LMFDB - Embedded newform 672.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(1,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{137}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 34 $ x^2 - x - 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.35235$ of defining polynomial
Character	$\chi$	$=$	672.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +16.7047 q^{5} +7.00000 q^{7} +9.00000 q^{9} +22.7047 q^{11} -11.4094 q^{13} +50.1141 q^{15} +3.29530 q^{17} +62.8188 q^{19} +21.0000 q^{21} +76.1141 q^{23} +154.047 q^{25} +27.0000 q^{27} -216.094 q^{29} +149.638 q^{31} +68.1141 q^{33} +116.933 q^{35} +187.866 q^{37} -34.2282 q^{39} -23.2953 q^{41} -214.094 q^{43} +150.342 q^{45} -187.960 q^{47} +49.0000 q^{49} +9.88590 q^{51} -122.685 q^{53} +379.275 q^{55} +188.456 q^{57} +225.409 q^{59} -546.188 q^{61} +63.0000 q^{63} -190.591 q^{65} -353.141 q^{67} +228.342 q^{69} +465.523 q^{71} +430.416 q^{73} +462.141 q^{75} +158.933 q^{77} +532.497 q^{79} +81.0000 q^{81} +185.181 q^{83} +55.0470 q^{85} -648.282 q^{87} -1597.31 q^{89} -79.8658 q^{91} +448.913 q^{93} +1049.37 q^{95} -1605.69 q^{97} +204.342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} + 24 q^{13} + 30 q^{15} + 30 q^{17} + 32 q^{19} + 42 q^{21} + 82 q^{23} + 74 q^{25} + 54 q^{27} + 36 q^{29} + 112 q^{31} + 66 q^{33} + 70 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 + 24 * q^13 + 30 * q^15 + 30 * q^17 + 32 * q^19 + 42 * q^21 + 82 * q^23 + 74 * q^25 + 54 * q^27 + 36 * q^29 + 112 * q^31 + 66 * q^33 + 70 * q^35 + 48 * q^37 + 72 * q^39 - 70 * q^41 + 40 * q^43 + 90 * q^45 + 420 * q^47 + 98 * q^49 + 90 * q^51 + 176 * q^53 + 384 * q^55 + 96 * q^57 + 404 * q^59 - 156 * q^61 + 126 * q^63 - 428 * q^65 - 4 * q^67 + 246 * q^69 + 814 * q^71 - 216 * q^73 + 222 * q^75 + 154 * q^77 + 1580 * q^79 + 162 * q^81 + 464 * q^83 - 124 * q^85 + 108 * q^87 - 1158 * q^89 + 168 * q^91 + 336 * q^93 + 1256 * q^95 - 1760 * q^97 + 198 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	16.7047	1.49411	0.747057	−	0.664760i	$-0.231466\pi$
$5$	16.7047	1.49411	0.747057	+	0.664760i	$0.231466\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	22.7047	0.622339	0.311169	−	0.950354i	$-0.399279\pi$
$11$	22.7047	0.622339	0.311169	+	0.950354i	$0.399279\pi$
$12$	0	0
$13$	−11.4094	−0.243415	−0.121708	−	0.992566i	$-0.538837\pi$
$13$	−11.4094	−0.243415	−0.121708	+	0.992566i	$0.538837\pi$
$14$	0	0
$15$	50.1141	0.862627
$16$	0	0
$17$	3.29530	0.0470134	0.0235067	−	0.999724i	$-0.492517\pi$
$17$	3.29530	0.0470134	0.0235067	+	0.999724i	$0.492517\pi$
$18$	0	0
$19$	62.8188	0.758506	0.379253	−	0.925293i	$-0.376181\pi$
$19$	62.8188	0.758506	0.379253	+	0.925293i	$0.376181\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	76.1141	0.690039	0.345019	−	0.938596i	$-0.387872\pi$
$23$	76.1141	0.690039	0.345019	+	0.938596i	$0.387872\pi$
$24$	0	0
$25$	154.047	1.23238
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−216.094	−1.38371	−0.691856	−	0.722036i	$-0.743207\pi$
$29$	−216.094	−1.38371	−0.691856	+	0.722036i	$0.743207\pi$
$30$	0	0
$31$	149.638	0.866958	0.433479	−	0.901164i	$-0.357286\pi$
$31$	149.638	0.866958	0.433479	+	0.901164i	$0.357286\pi$
$32$	0	0
$33$	68.1141	0.359307
$34$	0	0
$35$	116.933	0.564722
$36$	0	0
$37$	187.866	0.834728	0.417364	−	0.908739i	$-0.362954\pi$
$37$	187.866	0.834728	0.417364	+	0.908739i	$0.362954\pi$
$38$	0	0
$39$	−34.2282	−0.140536
$40$	0	0
$41$	−23.2953	−0.0887345	−0.0443673	−	0.999015i	$-0.514127\pi$
$41$	−23.2953	−0.0887345	−0.0443673	+	0.999015i	$0.514127\pi$
$42$	0	0
$43$	−214.094	−0.759280	−0.379640	−	0.925134i	$-0.623952\pi$
$43$	−214.094	−0.759280	−0.379640	+	0.925134i	$0.623952\pi$
$44$	0	0
$45$	150.342	0.498038
$46$	0	0
$47$	−187.960	−0.583335	−0.291668	−	0.956520i	$-0.594210\pi$
$47$	−187.960	−0.583335	−0.291668	+	0.956520i	$0.594210\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	9.88590	0.0271432
$52$	0	0
$53$	−122.685	−0.317963	−0.158981	−	0.987282i	$-0.550821\pi$
$53$	−122.685	−0.317963	−0.158981	+	0.987282i	$0.550821\pi$
$54$	0	0
$55$	379.275	0.929845
$56$	0	0
$57$	188.456	0.437924
$58$	0	0
$59$	225.409	0.497387	0.248693	−	0.968582i	$-0.419999\pi$
$59$	225.409	0.497387	0.248693	+	0.968582i	$0.419999\pi$
$60$	0	0
$61$	−546.188	−1.14643	−0.573215	−	0.819405i	$-0.694304\pi$
$61$	−546.188	−1.14643	−0.573215	+	0.819405i	$0.694304\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−190.591	−0.363690
$66$	0	0
$67$	−353.141	−0.643926	−0.321963	−	0.946752i	$-0.604343\pi$
$67$	−353.141	−0.643926	−0.321963	+	0.946752i	$0.604343\pi$
$68$	0	0
$69$	228.342	0.398394
$70$	0	0
$71$	465.523	0.778133	0.389067	−	0.921210i	$-0.372798\pi$
