LMFDB - Embedded newform 1680.4.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,4,Mod(1,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,5,0,-7,0,9,0,-12,0,30,0,15,0,-134] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$99.1232088096$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +30.0000 q^{13} +15.0000 q^{15} -134.000 q^{17} +92.0000 q^{19} -21.0000 q^{21} -112.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -58.0000 q^{29} +224.000 q^{31} -36.0000 q^{33} -35.0000 q^{35} -146.000 q^{37} +90.0000 q^{39} +18.0000 q^{41} -340.000 q^{43} +45.0000 q^{45} -208.000 q^{47} +49.0000 q^{49} -402.000 q^{51} -754.000 q^{53} -60.0000 q^{55} +276.000 q^{57} -380.000 q^{59} +718.000 q^{61} -63.0000 q^{63} +150.000 q^{65} -412.000 q^{67} -336.000 q^{69} +960.000 q^{71} +1066.00 q^{73} +75.0000 q^{75} +84.0000 q^{77} -896.000 q^{79} +81.0000 q^{81} -436.000 q^{83} -670.000 q^{85} -174.000 q^{87} -1038.00 q^{89} -210.000 q^{91} +672.000 q^{93} +460.000 q^{95} -702.000 q^{97} -108.000 q^{99} +O(q^{100})$

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
$5$	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
$9$	9.00000	0.333333
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	30.0000	0.640039	0.320019	−	0.947411i	$-0.396311\pi$
$13$	30.0000	0.640039	0.320019	+	0.947411i	$0.396311\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−134.000	−1.91175	−0.955876	−	0.293771i	$-0.905090\pi$
$17$	−134.000	−1.91175	−0.955876	+	0.293771i	$0.905090\pi$
$18$	0	0
$19$	92.0000	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−112.000	−1.01537	−0.507687	−	0.861541i	$-0.669499\pi$
$23$	−112.000	−1.01537	−0.507687	+	0.861541i	$0.669499\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−58.0000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	0	0
$39$	90.0000	0.369527
$40$	0	0
$41$	18.0000	0.0685641	0.0342820	−	0.999412i	$-0.489086\pi$
$41$	18.0000	0.0685641	0.0342820	+	0.999412i	$0.489086\pi$
$42$	0	0
$43$	−340.000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$43$	−340.000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−208.000	−0.645530	−0.322765	−	0.946479i	$-0.604612\pi$
$47$	−208.000	−0.645530	−0.322765	+	0.946479i	$0.604612\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−402.000	−1.10375
$52$	0	0
$53$	−754.000	−1.95415	−0.977074	−	0.212899i	$-0.931709\pi$
$53$	−754.000	−1.95415	−0.977074	+	0.212899i	$0.931709\pi$
$54$	0	0
$55$	−60.0000	−0.147098
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	−380.000	−0.838505	−0.419252	−	0.907870i	$-0.637708\pi$
$59$	−380.000	−0.838505	−0.419252	+	0.907870i	$0.637708\pi$
$60$	0	0
$61$	718.000	1.50706	0.753529	−	0.657415i	$-0.228350\pi$
$61$	718.000	1.50706	0.753529	+	0.657415i	$0.228350\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	150.000	0.286234
$66$	0	0
$67$	−412.000	−0.751251	−0.375625	−	0.926772i	$-0.622572\pi$
$67$	−412.000	−0.751251	−0.375625	+	0.926772i	$0.622572\pi$
$68$	0	0
$69$	−336.000	−0.586227
$70$	0	0
$71$	960.000	1.60466	0.802331	−	0.596879i	$-0.203593\pi$
$71$	960.000	1.60466	0.802331	+	0.596879i	$0.203593\pi$
$72$	0	0
$73$	1066.00	1.70912	0.854561	−	0.519352i	$-0.173826\pi$
