q^5 + 32 * q^7 + 79 * q^9 + 56 * q^11 - 139 * q^13 - 230 * q^15 - 6 * q^17 - 108 * q^21 - 92 * q^23 + 160 * q^25 + 119 * q^27 - 489 * q^29 + 127 * q^31 + 438 * q^33 - 408 * q^35 - 826 * q^37 - 323 * q^39 + 91 * q^41 - 762 * q^43 - 1240 * q^45 - 397 * q^47 + 748 * q^49 - 1132 * q^51 + 430 * q^53 + 380 * q^55 - 918 * q^57 + 340 * q^59 - 714 * q^61 + 2348 * q^63 - 194 * q^65 - 1832 * q^67 - 115 * q^69 + 1047 * q^71 - 27 * q^73 + 1539 * q^75 - 568 * q^77 + 794 * q^79 - 116 * q^81 - 502 * q^83 + 3752 * q^85 + 2191 * q^87 + 6 * q^89 - 2532 * q^91 + 217 * q^93 - 172 * q^95 - 1970 * q^97 - 1054 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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