$71$	465.523	0.778133	0.389067	+	0.921210i	$0.372798\pi$
$72$	0	0
$73$	430.416	0.690088	0.345044	−	0.938587i	$-0.387864\pi$
$73$	430.416	0.690088	0.345044	+	0.938587i	$0.387864\pi$
$74$	0	0
$75$	462.141	0.711513
$76$	0	0
$77$	158.933	0.235222
$78$	0	0
$79$	532.497	0.758361	0.379181	−	0.925323i	$-0.376206\pi$
$79$	532.497	0.758361	0.379181	+	0.925323i	$0.376206\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	185.181	0.244895	0.122447	−	0.992475i	$-0.460926\pi$
$83$	185.181	0.244895	0.122447	+	0.992475i	$0.460926\pi$
$84$	0	0
$85$	55.0470	0.0702434
$86$	0	0
$87$	−648.282	−0.798886
$88$	0	0
$89$	−1597.31	−1.90241	−0.951205	−	0.308561i	$-0.900153\pi$
$89$	−1597.31	−1.90241	−0.951205	+	0.308561i	$0.900153\pi$
$90$	0	0
$91$	−79.8658	−0.0920023
$92$	0	0
$93$	448.913	0.500539
$94$	0	0
$95$	1049.37	1.13329
$96$	0	0
$97$	−1605.69	−1.68076	−0.840378	−	0.542001i	$-0.817667\pi$
$97$	−1605.69	−1.68076	−0.840378	+	0.542001i	$0.817667\pi$
$98$	0	0
$99$	204.342	0.207446
$100$	0	0
$101$	1359.94	1.33979	0.669896	−	0.742455i	$-0.266339\pi$
$101$	1359.94	1.33979	0.669896	+	0.742455i	$0.266339\pi$
$102$	0	0
$103$	1821.56	1.74256	0.871278	−	0.490789i	$-0.163291\pi$
$103$	1821.56	1.74256	0.871278	+	0.490789i	$0.163291\pi$
$104$	0	0
$105$	350.799	0.326042
$106$	0	0
$107$	1440.53	1.30151	0.650754	−	0.759289i	$-0.274453\pi$
$107$	1440.53	1.30151	0.650754	+	0.759289i	$0.274453\pi$
$108$	0	0
$109$	884.014	0.776818	0.388409	−	0.921487i	$-0.373025\pi$
$109$	884.014	0.776818	0.388409	+	0.921487i	$0.373025\pi$
$110$	0	0
$111$	563.597	0.481931
$112$	0	0
$113$	−1493.46	−1.24330	−0.621651	−	0.783294i	$-0.713538\pi$
$113$	−1493.46	−1.24330	−0.621651	+	0.783294i	$0.713538\pi$
$114$	0	0
$115$	1271.46	1.03100
$116$	0	0
$117$	−102.685	−0.0811384
$118$	0	0
$119$	23.0671	0.0177694
$120$	0	0
$121$	−815.497	−0.612695
$122$	0	0
$123$	−69.8859	−0.0512309
$124$	0	0
$125$	485.221	0.347196
$126$	0	0
$127$	1421.41	0.993147	0.496574	−	0.867995i	$-0.334591\pi$
$127$	1421.41	0.993147	0.496574	+	0.867995i	$0.334591\pi$
$128$	0	0
$129$	−642.282	−0.438370
$130$	0	0
$131$	755.087	0.503605	0.251803	−	0.967779i	$-0.418977\pi$
$131$	755.087	0.503605	0.251803	+	0.967779i	$0.418977\pi$
$132$	0	0
$133$	439.732	0.286688
$134$	0	0
$135$	451.027	0.287542
$136$	0	0
$137$	1618.98	1.00963	0.504813	−	0.863229i	$-0.331561\pi$
$137$	1618.98	1.00963	0.504813	+	0.863229i	$0.331561\pi$
$138$	0	0
$139$	−1458.93	−0.890248	−0.445124	−	0.895469i	$-0.646840\pi$
$139$	−1458.93	−0.890248	−0.445124	+	0.895469i	$0.646840\pi$
$140$	0	0
$141$	−563.879	−0.336789
$142$	0	0
$143$	−259.047	−0.151487
$144$	0	0
$145$	−3609.79	−2.06742
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	3270.07	1.79795	0.898974	−	0.438001i	$-0.144313\pi$
$149$	3270.07	1.79795	0.898974	+	0.438001i	$0.144313\pi$
$150$	0	0
$151$	1198.01	0.645649	0.322824	−	0.946459i	$-0.395368\pi$
$151$	1198.01	0.645649	0.322824	+	0.946459i	$0.395368\pi$
$152$	0	0
$153$	29.6577	0.0156711
$154$	0	0
$155$	2499.65	1.29533
$156$	0	0
$157$	−1255.75	−0.638340	−0.319170	−	0.947697i	$-0.603404\pi$
$157$	−1255.75	−0.638340	−0.319170	+	0.947697i	$0.603404\pi$
$158$	0	0
$159$	−368.054	−0.183576
$160$	0	0
$161$	532.799	0.260810
$162$	0	0
$163$	1208.23	0.580587	0.290294	−	0.956938i	$-0.406247\pi$
$163$	1208.23	0.580587	0.290294	+	0.956938i	$0.406247\pi$
$164$	0	0
$165$	1137.83	0.536846
$166$	0	0
$167$	−2785.14	−1.29054	−0.645271	−	0.763953i	$-0.723256\pi$
$167$	−2785.14	−1.29054	−0.645271	+	0.763953i	$0.723256\pi$
$168$	0	0
$169$	−2066.83	−0.940749
$170$	0	0
$171$	565.369	0.252835
$172$	0	0
$173$	875.832	0.384903	0.192452	−	0.981306i	$-0.438356\pi$
$173$	875.832	0.384903	0.192452	+	0.981306i	$0.438356\pi$
$174$	0	0
$175$	1078.33	0.465794
$176$	0	0
$177$	676.228	0.287166
$178$	0	0
$179$	−1148.72	−0.479661	−0.239830	−	0.970815i	$-0.577092\pi$
$179$	−1148.72	−0.479661	−0.239830	+	0.970815i	$0.577092\pi$
$180$	0	0
$181$	1738.15	0.713787	0.356894	−	0.934145i	$-0.383836\pi$
$181$	1738.15	0.713787	0.356894	+	0.934145i	$0.383836\pi$
$182$	0	0
$183$	−1638.56	−0.661891
$184$	0	0
$185$	3138.24	1.24718
$186$	0	0
$187$	74.8188	0.0292582
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−3779.34	−1.43174	−0.715872	−	0.698231i	$-0.753971\pi$
$191$	−3779.34	−1.43174	−0.715872	+	0.698231i	$0.753971\pi$
$192$	0	0
$193$	−3966.15	−1.47922	−0.739610	−	0.673035i	$-0.764990\pi$
$193$	−3966.15	−1.47922	−0.739610	+	0.673035i	$0.764990\pi$
$194$	0	0
$195$	−571.772	−0.209977
$196$	0	0
$197$	2593.05	0.937802	0.468901	−	0.883251i	$-0.344650\pi$
$197$	2593.05	0.937802	0.468901	+	0.883251i	$0.344650\pi$
$198$	0	0
$199$	−4407.76	−1.57014	−0.785070	−	0.619407i	$-0.787373\pi$