$73$	1066.00	1.70912	0.854561	+	0.519352i	$0.173826\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	84.0000	0.124321
$78$	0	0
$79$	−896.000	−1.27605	−0.638025	−	0.770016i	$-0.720248\pi$
$79$	−896.000	−1.27605	−0.638025	+	0.770016i	$0.720248\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−436.000	−0.576593	−0.288296	−	0.957541i	$-0.593089\pi$
$83$	−436.000	−0.576593	−0.288296	+	0.957541i	$0.593089\pi$
$84$	0	0
$85$	−670.000	−0.854961
$86$	0	0
$87$	−174.000	−0.214423
$88$	0	0
$89$	−1038.00	−1.23627	−0.618134	−	0.786073i	$-0.712111\pi$
$89$	−1038.00	−1.23627	−0.618134	+	0.786073i	$0.712111\pi$
$90$	0	0
$91$	−210.000	−0.241912
$92$	0	0
$93$	672.000	0.749281
$94$	0	0
$95$	460.000	0.496790
$96$	0	0
$97$	−702.000	−0.734818	−0.367409	−	0.930060i	$-0.619755\pi$
$97$	−702.000	−0.734818	−0.367409	+	0.930060i	$0.619755\pi$
$98$	0	0
$99$	−108.000	−0.109640
$100$	0	0
$101$	46.0000	0.0453185	0.0226593	−	0.999743i	$-0.492787\pi$
$101$	46.0000	0.0453185	0.0226593	+	0.999743i	$0.492787\pi$
$102$	0	0
$103$	−1880.00	−1.79847	−0.899233	−	0.437471i	$-0.855874\pi$
$103$	−1880.00	−1.79847	−0.899233	+	0.437471i	$0.855874\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−732.000	−0.661356	−0.330678	−	0.943744i	$-0.607277\pi$
$107$	−732.000	−0.661356	−0.330678	+	0.943744i	$0.607277\pi$
$108$	0	0
$109$	−378.000	−0.332164	−0.166082	−	0.986112i	$-0.553112\pi$
$109$	−378.000	−0.332164	−0.166082	+	0.986112i	$0.553112\pi$
$110$	0	0
$111$	−438.000	−0.374533
$112$	0	0
$113$	1458.00	1.21378	0.606890	−	0.794786i	$-0.292417\pi$
$113$	1458.00	1.21378	0.606890	+	0.794786i	$0.292417\pi$
$114$	0	0
$115$	−560.000	−0.454089
$116$	0	0
$117$	270.000	0.213346
$118$	0	0
$119$	938.000	0.722574
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	54.0000	0.0395855
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−608.000	−0.424813	−0.212407	−	0.977181i	$-0.568130\pi$
$127$	−608.000	−0.424813	−0.212407	+	0.977181i	$0.568130\pi$
$128$	0	0
$129$	−1020.00	−0.696170
$130$	0	0
$131$	956.000	0.637604	0.318802	−	0.947821i	$-0.396720\pi$
$131$	956.000	0.637604	0.318802	+	0.947821i	$0.396720\pi$
$132$	0	0
$133$	−644.000	−0.419864
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−374.000	−0.233233	−0.116617	−	0.993177i	$-0.537205\pi$
$137$	−374.000	−0.233233	−0.116617	+	0.993177i	$0.537205\pi$
$138$	0	0
$139$	−396.000	−0.241642	−0.120821	−	0.992674i	$-0.538553\pi$
$139$	−396.000	−0.241642	−0.120821	+	0.992674i	$0.538553\pi$
$140$	0	0
$141$	−624.000	−0.372697
$142$	0	0
$143$	−360.000	−0.210522
$144$	0	0
$145$	−290.000	−0.166091
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−1874.00	−1.03036	−0.515181	−	0.857081i	$-0.672275\pi$
$149$	−1874.00	−1.03036	−0.515181	+	0.857081i	$0.672275\pi$
$150$	0	0
$151$	1096.00	0.590670	0.295335	−	0.955394i	$-0.404569\pi$
$151$	1096.00	0.590670	0.295335	+	0.955394i	$0.404569\pi$
$152$	0	0
$153$	−1206.00	−0.637250
$154$	0	0
$155$	1120.00	0.580391
$156$	0	0
$157$	1918.00	0.974988	0.487494	−	0.873126i	$-0.337911\pi$
$157$	1918.00	0.974988	0.487494	+	0.873126i	$0.337911\pi$
$158$	0	0
$159$	−2262.00	−1.12823
$160$	0	0
$161$	784.000	0.383776
$162$	0	0
$163$	−2316.00	−1.11290	−0.556451	−	0.830880i	$-0.687837\pi$