$199$	−4407.76	−1.57014	−0.785070	+	0.619407i	$0.787373\pi$
$200$	0	0
$201$	−1059.42	−0.371771
$202$	0	0
$203$	−1512.66	−0.522994
$204$	0	0
$205$	−389.141	−0.132579
$206$	0	0
$207$	685.027	0.230013
$208$	0	0
$209$	1426.28	0.472048
$210$	0	0
$211$	−3745.13	−1.22192	−0.610960	−	0.791661i	$-0.709217\pi$
$211$	−3745.13	−1.22192	−0.610960	+	0.791661i	$0.709217\pi$
$212$	0	0
$213$	1396.57	0.449256
$214$	0	0
$215$	−3576.38	−1.13445
$216$	0	0
$217$	1047.46	0.327679
$218$	0	0
$219$	1291.25	0.398422
$220$	0	0
$221$	−37.5974	−0.0114438
$222$	0	0
$223$	−1199.17	−0.360100	−0.180050	−	0.983657i	$-0.557626\pi$
$223$	−1199.17	−0.360100	−0.180050	+	0.983657i	$0.557626\pi$
$224$	0	0
$225$	1386.42	0.410792
$226$	0	0
$227$	4378.08	1.28010	0.640052	−	0.768332i	$-0.278913\pi$
$227$	4378.08	1.28010	0.640052	+	0.768332i	$0.278913\pi$
$228$	0	0
$229$	−5518.82	−1.59255	−0.796275	−	0.604935i	$-0.793199\pi$
$229$	−5518.82	−1.59255	−0.796275	+	0.604935i	$0.793199\pi$
$230$	0	0
$231$	476.799	0.135805
$232$	0	0
$233$	2511.79	0.706234	0.353117	−	0.935579i	$-0.385122\pi$
$233$	2511.79	0.706234	0.353117	+	0.935579i	$0.385122\pi$
$234$	0	0
$235$	−3139.81	−0.871569
$236$	0	0
$237$	1597.49	0.437840
$238$	0	0
$239$	7146.56	1.93419	0.967097	−	0.254408i	$-0.0818805\pi$
$239$	7146.56	1.93419	0.967097	+	0.254408i	$0.0818805\pi$
$240$	0	0
$241$	−332.563	−0.0888892	−0.0444446	−	0.999012i	$-0.514152\pi$
$241$	−332.563	−0.0888892	−0.0444446	+	0.999012i	$0.514152\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	818.530	0.213445
$246$	0	0
$247$	−716.725	−0.184632
$248$	0	0
$249$	555.544	0.141390
$250$	0	0
$251$	−4833.25	−1.21543	−0.607713	−	0.794157i	$-0.707913\pi$
$251$	−4833.25	−1.21543	−0.607713	+	0.794157i	$0.707913\pi$
$252$	0	0
$253$	1728.15	0.429438
$254$	0	0
$255$	165.141	0.0405550
$256$	0	0
$257$	−1372.18	−0.333052	−0.166526	−	0.986037i	$-0.553255\pi$
$257$	−1372.18	−0.333052	−0.166526	+	0.986037i	$0.553255\pi$
$258$	0	0
$259$	1315.06	0.315498
$260$	0	0
$261$	−1944.85	−0.461237
$262$	0	0
$263$	−1170.60	−0.274457	−0.137228	−	0.990539i	$-0.543819\pi$
$263$	−1170.60	−0.274457	−0.137228	+	0.990539i	$0.543819\pi$
$264$	0	0
$265$	−2049.41	−0.475073
$266$	0	0
$267$	−4791.93	−1.09836
$268$	0	0
$269$	4260.36	0.965645	0.482822	−	0.875718i	$-0.339612\pi$
$269$	4260.36	0.965645	0.482822	+	0.875718i	$0.339612\pi$
$270$	0	0
$271$	−2574.85	−0.577161	−0.288581	−	0.957456i	$-0.593183\pi$
$271$	−2574.85	−0.577161	−0.288581	+	0.957456i	$0.593183\pi$
$272$	0	0
$273$	−239.597	−0.0531176
$274$	0	0
$275$	3497.59	0.766955
$276$	0	0
$277$	−6453.13	−1.39975	−0.699875	−	0.714265i	$-0.746761\pi$
$277$	−6453.13	−1.39975	−0.699875	+	0.714265i	$0.746761\pi$
$278$	0	0
$279$	1346.74	0.288986
$280$	0	0
$281$	349.664	0.0742321	0.0371160	−	0.999311i	$-0.488183\pi$
$281$	349.664	0.0742321	0.0371160	+	0.999311i	$0.488183\pi$
$282$	0	0
$283$	−699.865	−0.147006	−0.0735029	−	0.997295i	$-0.523418\pi$
$283$	−699.865	−0.147006	−0.0735029	+	0.997295i	$0.523418\pi$
$284$	0	0
$285$	3148.11	0.654308
$286$	0	0
$287$	−163.067	−0.0335385
$288$	0	0
$289$	−4902.14	−0.997790
$290$	0	0
$291$	−4817.07	−0.970385
$292$	0	0
$293$	162.087	0.0323182	0.0161591	−	0.999869i	$-0.494856\pi$
$293$	162.087	0.0323182	0.0161591	+	0.999869i	$0.494856\pi$
$294$	0	0
$295$	3765.40	0.743152
$296$	0	0
$297$	613.027	0.119769
$298$	0	0
$299$	−868.416	−0.167966
$300$	0	0
$301$	−1498.66	−0.286981
$302$	0	0
$303$	4079.82	0.773530
$304$	0	0
$305$	−9123.91	−1.71290
$306$	0	0
$307$	2185.16	0.406233	0.203116	−	0.979155i	$-0.434893\pi$
$307$	2185.16	0.406233	0.203116	+	0.979155i	$0.434893\pi$
$308$	0	0
$309$	5464.67	1.00607
$310$	0	0
$311$	−4763.42	−0.868517	−0.434259	−	0.900788i	$-0.642990\pi$
$311$	−4763.42	−0.868517	−0.434259	+	0.900788i	$0.642990\pi$
$312$	0	0
$313$	−1080.84	−0.195185	−0.0975926	−	0.995226i	$-0.531114\pi$
$313$	−1080.84	−0.195185	−0.0975926	+	0.995226i	$0.531114\pi$
$314$	0	0
$315$	1052.40	0.188241
$316$	0	0
$317$	−5832.91	−1.03347	−0.516734	−	0.856146i	$-0.672852\pi$
$317$	−5832.91	−1.03347	−0.516734	+	0.856146i	$0.672852\pi$
$318$	0	0
$319$	−4906.35	−0.861137
$320$	0	0
$321$	4321.59	0.751426
$322$	0	0
$323$	207.007	0.0356600
$324$	0	0
$325$	−1757.58	−0.299979
$326$	0	0
$327$	2652.04	0.448496
$328$	0	0
$329$	−1315.72	−0.220480
$330$	0	0
$331$	−3483.11	−0.578397	−0.289198	−	0.957269i	$-0.593389\pi$
$331$	−3483.11	−0.578397	−0.289198	+	0.957269i	$0.593389\pi$
$332$	0	0
$333$	1690.79	0.278243
$334$	0	0
$335$	−5899.11	−0.962099
$336$	0	0
$337$	−6625.03	−1.07089	−0.535443	−	0.844571i	$-0.679855\pi$
$337$	−6625.03	−1.07089	−0.535443	+	0.844571i	$0.679855\pi$
$338$	0	0
$339$	−4480.39	−0.717821
$340$	0	0
$341$	3397.48	0.539542
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	3814.39	0.595246
$346$	0	0
$347$	−5567.83	−0.861374	−0.430687	−	0.902501i	$-0.641729\pi$
$347$	−5567.83	−0.861374	−0.430687	+	0.902501i	$0.641729\pi$
$348$	0	0
$349$	4552.55	0.698259	0.349130	−	0.937074i	$-0.386477\pi$
$349$	4552.55	0.698259	0.349130	+	0.937074i	$0.386477\pi$
$350$	0	0
$351$	−308.054	−0.0468453
$352$	0	0
$353$	−4627.15	−0.697672	−0.348836	−	0.937184i	$-0.613423\pi$
$353$	−4627.15	−0.697672	−0.348836	+	0.937184i	$0.613423\pi$
$354$	0	0
$355$	7776.43	1.16262
$356$	0	0
$357$	69.2013	0.0102592
$358$	0	0
$359$	−8347.95	−1.22726	−0.613632	−	0.789592i	$-0.710292\pi$
$359$	−8347.95	−1.22726	−0.613632	+	0.789592i	$0.710292\pi$
$360$	0	0
$361$	−2912.80	−0.424668
$362$	0	0
$363$	−2446.49	−0.353739
$364$	0	0
$365$	7189.97	1.03107
$366$	0	0
$367$	−11015.9	−1.56683	−0.783417	−	0.621497i	$-0.786525\pi$
$367$	−11015.9	−1.56683	−0.783417	+	0.621497i	$0.786525\pi$
$368$	0	0
$369$	−209.658	−0.0295782
$370$	0	0
$371$	−858.792	−0.120179
$372$	0	0
$373$	−3459.93	−0.480291	−0.240145	−	0.970737i	$-0.577195\pi$
$373$	−3459.93	−0.480291	−0.240145	+	0.970737i	$0.577195\pi$
$374$	0	0
$375$	1455.66	0.200454
$376$	0	0
$377$	2465.50	0.336817
$378$	0	0
$379$	−13137.3	−1.78053	−0.890264	−	0.455445i	$-0.849480\pi$
$379$	−13137.3	−1.78053	−0.890264	+	0.455445i	$0.849480\pi$
$380$	0	0
$381$	4264.23	0.573394
$382$	0	0
$383$	−11542.6	−1.53995	−0.769973	−	0.638076i	$-0.779731\pi$
$383$	−11542.6	−1.53995	−0.769973	+	0.638076i	$0.779731\pi$
$384$	0	0
$385$	2654.93	0.351448
$386$	0	0
$387$	−1926.85	−0.253093
$388$	0	0
$389$	−13580.9	−1.77012	−0.885061	−	0.465475i	$-0.845884\pi$
$389$	−13580.9	−1.77012	−0.885061	+	0.465475i	$0.845884\pi$
$390$	0	0
$391$	250.819	0.0324411
$392$	0	0
$393$	2265.26	0.290757
$394$	0	0
$395$	8895.20	1.13308
$396$	0	0
$397$	15560.6	1.96717	0.983584	−	0.180450i	$-0.0577555\pi$
$397$	15560.6	1.96717	0.983584	+	0.180450i	$0.0577555\pi$
$398$	0	0
$399$	1319.19	0.165520
$400$	0	0
$401$	320.322	0.0398906	0.0199453	−	0.999801i	$-0.493651\pi$
$401$	320.322	0.0398906	0.0199453	+	0.999801i	$0.493651\pi$
$402$	0	0
$403$	−1707.28	−0.211031
$404$	0	0
$405$	1353.08	0.166013
$406$	0	0
$407$	4265.44	0.519484
$408$	0	0
$409$	−2303.79	−0.278521	−0.139260	−	0.990256i	$-0.544472\pi$
$409$	−2303.79	−0.278521	−0.139260	+	0.990256i	$0.544472\pi$
$410$	0	0
$411$	4856.94	0.582908
$412$	0	0
$413$	1577.87	0.187994
$414$	0	0
$415$	3093.40	0.365901
$416$	0	0
$417$	−4376.78	−0.513985
$418$	0	0
$419$	−9770.19	−1.13915	−0.569576	−	0.821938i	$-0.692893\pi$
$419$	−9770.19	−1.13915	−0.569576	+	0.821938i	$0.692893\pi$
$420$	0	0
$421$	−918.792	−0.106364	−0.0531819	−	0.998585i	$-0.516936\pi$
$421$	−918.792	−0.106364	−0.0531819	+	0.998585i	$0.516936\pi$
$422$	0	0
$423$	−1691.64	−0.194445
$424$	0	0
$425$	507.631	0.0579382
$426$	0	0
$427$	−3823.32	−0.433310
$428$	0	0
$429$	−777.141	−0.0874609
$430$	0	0
$431$	12570.7	1.40489	0.702446	−	0.711737i	$-0.252091\pi$
$431$	12570.7	1.40489	0.702446	+	0.711737i	$0.252091\pi$
$432$	0	0
$433$	8107.80	0.899852	0.449926	−	0.893066i	$-0.351450\pi$
$433$	8107.80	0.899852	0.449926	+	0.893066i	$0.351450\pi$
$434$	0	0
$435$	−10829.4	−1.19363
$436$	0	0
$437$	4781.40	0.523399
$438$	0	0
$439$	12395.2	1.34758	0.673792	−	0.738921i	$-0.264664\pi$
$439$	12395.2	1.34758	0.673792	+	0.738921i	$0.264664\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	934.503	0.100225	0.0501124	−	0.998744i	$-0.484042\pi$
$443$	934.503	0.100225	0.0501124	+	0.998744i	$0.484042\pi$
$444$	0	0
$445$	−26682.6	−2.84242
$446$	0	0
$447$	9810.20	1.03805
$448$	0	0
$449$	13725.7	1.44266	0.721331	−	0.692591i	$-0.243531\pi$
$449$	13725.7	1.44266	0.721331	+	0.692591i	$0.243531\pi$
$450$	0	0
$451$	−528.913	−0.0552229
$452$	0	0
$453$	3594.04	0.372765
$454$	0	0
$455$	−1334.13	−0.137462
$456$	0	0
$457$	−15676.9	−1.60467	−0.802337	−	0.596872i	$-0.796410\pi$
$457$	−15676.9	−1.60467	−0.802337	+	0.596872i	$0.796410\pi$
$458$	0	0
$459$	88.9731	0.00904773
$460$	0	0
$461$	19210.4	1.94082	0.970410	−	0.241463i	$-0.0776273\pi$
$461$	19210.4	1.94082	0.970410	+	0.241463i	$0.0776273\pi$
$462$	0	0
$463$	−2935.18	−0.294621	−0.147310	−	0.989090i	$-0.547062\pi$
$463$	−2935.18	−0.294621	−0.147310	+	0.989090i	$0.547062\pi$
$464$	0	0
$465$	7498.95	0.747862
$466$	0	0
$467$	−10685.1	−1.05878	−0.529389	−	0.848379i	$-0.677579\pi$
$467$	−10685.1	−1.05878	−0.529389	+	0.848379i	$0.677579\pi$
$468$	0	0
$469$	−2471.99	−0.243381
$470$	0	0
$471$	−3767.24	−0.368546
$472$	0	0
$473$	−4860.94	−0.472529
$474$	0	0
$475$	9677.05	0.934765
$476$	0	0
$477$	−1104.16	−0.105988
$478$	0	0