$163$	−2316.00	−1.11290	−0.556451	+	0.830880i	$0.687837\pi$
$164$	0	0
$165$	−180.000	−0.0849272
$166$	0	0
$167$	1736.00	0.804405	0.402203	−	0.915551i	$-0.368245\pi$
$167$	1736.00	0.804405	0.402203	+	0.915551i	$0.368245\pi$
$168$	0	0
$169$	−1297.00	−0.590350
$170$	0	0
$171$	828.000	0.370285
$172$	0	0
$173$	−2442.00	−1.07319	−0.536595	−	0.843840i	$-0.680290\pi$
$173$	−2442.00	−1.07319	−0.536595	+	0.843840i	$0.680290\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−1140.00	−0.484111
$178$	0	0
$179$	4092.00	1.70866	0.854331	−	0.519730i	$-0.173967\pi$
$179$	4092.00	1.70866	0.854331	+	0.519730i	$0.173967\pi$
$180$	0	0
$181$	1270.00	0.521538	0.260769	−	0.965401i	$-0.416024\pi$
$181$	1270.00	0.521538	0.260769	+	0.965401i	$0.416024\pi$
$182$	0	0
$183$	2154.00	0.870100
$184$	0	0
$185$	−730.000	−0.290112
$186$	0	0
$187$	1608.00	0.628816
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−4904.00	−1.85781	−0.928903	−	0.370323i	$-0.879247\pi$
$191$	−4904.00	−1.85781	−0.928903	+	0.370323i	$0.879247\pi$
$192$	0	0
$193$	2178.00	0.812310	0.406155	−	0.913804i	$-0.366869\pi$
$193$	2178.00	0.812310	0.406155	+	0.913804i	$0.366869\pi$
$194$	0	0
$195$	450.000	0.165257
$196$	0	0
$197$	−2850.00	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$197$	−2850.00	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$198$	0	0
$199$	1144.00	0.407518	0.203759	−	0.979021i	$-0.434684\pi$
$199$	1144.00	0.407518	0.203759	+	0.979021i	$0.434684\pi$
$200$	0	0
$201$	−1236.00	−0.433735
$202$	0	0
$203$	406.000	0.140372
$204$	0	0
$205$	90.0000	0.0306628
$206$	0	0
$207$	−1008.00	−0.338458
$208$	0	0
$209$	−1104.00	−0.365384
$210$	0	0
$211$	−412.000	−0.134423	−0.0672115	−	0.997739i	$-0.521410\pi$
$211$	−412.000	−0.134423	−0.0672115	+	0.997739i	$0.521410\pi$
$212$	0	0
$213$	2880.00	0.926452
$214$	0	0
$215$	−1700.00	−0.539251
$216$	0	0
$217$	−1568.00	−0.490520
$218$	0	0
$219$	3198.00	0.986762
$220$	0	0
$221$	−4020.00	−1.22359
$222$	0	0
$223$	1632.00	0.490075	0.245038	−	0.969514i	$-0.421200\pi$
$223$	1632.00	0.490075	0.245038	+	0.969514i	$0.421200\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−4084.00	−1.19412	−0.597059	−	0.802198i	$-0.703664\pi$
$227$	−4084.00	−1.19412	−0.597059	+	0.802198i	$0.703664\pi$
$228$	0	0
$229$	−3386.00	−0.977088	−0.488544	−	0.872539i	$-0.662472\pi$
$229$	−3386.00	−0.977088	−0.488544	+	0.872539i	$0.662472\pi$
$230$	0	0
$231$	252.000	0.0717765
$232$	0	0
$233$	5322.00	1.49638	0.748188	−	0.663486i	$-0.230924\pi$
$233$	5322.00	1.49638	0.748188	+	0.663486i	$0.230924\pi$
$234$	0	0
$235$	−1040.00	−0.288690
$236$	0	0
$237$	−2688.00	−0.736727
$238$	0	0
$239$	−3736.00	−1.01114	−0.505569	−	0.862786i	$-0.668717\pi$
$239$	−3736.00	−1.01114	−0.505569	+	0.862786i	$0.668717\pi$
$240$	0	0
$241$	210.000	0.0561298	0.0280649	−	0.999606i	$-0.491065\pi$
$241$	210.000	0.0561298	0.0280649	+	0.999606i	$0.491065\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	2760.00	0.710990
$248$	0	0
$249$	−1308.00	−0.332896
$250$	0	0
$251$	4212.00	1.05920	0.529600	−	0.848248i	$-0.322342\pi$
$251$	4212.00	1.05920	0.529600	+	0.848248i	$0.322342\pi$
$252$	0	0
$253$	1344.00	0.333978
$254$	0	0
$255$	−2010.00	−0.493612