$479$	6984.90	0.666280	0.333140	−	0.942877i	$-0.391892\pi$
$479$	6984.90	0.666280	0.333140	+	0.942877i	$0.391892\pi$
$480$	0	0
$481$	−2143.44	−0.203186
$482$	0	0
$483$	1598.40	0.150579
$484$	0	0
$485$	−26822.6	−2.51124
$486$	0	0
$487$	1157.50	0.107703	0.0538516	−	0.998549i	$-0.482850\pi$
$487$	1157.50	0.107703	0.0538516	+	0.998549i	$0.482850\pi$
$488$	0	0
$489$	3624.68	0.335202
$490$	0	0
$491$	19036.2	1.74968	0.874839	−	0.484414i	$-0.160967\pi$
$491$	19036.2	1.74968	0.874839	+	0.484414i	$0.160967\pi$
$492$	0	0
$493$	−712.095	−0.0650530
$494$	0	0
$495$	3413.48	0.309948
$496$	0	0
$497$	3258.66	0.294107
$498$	0	0
$499$	13623.3	1.22217	0.611085	−	0.791565i	$-0.290733\pi$
$499$	13623.3	1.22217	0.611085	+	0.791565i	$0.290733\pi$
$500$	0	0
$501$	−8355.42	−0.745095
$502$	0	0
$503$	−3211.73	−0.284700	−0.142350	−	0.989816i	$-0.545466\pi$
$503$	−3211.73	−0.284700	−0.142350	+	0.989816i	$0.545466\pi$
$504$	0	0
$505$	22717.4	2.00180
$506$	0	0
$507$	−6200.48	−0.543142
$508$	0	0
$509$	5483.82	0.477536	0.238768	−	0.971077i	$-0.423256\pi$
$509$	5483.82	0.477536	0.238768	+	0.971077i	$0.423256\pi$
$510$	0	0
$511$	3012.91	0.260829
$512$	0	0
$513$	1696.11	0.145975
$514$	0	0
$515$	30428.6	2.60358
$516$	0	0
$517$	−4267.57	−0.363032
$518$	0	0
$519$	2627.50	0.222224
$520$	0	0
$521$	16008.9	1.34618	0.673091	−	0.739560i	$-0.264966\pi$
$521$	16008.9	1.34618	0.673091	+	0.739560i	$0.264966\pi$
$522$	0	0
$523$	12791.7	1.06948	0.534742	−	0.845016i	$-0.320409\pi$
$523$	12791.7	1.06948	0.534742	+	0.845016i	$0.320409\pi$
$524$	0	0
$525$	3234.99	0.268926
$526$	0	0
$527$	493.101	0.0407586
$528$	0	0
$529$	−6373.64	−0.523847
$530$	0	0
$531$	2028.68	0.165796
$532$	0	0
$533$	265.785	0.0215993
$534$	0	0
$535$	24063.6	1.94460
$536$	0	0
$537$	−3446.15	−0.276932
$538$	0	0
$539$	1112.53	0.0889055
$540$	0	0
$541$	7091.14	0.563534	0.281767	−	0.959483i	$-0.409079\pi$
$541$	7091.14	0.563534	0.281767	+	0.959483i	$0.409079\pi$
$542$	0	0
$543$	5214.44	0.412105
$544$	0	0
$545$	14767.2	1.16065
$546$	0	0
$547$	−4704.93	−0.367766	−0.183883	−	0.982948i	$-0.558867\pi$
$547$	−4704.93	−0.367766	−0.183883	+	0.982948i	$0.558867\pi$
$548$	0	0
$549$	−4915.69	−0.382143
$550$	0	0
$551$	−13574.8	−1.04955
$552$	0	0
$553$	3727.48	0.286634
$554$	0	0
$555$	9414.73	0.720059
$556$	0	0
$557$	3295.22	0.250670	0.125335	−	0.992114i	$-0.459999\pi$
$557$	3295.22	0.250670	0.125335	+	0.992114i	$0.459999\pi$
$558$	0	0
$559$	2442.68	0.184820
$560$	0	0
$561$	224.456	0.0168923
$562$	0	0
$563$	6620.17	0.495572	0.247786	−	0.968815i	$-0.420297\pi$
$563$	6620.17	0.495572	0.247786	+	0.968815i	$0.420297\pi$
$564$	0	0
$565$	−24947.9	−1.85764
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	16960.6	1.24961	0.624804	−	0.780781i	$-0.285179\pi$
$569$	16960.6	1.24961	0.624804	+	0.780781i	$0.285179\pi$
$570$	0	0
$571$	−15713.6	−1.15165	−0.575826	−	0.817572i	$-0.695320\pi$
$571$	−15713.6	−1.15165	−0.575826	+	0.817572i	$0.695320\pi$
$572$	0	0
$573$	−11338.0	−0.826618
$574$	0	0
$575$	11725.1	0.850387
$576$	0	0
$577$	26793.7	1.93317	0.966583	−	0.256354i	$-0.0825214\pi$
$577$	26793.7	1.93317	0.966583	+	0.256354i	$0.0825214\pi$
$578$	0	0
$579$	−11898.4	−0.854028
$580$	0	0
$581$	1296.27	0.0925616
$582$	0	0
$583$	−2785.52	−0.197880
$584$	0	0
$585$	−1715.32	−0.121230
$586$	0	0
$587$	25425.5	1.78777	0.893885	−	0.448296i	$-0.147969\pi$
$587$	25425.5	1.78777	0.893885	+	0.448296i	$0.147969\pi$
$588$	0	0
$589$	9400.05	0.657593
$590$	0	0
$591$	7779.14	0.541440
$592$	0	0
$593$	−10089.0	−0.698660	−0.349330	−	0.937000i	$-0.613591\pi$
$593$	−10089.0	−0.698660	−0.349330	+	0.937000i	$0.613591\pi$
$594$	0	0
$595$	385.329	0.0265495
$596$	0	0
$597$	−13223.3	−0.906520
$598$	0	0
$599$	−17402.3	−1.18704	−0.593520	−	0.804819i	$-0.702262\pi$
$599$	−17402.3	−1.18704	−0.593520	+	0.804819i	$0.702262\pi$
$600$	0	0
$601$	2731.58	0.185397	0.0926985	−	0.995694i	$-0.470451\pi$
$601$	2731.58	0.185397	0.0926985	+	0.995694i	$0.470451\pi$
$602$	0	0
$603$	−3178.27	−0.214642
$604$	0	0
$605$	−13622.6	−0.915436
$606$	0	0
$607$	−5374.07	−0.359352	−0.179676	−	0.983726i	$-0.557505\pi$
$607$	−5374.07	−0.359352	−0.179676	+	0.983726i	$0.557505\pi$
$608$	0	0
$609$	−4537.97	−0.301951
$610$	0	0
$611$	2144.51	0.141993
$612$	0	0
$613$	−24411.6	−1.60844	−0.804222	−	0.594330i	$-0.797417\pi$
$613$	−24411.6	−1.60844	−0.804222	+	0.594330i	$0.797417\pi$
$614$	0	0
$615$	−1167.42	−0.0765448
$616$	0	0
$617$	8070.23	0.526572	0.263286	−	0.964718i	$-0.415194\pi$
$617$	8070.23	0.526572	0.263286	+	0.964718i	$0.415194\pi$
$618$	0	0
$619$	−22451.4	−1.45783	−0.728915	−	0.684604i	$-0.759975\pi$
$619$	−22451.4	−1.45783	−0.728915	+	0.684604i	$0.759975\pi$
$620$	0	0
$621$	2055.08	0.132798
$622$	0	0
$623$	−11181.2	−0.719043
$624$	0	0
$625$	−11150.4	−0.713625
$626$	0	0
$627$	4278.85	0.272537
$628$	0	0
$629$	619.074	0.0392434
$630$	0	0
$631$	23936.0	1.51011	0.755054	−	0.655662i	$-0.227610\pi$
$631$	23936.0	1.51011	0.755054	+	0.655662i	$0.227610\pi$
$632$	0	0
$633$	−11235.4	−0.705476
$634$	0	0
$635$	23744.2	1.48387
$636$	0	0
$637$	−559.061	−0.0347736
$638$	0	0
$639$	4189.71	0.259378
$640$	0	0
$641$	−10470.3	−0.645164	−0.322582	−	0.946542i	$-0.604551\pi$
$641$	−10470.3	−0.645164	−0.322582	+	0.946542i	$0.604551\pi$
$642$	0	0
$643$	7322.20	0.449081	0.224541	−	0.974465i	$-0.427912\pi$
$643$	7322.20	0.449081	0.224541	+	0.974465i	$0.427912\pi$
$644$	0	0
$645$	−10729.1	−0.654975
$646$	0	0
$647$	10714.9	0.651074	0.325537	−	0.945529i	$-0.394455\pi$
$647$	10714.9	0.651074	0.325537	+	0.945529i	$0.394455\pi$
$648$	0	0
$649$	5117.85	0.309543
$650$	0	0
$651$	3142.39	0.189186
$652$	0	0
$653$	265.840	0.0159313	0.00796565	−	0.999968i	$-0.497464\pi$
$653$	265.840	0.0159313	0.00796565	+	0.999968i	$0.497464\pi$
$654$	0	0
$655$	12613.5	0.752443
$656$	0	0
$657$	3873.75	0.230029
$658$	0	0
$659$	−1989.35	−0.117593	−0.0587967	−	0.998270i	$-0.518726\pi$
$659$	−1989.35	−0.117593	−0.0587967	+	0.998270i	$0.518726\pi$
$660$	0	0
$661$	−5879.26	−0.345956	−0.172978	−	0.984926i	$-0.555339\pi$
$661$	−5879.26	−0.345956	−0.172978	+	0.984926i	$0.555339\pi$
$662$	0	0
$663$	−112.792	−0.00660707
$664$	0	0
$665$	7345.58	0.428345
$666$	0	0
$667$	−16447.8	−0.954815
$668$	0	0
$669$	−3597.50	−0.207904
$670$	0	0
$671$	−12401.0	−0.713467
$672$	0	0
$673$	−16670.1	−0.954805	−0.477403	−	0.878685i	$-0.658422\pi$
$673$	−16670.1	−0.954805	−0.477403	+	0.878685i	$0.658422\pi$
$674$	0	0
$675$	4159.27	0.237171
$676$	0	0
$677$	−13648.0	−0.774795	−0.387398	−	0.921913i	$-0.626626\pi$
$677$	−13648.0	−0.774795	−0.387398	+	0.921913i	$0.626626\pi$
$678$	0	0
$679$	−11239.8	−0.635266
$680$	0	0
$681$	13134.2	0.739068
$682$	0	0
$683$	811.575	0.0454671	0.0227336	−	0.999742i	$-0.492763\pi$
$683$	811.575	0.0454671	0.0227336	+	0.999742i	$0.492763\pi$
$684$	0	0
$685$	27044.6	1.50850
$686$	0	0
$687$	−16556.5	−0.919459
$688$	0	0
$689$	1399.76	0.0773970
$690$	0	0
$691$	−4615.20	−0.254082	−0.127041	−	0.991897i	$-0.540548\pi$
$691$	−4615.20	−0.254082	−0.127041	+	0.991897i	$0.540548\pi$
$692$	0	0
$693$	1430.40	0.0784073
$694$	0	0
$695$	−24370.9	−1.33013
$696$	0	0
$697$	−76.7650	−0.00417171
$698$	0	0
$699$	7535.36	0.407744
$700$	0	0
$701$	−6054.51	−0.326214	−0.163107	−	0.986608i	$-0.552152\pi$
$701$	−6054.51	−0.326214	−0.163107	+	0.986608i	$0.552152\pi$
$702$	0	0
$703$	11801.5	0.633147
$704$	0	0
$705$	−9419.44	−0.503201
$706$	0	0
$707$	9519.58	0.506394
$708$	0	0
$709$	19774.2	1.04744	0.523721	−	0.851890i	$-0.324543\pi$
$709$	19774.2	1.04744	0.523721	+	0.851890i	$0.324543\pi$
$710$	0	0
$711$	4792.47	0.252787
$712$	0	0
$713$	11389.5	0.598235
$714$	0	0
$715$	−4327.30	−0.226338
$716$	0	0
$717$	21439.7	1.11671
$718$	0	0
$719$	−26139.8	−1.35584	−0.677920	−	0.735135i	$-0.737119\pi$
$719$	−26139.8	−1.35584	−0.677920	+	0.735135i	$0.737119\pi$
$720$	0	0
$721$	12750.9	0.658625
$722$	0	0
$723$	−997.690	−0.0513202
$724$	0	0
$725$	−33288.6	−1.70525
$726$	0	0
$727$	19081.7	0.973454	0.486727	−	0.873554i	$-0.338191\pi$
$727$	19081.7	0.973454	0.486727	+	0.873554i	$0.338191\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−705.504	−0.0356963
$732$	0	0
$733$	−34359.5	−1.73138	−0.865688	−	0.500584i	$-0.833119\pi$
$733$	−34359.5	−1.73138	−0.865688	+	0.500584i	$0.833119\pi$
$734$	0	0
$735$	2455.59	0.123232
$736$	0	0
$737$	−8017.96	−0.400740
$738$	0	0
$739$	4097.73	0.203975	0.101988	−	0.994786i	$-0.467480\pi$
$739$	4097.73	0.203975	0.101988	+	0.994786i	$0.467480\pi$
$740$	0	0
$741$	−2150.17	−0.106597
$742$	0	0
$743$	5253.98	0.259421	0.129711	−	0.991552i	$-0.458595\pi$
$743$	5253.98	0.259421	0.129711	+	0.991552i	$0.458595\pi$
$744$	0	0
$745$	54625.5	2.68634
$746$	0	0
$747$	1666.63	0.0816316
$748$	0	0
$749$	10083.7	0.491924
$750$	0	0
$751$	−128.052	−0.00622195	−0.00311098	−	0.999995i	$-0.500990\pi$
$751$	−128.052	−0.00622195	−0.00311098	+	0.999995i	$0.500990\pi$
$752$	0	0
$753$	−14499.7	−0.701727
$754$	0	0
$755$	20012.5	0.964673
$756$	0	0
$757$	16625.5	0.798235	0.399117	−	0.916900i	$-0.369317\pi$
$757$	16625.5	0.798235	0.399117	+	0.916900i	$0.369317\pi$
$758$	0	0
$759$	5184.44	0.247936
$760$	0	0
$761$	−29091.6	−1.38577	−0.692885	−	0.721048i	$-0.743661\pi$
$761$	−29091.6	−1.38577	−0.692885	+	0.721048i	$0.743661\pi$
$762$	0	0
$763$	6188.10	0.293610
$764$	0	0
$765$	495.423	0.0234145
$766$	0	0
$767$	−2571.79	−0.121071
$768$	0	0
$769$	−2004.69	−0.0940063	−0.0470032	−	0.998895i	$-0.514967\pi$