$256$	0	0
$257$	5130.00	1.24514	0.622569	−	0.782565i	$-0.286089\pi$
$257$	5130.00	1.24514	0.622569	+	0.782565i	$0.286089\pi$
$258$	0	0
$259$	1022.00	0.245189
$260$	0	0
$261$	−522.000	−0.123797
$262$	0	0
$263$	−848.000	−0.198821	−0.0994105	−	0.995047i	$-0.531696\pi$
$263$	−848.000	−0.198821	−0.0994105	+	0.995047i	$0.531696\pi$
$264$	0	0
$265$	−3770.00	−0.873922
$266$	0	0
$267$	−3114.00	−0.713759
$268$	0	0
$269$	−1274.00	−0.288763	−0.144381	−	0.989522i	$-0.546119\pi$
$269$	−1274.00	−0.288763	−0.144381	+	0.989522i	$0.546119\pi$
$270$	0	0
$271$	−864.000	−0.193669	−0.0968344	−	0.995301i	$-0.530872\pi$
$271$	−864.000	−0.193669	−0.0968344	+	0.995301i	$0.530872\pi$
$272$	0	0
$273$	−630.000	−0.139668
$274$	0	0
$275$	−300.000	−0.0657843
$276$	0	0
$277$	−8530.00	−1.85025	−0.925123	−	0.379668i	$-0.876038\pi$
$277$	−8530.00	−1.85025	−0.925123	+	0.379668i	$0.876038\pi$
$278$	0	0
$279$	2016.00	0.432598
$280$	0	0
$281$	−5382.00	−1.14257	−0.571287	−	0.820750i	$-0.693556\pi$
$281$	−5382.00	−1.14257	−0.571287	+	0.820750i	$0.693556\pi$
$282$	0	0
$283$	−6236.00	−1.30986	−0.654932	−	0.755687i	$-0.727303\pi$
$283$	−6236.00	−1.30986	−0.654932	+	0.755687i	$0.727303\pi$
$284$	0	0
$285$	1380.00	0.286822
$286$	0	0
$287$	−126.000	−0.0259148
$288$	0	0
$289$	13043.0	2.65479
$290$	0	0
$291$	−2106.00	−0.424247
$292$	0	0
$293$	−818.000	−0.163099	−0.0815496	−	0.996669i	$-0.525987\pi$
$293$	−818.000	−0.163099	−0.0815496	+	0.996669i	$0.525987\pi$
$294$	0	0
$295$	−1900.00	−0.374991
$296$	0	0
$297$	−324.000	−0.0633010
$298$	0	0
$299$	−3360.00	−0.649879
$300$	0	0
$301$	2380.00	0.455751
$302$	0	0
$303$	138.000	0.0261647
$304$	0	0
$305$	3590.00	0.673976
$306$	0	0
$307$	2268.00	0.421634	0.210817	−	0.977526i	$-0.432388\pi$
$307$	2268.00	0.421634	0.210817	+	0.977526i	$0.432388\pi$
$308$	0	0
$309$	−5640.00	−1.03834
$310$	0	0
$311$	−6648.00	−1.21213	−0.606067	−	0.795414i	$-0.707254\pi$
$311$	−6648.00	−1.21213	−0.606067	+	0.795414i	$0.707254\pi$
$312$	0	0
$313$	9818.00	1.77299	0.886495	−	0.462737i	$-0.153133\pi$
$313$	9818.00	1.77299	0.886495	+	0.462737i	$0.153133\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	934.000	0.165485	0.0827424	−	0.996571i	$-0.473632\pi$
$317$	934.000	0.165485	0.0827424	+	0.996571i	$0.473632\pi$
$318$	0	0
$319$	696.000	0.122158
$320$	0	0
$321$	−2196.00	−0.381834
$322$	0	0
$323$	−12328.0	−2.12368
$324$	0	0
$325$	750.000	0.128008
$326$	0	0
$327$	−1134.00	−0.191775
$328$	0	0
$329$	1456.00	0.243987
$330$	0	0
$331$	−2292.00	−0.380603	−0.190302	−	0.981726i	$-0.560947\pi$
$331$	−2292.00	−0.380603	−0.190302	+	0.981726i	$0.560947\pi$
$332$	0	0
$333$	−1314.00	−0.216237
$334$	0	0
$335$	−2060.00	−0.335970
$336$	0	0
$337$	−6062.00	−0.979876	−0.489938	−	0.871757i	$-0.662981\pi$
$337$	−6062.00	−0.979876	−0.489938	+	0.871757i	$0.662981\pi$
$338$	0	0
$339$	4374.00	0.700776
$340$	0	0
$341$	−2688.00	−0.426872
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1680.00	−0.262169
$346$	0	0
$347$	−1484.00	−0.229583	−0.114791	−	0.993390i	$-0.536620\pi$
$347$	−1484.00	−0.229583	−0.114791	+	0.993390i	$0.536620\pi$
$348$	0	0
$349$	254.000	0.0389579	0.0194790	−	0.999810i	$-0.493799\pi$
$349$	254.000	0.0389579	0.0194790	+	0.999810i	$0.493799\pi$
$350$	0	0
$351$	810.000	0.123176
$352$	0	0
$353$	−10950.0	−1.65102	−0.825509	−	0.564388i	$-0.809112\pi$
$353$	−10950.0	−1.65102	−0.825509	+	0.564388i	$0.809112\pi$
$354$	0	0
$355$	4800.00	0.717627
$356$	0	0
$357$	2814.00	0.417178
$358$	0	0
$359$	−11376.0	−1.67243	−0.836215	−	0.548402i	$-0.815236\pi$
$359$	−11376.0	−1.67243	−0.836215	+	0.548402i	$0.815236\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−3561.00	−0.514887
$364$	0	0
$365$	5330.00	0.764342
$366$	0	0
$367$	1136.00	0.161577	0.0807884	−	0.996731i	$-0.474256\pi$
$367$	1136.00	0.161577	0.0807884	+	0.996731i	$0.474256\pi$
$368$	0	0
$369$	162.000	0.0228547
$370$	0	0
$371$	5278.00	0.738599
$372$	0	0
$373$	−8242.00	−1.14411	−0.572057	−	0.820214i	$-0.693854\pi$
$373$	−8242.00	−1.14411	−0.572057	+	0.820214i	$0.693854\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−1740.00	−0.237704
$378$	0	0
$379$	−3620.00	−0.490625	−0.245313	−	0.969444i	$-0.578891\pi$
$379$	−3620.00	−0.490625	−0.245313	+	0.969444i	$0.578891\pi$
$380$	0	0
$381$	−1824.00	−0.245266
$382$	0	0
$383$	8464.00	1.12922	0.564609	−	0.825359i	$-0.309027\pi$
$383$	8464.00	1.12922	0.564609	+	0.825359i	$0.309027\pi$
$384$	0	0
$385$	420.000	0.0555979
$386$	0	0
$387$	−3060.00	−0.401934
$388$	0	0
$389$	3678.00	0.479388	0.239694	−	0.970848i	$-0.422953\pi$
$389$	3678.00	0.479388	0.239694	+	0.970848i	$0.422953\pi$
$390$	0	0
$391$	15008.0	1.94114
$392$	0	0
$393$	2868.00	0.368121
$394$	0	0
$395$	−4480.00	−0.570666
$396$	0	0
$397$	12590.0	1.59162	0.795811	−	0.605545i	$-0.207045\pi$
$397$	12590.0	1.59162	0.795811	+	0.605545i	$0.207045\pi$
$398$	0	0
$399$	−1932.00	−0.242408
$400$	0	0
$401$	2850.00	0.354918	0.177459	−	0.984128i	$-0.443212\pi$
$401$	2850.00	0.354918	0.177459	+	0.984128i	$0.443212\pi$
$402$	0	0
$403$	6720.00	0.830638
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	1752.00	0.213374
$408$	0	0
$409$	1226.00	0.148220	0.0741098	−	0.997250i	$-0.476388\pi$
$409$	1226.00	0.148220	0.0741098	+	0.997250i	$0.476388\pi$
$410$	0	0
$411$	−1122.00	−0.134657
$412$	0	0
$413$	2660.00	0.316925
$414$	0	0
$415$	−2180.00	−0.257860
$416$	0	0
$417$	−1188.00	−0.139512
$418$	0	0
$419$	−612.000	−0.0713560	−0.0356780	−	0.999363i	$-0.511359\pi$
$419$	−612.000	−0.0713560	−0.0356780	+	0.999363i	$0.511359\pi$
$420$	0	0
$421$	5182.00	0.599894	0.299947	−	0.953956i	$-0.403031\pi$
$421$	5182.00	0.599894	0.299947	+	0.953956i	$0.403031\pi$
$422$	0	0
$423$	−1872.00	−0.215177
$424$	0	0
$425$	−3350.00	−0.382350
$426$	0	0
$427$	−5026.00	−0.569614
$428$	0	0
$429$	−1080.00	−0.121545
$430$	0	0
$431$	4984.00	0.557009	0.278504	−	0.960435i	$-0.410161\pi$
$431$	4984.00	0.557009	0.278504	+	0.960435i	$0.410161\pi$
$432$	0	0
$433$	−1694.00	−0.188010	−0.0940051	−	0.995572i	$-0.529967\pi$
$433$	−1694.00	−0.188010	−0.0940051	+	0.995572i	$0.529967\pi$
$434$	0	0
$435$	−870.000	−0.0958927
$436$	0	0
$437$	−10304.0	−1.12793
$438$	0	0
$439$	−13864.0	−1.50727	−0.753636	−	0.657292i	$-0.771702\pi$
$439$	−13864.0	−1.50727	−0.753636	+	0.657292i	$0.771702\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	4644.00	0.498066	0.249033	−	0.968495i	$-0.419887\pi$
$443$	4644.00	0.498066	0.249033	+	0.968495i	$0.419887\pi$