$769$	−2004.69	−0.0940063	−0.0470032	+	0.998895i	$0.514967\pi$
$770$	0	0
$771$	−4116.54	−0.192288
$772$	0	0
$773$	10983.8	0.511071	0.255536	−	0.966800i	$-0.417748\pi$
$773$	10983.8	0.511071	0.255536	+	0.966800i	$0.417748\pi$
$774$	0	0
$775$	23051.2	1.06842
$776$	0	0
$777$	3945.18	0.182153
$778$	0	0
$779$	−1463.38	−0.0673057
$780$	0	0
$781$	10569.6	0.484262
$782$	0	0
$783$	−5834.54	−0.266295
$784$	0	0
$785$	−20976.8	−0.953753
$786$	0	0
$787$	−21513.7	−0.974436	−0.487218	−	0.873280i	$-0.661988\pi$
$787$	−21513.7	−0.974436	−0.487218	+	0.873280i	$0.661988\pi$
$788$	0	0
$789$	−3511.79	−0.158458
$790$	0	0
$791$	−10454.2	−0.469924
$792$	0	0
$793$	6231.68	0.279058
$794$	0	0
$795$	−6148.23	−0.274283
$796$	0	0
$797$	5231.43	0.232505	0.116253	−	0.993220i	$-0.462912\pi$
$797$	5231.43	0.232505	0.116253	+	0.993220i	$0.462912\pi$
$798$	0	0
$799$	−619.384	−0.0274246
$800$	0	0
$801$	−14375.8	−0.634136
$802$	0	0
$803$	9772.47	0.429468
$804$	0	0
$805$	8900.24	0.389680
$806$	0	0
$807$	12781.1	0.557515
$808$	0	0
$809$	−18118.0	−0.787384	−0.393692	−	0.919242i	$-0.628802\pi$
$809$	−18118.0	−0.787384	−0.393692	+	0.919242i	$0.628802\pi$
$810$	0	0
$811$	34536.2	1.49535	0.747676	−	0.664064i	$-0.231170\pi$
$811$	34536.2	1.49535	0.747676	+	0.664064i	$0.231170\pi$
$812$	0	0
$813$	−7724.54	−0.333224
$814$	0	0
$815$	20183.1	0.867464
$816$	0	0
$817$	−13449.1	−0.575919
$818$	0	0
$819$	−718.792	−0.0306674
$820$	0	0
$821$	−10122.4	−0.430297	−0.215148	−	0.976581i	$-0.569024\pi$
$821$	−10122.4	−0.430297	−0.215148	+	0.976581i	$0.569024\pi$
$822$	0	0
$823$	−1623.67	−0.0687696	−0.0343848	−	0.999409i	$-0.510947\pi$
$823$	−1623.67	−0.0687696	−0.0343848	+	0.999409i	$0.510947\pi$
$824$	0	0
$825$	10492.8	0.442802
$826$	0	0
$827$	−23112.3	−0.971819	−0.485910	−	0.874009i	$-0.661512\pi$
$827$	−23112.3	−0.971819	−0.485910	+	0.874009i	$0.661512\pi$
$828$	0	0
$829$	23292.5	0.975851	0.487925	−	0.872885i	$-0.337754\pi$
$829$	23292.5	0.975851	0.487925	+	0.872885i	$0.337754\pi$
$830$	0	0
$831$	−19359.4	−0.808146
$832$	0	0
$833$	161.470	0.00671620
$834$	0	0
$835$	−46524.9	−1.92822
$836$	0	0
$837$	4040.22	0.166846
$838$	0	0
$839$	−21603.7	−0.888968	−0.444484	−	0.895787i	$-0.646613\pi$
$839$	−21603.7	−0.888968	−0.444484	+	0.895787i	$0.646613\pi$
$840$	0	0
$841$	22307.6	0.914659
$842$	0	0
$843$	1048.99	0.0428579
$844$	0	0
$845$	−34525.7	−1.40559
$846$	0	0
$847$	−5708.48	−0.231577
$848$	0	0
$849$	−2099.60	−0.0848739
$850$	0	0
$851$	14299.2	0.575995
$852$	0	0
$853$	38664.5	1.55199	0.775995	−	0.630739i	$-0.217248\pi$
$853$	38664.5	1.55199	0.775995	+	0.630739i	$0.217248\pi$
$854$	0	0
$855$	9444.32	0.377765
$856$	0	0
$857$	9703.63	0.386779	0.193390	−	0.981122i	$-0.438052\pi$
$857$	9703.63	0.386779	0.193390	+	0.981122i	$0.438052\pi$
$858$	0	0
$859$	21634.2	0.859312	0.429656	−	0.902993i	$-0.358635\pi$
$859$	21634.2	0.859312	0.429656	+	0.902993i	$0.358635\pi$
$860$	0	0
$861$	−489.201	−0.0193635
$862$	0	0
$863$	27307.4	1.07712	0.538561	−	0.842587i	$-0.318968\pi$
$863$	27307.4	1.07712	0.538561	+	0.842587i	$0.318968\pi$
$864$	0	0
$865$	14630.5	0.575089
$866$	0	0
$867$	−14706.4	−0.576074
$868$	0	0
$869$	12090.2	0.471958
$870$	0	0
$871$	4029.13	0.156741
$872$	0	0
$873$	−14451.2	−0.560252
$874$	0	0
$875$	3396.55	0.131228
$876$	0	0
$877$	32605.7	1.25544	0.627718	−	0.778441i	$-0.283989\pi$
$877$	32605.7	1.25544	0.627718	+	0.778441i	$0.283989\pi$
$878$	0	0
$879$	486.261	0.0186589
$880$	0	0
$881$	−19306.3	−0.738303	−0.369151	−	0.929369i	$-0.620352\pi$
$881$	−19306.3	−0.738303	−0.369151	+	0.929369i	$0.620352\pi$
$882$	0	0
$883$	−14203.0	−0.541299	−0.270650	−	0.962678i	$-0.587239\pi$
$883$	−14203.0	−0.541299	−0.270650	+	0.962678i	$0.587239\pi$
$884$	0	0
$885$	11296.2	0.429059
$886$	0	0
$887$	1468.77	0.0555991	0.0277996	−	0.999614i	$-0.491150\pi$
$887$	1468.77	0.0555991	0.0277996	+	0.999614i	$0.491150\pi$
$888$	0	0
$889$	9949.87	0.375374
$890$	0	0
$891$	1839.08	0.0691487
$892$	0	0
$893$	−11807.4	−0.442463
$894$	0	0
$895$	−19189.0	−0.716667
$896$	0	0
$897$	−2605.25	−0.0969752
$898$	0	0
$899$	−32335.8	−1.19962
$900$	0	0
$901$	−404.283	−0.0149485
$902$	0	0
$903$	−4495.97	−0.165688
$904$	0	0
$905$	29035.2	1.06648
$906$	0	0
$907$	−4131.84	−0.151263	−0.0756314	−	0.997136i	$-0.524097\pi$
$907$	−4131.84	−0.151263	−0.0756314	+	0.997136i	$0.524097\pi$
$908$	0	0
$909$	12239.5	0.446598
$910$	0	0
$911$	−45434.5	−1.65237	−0.826187	−	0.563395i	$-0.809495\pi$
$911$	−45434.5	−1.65237	−0.826187	+	0.563395i	$0.809495\pi$
$912$	0	0
$913$	4204.48	0.152408
$914$	0	0
$915$	−27371.7	−0.988941
$916$	0	0
$917$	5285.61	0.190345
$918$	0	0
$919$	−14032.4	−0.503685	−0.251842	−	0.967768i	$-0.581036\pi$