$444$	0	0
$445$	−5190.00	−0.552875
$446$	0	0
$447$	−5622.00	−0.594880
$448$	0	0
$449$	−4926.00	−0.517756	−0.258878	−	0.965910i	$-0.583353\pi$
$449$	−4926.00	−0.517756	−0.258878	+	0.965910i	$0.583353\pi$
$450$	0	0
$451$	−216.000	−0.0225522
$452$	0	0
$453$	3288.00	0.341024
$454$	0	0
$455$	−1050.00	−0.108186
$456$	0	0
$457$	−14694.0	−1.50406	−0.752031	−	0.659128i	$-0.770926\pi$
$457$	−14694.0	−1.50406	−0.752031	+	0.659128i	$0.770926\pi$
$458$	0	0
$459$	−3618.00	−0.367917
$460$	0	0
$461$	2006.00	0.202665	0.101333	−	0.994853i	$-0.467689\pi$
$461$	2006.00	0.202665	0.101333	+	0.994853i	$0.467689\pi$
$462$	0	0
$463$	−4896.00	−0.491439	−0.245720	−	0.969341i	$-0.579024\pi$
$463$	−4896.00	−0.491439	−0.245720	+	0.969341i	$0.579024\pi$
$464$	0	0
$465$	3360.00	0.335089
$466$	0	0
$467$	−2660.00	−0.263576	−0.131788	−	0.991278i	$-0.542072\pi$
$467$	−2660.00	−0.263576	−0.131788	+	0.991278i	$0.542072\pi$
$468$	0	0
$469$	2884.00	0.283946
$470$	0	0
$471$	5754.00	0.562909
$472$	0	0
$473$	4080.00	0.396614
$474$	0	0
$475$	2300.00	0.222171
$476$	0	0
$477$	−6786.00	−0.651383
$478$	0	0
$479$	5600.00	0.534176	0.267088	−	0.963672i	$-0.413938\pi$
$479$	5600.00	0.534176	0.267088	+	0.963672i	$0.413938\pi$
$480$	0	0
$481$	−4380.00	−0.415199
$482$	0	0
$483$	2352.00	0.221573
$484$	0	0
$485$	−3510.00	−0.328620
$486$	0	0
$487$	6424.00	0.597740	0.298870	−	0.954294i	$-0.403390\pi$
$487$	6424.00	0.597740	0.298870	+	0.954294i	$0.403390\pi$
$488$	0	0
$489$	−6948.00	−0.642535
$490$	0	0
$491$	18900.0	1.73716	0.868579	−	0.495550i	$-0.165033\pi$
$491$	18900.0	1.73716	0.868579	+	0.495550i	$0.165033\pi$
$492$	0	0
$493$	7772.00	0.710007
$494$	0	0
$495$	−540.000	−0.0490327
$496$	0	0
$497$	−6720.00	−0.606505
$498$	0	0
$499$	15364.0	1.37833	0.689165	−	0.724604i	$-0.257977\pi$
$499$	15364.0	1.37833	0.689165	+	0.724604i	$0.257977\pi$
$500$	0	0
$501$	5208.00	0.464424
$502$	0	0
$503$	−2216.00	−0.196435	−0.0982173	−	0.995165i	$-0.531314\pi$
$503$	−2216.00	−0.196435	−0.0982173	+	0.995165i	$0.531314\pi$
$504$	0	0
$505$	230.000	0.0202671
$506$	0	0
$507$	−3891.00	−0.340839
$508$	0	0
$509$	−3754.00	−0.326902	−0.163451	−	0.986551i	$-0.552263\pi$
$509$	−3754.00	−0.326902	−0.163451	+	0.986551i	$0.552263\pi$
$510$	0	0
$511$	−7462.00	−0.645987
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	−9400.00	−0.804298
$516$	0	0
$517$	2496.00	0.212329
$518$	0	0
$519$	−7326.00	−0.619606
$520$	0	0
$521$	−4702.00	−0.395390	−0.197695	−	0.980264i	$-0.563346\pi$
$521$	−4702.00	−0.395390	−0.197695	+	0.980264i	$0.563346\pi$
$522$	0	0
$523$	22660.0	1.89456	0.947278	−	0.320413i	$-0.103822\pi$
$523$	22660.0	1.89456	0.947278	+	0.320413i	$0.103822\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	−30016.0	−2.48106
$528$	0	0
$529$	377.000	0.0309855
$530$	0	0
$531$	−3420.00	−0.279502
$532$	0	0
$533$	540.000	0.0438837
$534$	0	0
$535$	−3660.00	−0.295767
$536$	0	0
$537$	12276.0	0.986496
$538$	0	0
$539$	−588.000	−0.0469888
$540$	0	0
$541$	−8634.00	−0.686145	−0.343073	−	0.939309i	$-0.611468\pi$
$541$	−8634.00	−0.686145	−0.343073	+	0.939309i	$0.611468\pi$
$542$	0	0
$543$	3810.00	0.301110
$544$	0	0
$545$	−1890.00	−0.148548
$546$	0	0
$547$	19284.0	1.50736	0.753679	−	0.657243i	$-0.228278\pi$
$547$	19284.0	1.50736	0.753679	+	0.657243i	$0.228278\pi$
$548$	0	0
$549$	6462.00	0.502352
$550$	0	0
$551$	−5336.00	−0.412561
$552$	0	0
$553$	6272.00	0.482301
$554$	0	0
$555$	−2190.00	−0.167496
$556$	0	0
$557$	−19658.0	−1.49540	−0.747699	−	0.664038i	$-0.768841\pi$
$557$	−19658.0	−1.49540	−0.747699	+	0.664038i	$0.768841\pi$
$558$	0	0
$559$	−10200.0	−0.771760
$560$	0	0
$561$	4824.00	0.363047
$562$	0	0
$563$	25612.0	1.91726	0.958630	−	0.284656i	$-0.0918793\pi$
$563$	25612.0	1.91726	0.958630	+	0.284656i	$0.0918793\pi$
$564$	0	0
$565$	7290.00	0.542819
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	7002.00	0.515886	0.257943	−	0.966160i	$-0.416955\pi$
$569$	7002.00	0.515886	0.257943	+	0.966160i	$0.416955\pi$
$570$	0	0
$571$	4524.00	0.331565	0.165782	−	0.986162i	$-0.446985\pi$
$571$	4524.00	0.331565	0.165782	+	0.986162i	$0.446985\pi$
$572$	0	0
$573$	−14712.0	−1.07260
$574$	0	0
$575$	−2800.00	−0.203075
$576$	0	0
$577$	−6014.00	−0.433910	−0.216955	−	0.976182i	$-0.569612\pi$
$577$	−6014.00	−0.433910	−0.216955	+	0.976182i	$0.569612\pi$
$578$	0	0
$579$	6534.00	0.468988
$580$	0	0
$581$	3052.00	0.217932
$582$	0	0
$583$	9048.00	0.642761
$584$	0	0
$585$	1350.00	0.0954113
$586$	0	0
$587$	11748.0	0.826051	0.413025	−	0.910719i	$-0.364472\pi$
$587$	11748.0	0.826051	0.413025	+	0.910719i	$0.364472\pi$
$588$	0	0
$589$	20608.0	1.44166
$590$	0	0
$591$	−8550.00	−0.595093
$592$	0	0
$593$	−9462.00	−0.655241	−0.327620	−	0.944809i	$-0.606247\pi$
$593$	−9462.00	−0.655241	−0.327620	+	0.944809i	$0.606247\pi$
$594$	0	0
$595$	4690.00	0.323145
$596$	0	0
$597$	3432.00	0.235280
$598$	0	0
$599$	−2320.00	−0.158251	−0.0791257	−	0.996865i	$-0.525213\pi$
$599$	−2320.00	−0.158251	−0.0791257	+	0.996865i	$0.525213\pi$
$600$	0	0
$601$	4650.00	0.315603	0.157802	−	0.987471i	$-0.449559\pi$
$601$	4650.00	0.315603	0.157802	+	0.987471i	$0.449559\pi$
$602$	0	0
$603$	−3708.00	−0.250417
$604$	0	0
$605$	−5935.00	−0.398830
$606$	0	0
$607$	14656.0	0.980014	0.490007	−	0.871718i	$-0.336994\pi$
$607$	14656.0	0.980014	0.490007	+	0.871718i	$0.336994\pi$
$608$	0	0
$609$	1218.00	0.0810441
$610$	0	0
$611$	−6240.00	−0.413164
$612$	0	0
$613$	29166.0	1.92170	0.960851	−	0.277065i	$-0.0893616\pi$
$613$	29166.0	1.92170	0.960851	+	0.277065i	$0.0893616\pi$
$614$	0	0
$615$	270.000	0.0177032
$616$	0	0
$617$	28554.0	1.86311	0.931557	−	0.363597i	$-0.118451\pi$
$617$	28554.0	1.86311	0.931557	+	0.363597i	$0.118451\pi$
$618$	0	0
$619$	3876.00	0.251679	0.125840	−	0.992051i	$-0.459837\pi$
$619$	3876.00	0.251679	0.125840	+	0.992051i	$0.459837\pi$
$620$	0	0
$621$	−3024.00	−0.195409
$622$	0	0
$623$	7266.00	0.467265
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−3312.00	−0.210955
$628$	0	0
$629$	19564.0	1.24017
$630$	0	0
$631$	−2904.00	−0.183211	−0.0916057	−	0.995795i	$-0.529200\pi$
$631$	−2904.00	−0.183211	−0.0916057	+	0.995795i	$0.529200\pi$
$632$	0	0
$633$	−1236.00	−0.0776091
$634$	0	0
$635$	−3040.00	−0.189982
$636$	0	0
$637$	1470.00	0.0914341
$638$	0	0
$639$	8640.00	0.534888
$640$	0	0
$641$	9330.00	0.574903	0.287452	−	0.957795i	$-0.407192\pi$
$641$	9330.00	0.574903	0.287452	+	0.957795i	$0.407192\pi$
$642$	0	0
$643$	18332.0	1.12433	0.562164	−	0.827025i	$-0.309969\pi$