$919$	−14032.4	−0.503685	−0.251842	+	0.967768i	$0.581036\pi$
$920$	0	0
$921$	6555.47	0.234538
$922$	0	0
$923$	−5311.34	−0.189410
$924$	0	0
$925$	28940.2	1.02870
$926$	0	0
$927$	16394.0	0.580852
$928$	0	0
$929$	−48732.4	−1.72105	−0.860527	−	0.509405i	$-0.829866\pi$
$929$	−48732.4	−1.72105	−0.860527	+	0.509405i	$0.829866\pi$
$930$	0	0
$931$	3078.12	0.108358
$932$	0	0
$933$	−14290.3	−0.501439
$934$	0	0
$935$	1249.83	0.0437152
$936$	0	0
$937$	12813.8	0.446755	0.223378	−	0.974732i	$-0.428292\pi$
$937$	12813.8	0.446755	0.223378	+	0.974732i	$0.428292\pi$
$938$	0	0
$939$	−3242.53	−0.112690
$940$	0	0
$941$	28586.5	0.990323	0.495162	−	0.868801i	$-0.335109\pi$
$941$	28586.5	0.990323	0.495162	+	0.868801i	$0.335109\pi$
$942$	0	0
$943$	−1773.10	−0.0612302
$944$	0	0
$945$	3157.19	0.108681
$946$	0	0
$947$	32669.6	1.12103	0.560517	−	0.828143i	$-0.310603\pi$
$947$	32669.6	1.12103	0.560517	+	0.828143i	$0.310603\pi$
$948$	0	0
$949$	−4910.79	−0.167978
$950$	0	0
$951$	−17498.7	−0.596672
$952$	0	0
$953$	−3583.93	−0.121821	−0.0609103	−	0.998143i	$-0.519400\pi$
$953$	−3583.93	−0.121821	−0.0609103	+	0.998143i	$0.519400\pi$
$954$	0	0
$955$	−63132.7	−2.13919
$956$	0	0
$957$	−14719.0	−0.497178
$958$	0	0
$959$	11332.9	0.381603
$960$	0	0
$961$	−7399.59	−0.248383
$962$	0	0
$963$	12964.8	0.433836
$964$	0	0
$965$	−66253.3	−2.21012
$966$	0	0
$967$	6332.28	0.210582	0.105291	−	0.994441i	$-0.466423\pi$
$967$	6332.28	0.210582	0.105291	+	0.994441i	$0.466423\pi$
$968$	0	0
$969$	621.020	0.0205883
$970$	0	0
$971$	−21547.7	−0.712152	−0.356076	−	0.934457i	$-0.615886\pi$
$971$	−21547.7	−0.712152	−0.356076	+	0.934457i	$0.615886\pi$
$972$	0	0
$973$	−10212.5	−0.336482
$974$	0	0
$975$	−5272.75	−0.173193
$976$	0	0
$977$	−15024.3	−0.491987	−0.245994	−	0.969271i	$-0.579114\pi$
$977$	−15024.3	−0.491987	−0.245994	+	0.969271i	$0.579114\pi$
$978$	0	0
$979$	−36266.4	−1.18394
$980$	0	0
$981$	7956.12	0.258939
$982$	0	0
$983$	−22229.5	−0.721271	−0.360636	−	0.932707i	$-0.617440\pi$
$983$	−22229.5	−0.721271	−0.360636	+	0.932707i	$0.617440\pi$
$984$	0	0
$985$	43316.1	1.40118
$986$	0	0
$987$	−3947.16	−0.127294
$988$	0	0
$989$	−16295.6	−0.523932
$990$	0	0
$991$	−37009.0	−1.18631	−0.593154	−	0.805089i	$-0.702117\pi$
$991$	−37009.0	−1.18631	−0.593154	+	0.805089i	$0.702117\pi$
$992$	0	0
$993$	−10449.3	−0.333937
$994$	0	0
$995$	−73630.3	−2.34597
$996$	0	0
$997$	−15364.0	−0.488048	−0.244024	−	0.969769i	$-0.578468\pi$
$997$	−15364.0	−0.488048	−0.244024	+	0.969769i	$0.578468\pi$
$998$	0	0
$999$	5072.38	0.160644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.a.m.1.2	yes	2
3.2	odd	2		2016.4.a.j.1.1		2
4.3	odd	2		672.4.a.h.1.2	✓	2
8.3	odd	2		1344.4.a.bm.1.1		2
8.5	even	2		1344.4.a.be.1.1		2
12.11	even	2		2016.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.h.1.2	✓	2	4.3	odd	2
672.4.a.m.1.2	yes	2	1.1	even	1	trivial
1344.4.a.be.1.1		2	8.5	even	2
1344.4.a.bm.1.1		2	8.3	odd	2
2016.4.a.i.1.1		2	12.11	even	2
2016.4.a.j.1.1		2	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +16.7047 q^{5} +7.00000 q^{7} +9.00000 q^{9} +22.7047 q^{11} -11.4094 q^{13} +50.1141 q^{15} +3.29530 q^{17} +62.8188 q^{19} +21.0000 q^{21} +76.1141 q^{23} +154.047 q^{25} +27.0000 q^{27} -216.094 q^{29} +149.638 q^{31} +68.1141 q^{33} +116.933 q^{35} +187.866 q^{37} -34.2282 q^{39} -23.2953 q^{41} -214.094 q^{43} +150.342 q^{45} -187.960 q^{47} +49.0000 q^{49} +9.88590 q^{51} -122.685 q^{53} +379.275 q^{55} +188.456 q^{57} +225.409 q^{59} -546.188 q^{61} +63.0000 q^{63} -190.591 q^{65} -353.141 q^{67} +228.342 q^{69} +465.523 q^{71} +430.416 q^{73} +462.141 q^{75} +158.933 q^{77} +532.497 q^{79} +81.0000 q^{81} +185.181 q^{83} +55.0470 q^{85} -648.282 q^{87} -1597.31 q^{89} -79.8658 q^{91} +448.913 q^{93} +1049.37 q^{95} -1605.69 q^{97} +204.342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} + 24 q^{13} + 30 q^{15} + 30 q^{17} + 32 q^{19} + 42 q^{21} + 82 q^{23} + 74 q^{25} + 54 q^{27} + 36 q^{29} + 112 q^{31} + 66 q^{33} + 70 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 + 24 * q^13 + 30 * q^15 + 30 * q^17 + 32 * q^19 + 42 * q^21 + 82 * q^23 + 74 * q^25 + 54 * q^27 + 36 * q^29 + 112 * q^31 + 66 * q^33 + 70 * q^35 + 48 * q^37 + 72 * q^39 - 70 * q^41 + 40 * q^43 + 90 * q^45 + 420 * q^47 + 98 * q^49 + 90 * q^51 + 176 * q^53 + 384 * q^55 + 96 * q^57 + 404 * q^59 - 156 * q^61 + 126 * q^63 - 428 * q^65 - 4 * q^67 + 246 * q^69 + 814 * q^71 - 216 * q^73 + 222 * q^75 + 154 * q^77 + 1580 * q^79 + 162 * q^81 + 464 * q^83 - 124 * q^85 + 108 * q^87 - 1158 * q^89 + 168 * q^91 + 336 * q^93 + 1256 * q^95 - 1760 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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