$643$	18332.0	1.12433	0.562164	+	0.827025i	$0.309969\pi$
$644$	0	0
$645$	−5100.00	−0.311337
$646$	0	0
$647$	2088.00	0.126874	0.0634372	−	0.997986i	$-0.479794\pi$
$647$	2088.00	0.126874	0.0634372	+	0.997986i	$0.479794\pi$
$648$	0	0
$649$	4560.00	0.275802
$650$	0	0
$651$	−4704.00	−0.283202
$652$	0	0
$653$	22.0000	0.00131842	0.000659209	−	1.00000i	$-0.499790\pi$
$653$	22.0000	0.00131842	0.000659209		1.00000i	$0.499790\pi$
$654$	0	0
$655$	4780.00	0.285145
$656$	0	0
$657$	9594.00	0.569707
$658$	0	0
$659$	−16260.0	−0.961153	−0.480576	−	0.876953i	$-0.659573\pi$
$659$	−16260.0	−0.961153	−0.480576	+	0.876953i	$0.659573\pi$
$660$	0	0
$661$	−23818.0	−1.40153	−0.700766	−	0.713391i	$-0.747158\pi$
$661$	−23818.0	−1.40153	−0.700766	+	0.713391i	$0.747158\pi$
$662$	0	0
$663$	−12060.0	−0.706443
$664$	0	0
$665$	−3220.00	−0.187769
$666$	0	0
$667$	6496.00	0.377101
$668$	0	0
$669$	4896.00	0.282945
$670$	0	0
$671$	−8616.00	−0.495703
$672$	0	0
$673$	31106.0	1.78165	0.890823	−	0.454350i	$-0.150128\pi$
$673$	31106.0	1.78165	0.890823	+	0.454350i	$0.150128\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−1090.00	−0.0618790	−0.0309395	−	0.999521i	$-0.509850\pi$
$677$	−1090.00	−0.0618790	−0.0309395	+	0.999521i	$0.509850\pi$
$678$	0	0
$679$	4914.00	0.277735
$680$	0	0
$681$	−12252.0	−0.689424
$682$	0	0
$683$	12372.0	0.693121	0.346560	−	0.938028i	$-0.387350\pi$
$683$	12372.0	0.693121	0.346560	+	0.938028i	$0.387350\pi$
$684$	0	0
$685$	−1870.00	−0.104305
$686$	0	0
$687$	−10158.0	−0.564122
$688$	0	0
$689$	−22620.0	−1.25073
$690$	0	0
$691$	−3252.00	−0.179033	−0.0895166	−	0.995985i	$-0.528532\pi$
$691$	−3252.00	−0.179033	−0.0895166	+	0.995985i	$0.528532\pi$
$692$	0	0
$693$	756.000	0.0414402
$694$	0	0
$695$	−1980.00	−0.108066
$696$	0	0
$697$	−2412.00	−0.131077
$698$	0	0
$699$	15966.0	0.863934
$700$	0	0
$701$	−5434.00	−0.292781	−0.146390	−	0.989227i	$-0.546766\pi$
$701$	−5434.00	−0.292781	−0.146390	+	0.989227i	$0.546766\pi$
$702$	0	0
$703$	−13432.0	−0.720622
$704$	0	0
$705$	−3120.00	−0.166675
$706$	0	0
$707$	−322.000	−0.0171288
$708$	0	0
$709$	−5330.00	−0.282331	−0.141165	−	0.989986i	$-0.545085\pi$
$709$	−5330.00	−0.282331	−0.141165	+	0.989986i	$0.545085\pi$
$710$	0	0
$711$	−8064.00	−0.425350
$712$	0	0
$713$	−25088.0	−1.31775
$714$	0	0
$715$	−1800.00	−0.0941485
$716$	0	0
$717$	−11208.0	−0.583780
$718$	0	0
$719$	7520.00	0.390054	0.195027	−	0.980798i	$-0.437521\pi$
$719$	7520.00	0.390054	0.195027	+	0.980798i	$0.437521\pi$
$720$	0	0
$721$	13160.0	0.679756
$722$	0	0
$723$	630.000	0.0324066
$724$	0	0
$725$	−1450.00	−0.0742781
$726$	0	0
$727$	−19336.0	−0.986427	−0.493214	−	0.869908i	$-0.664178\pi$
$727$	−19336.0	−0.986427	−0.493214	+	0.869908i	$0.664178\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	45560.0	2.30519
$732$	0	0
$733$	−22498.0	−1.13367	−0.566837	−	0.823830i	$-0.691833\pi$
$733$	−22498.0	−1.13367	−0.566837	+	0.823830i	$0.691833\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	4944.00	0.247103
$738$	0	0
$739$	18292.0	0.910531	0.455265	−	0.890356i	$-0.349544\pi$
$739$	18292.0	0.910531	0.455265	+	0.890356i	$0.349544\pi$
$740$	0	0
$741$	8280.00	0.410490
$742$	0	0
$743$	−17904.0	−0.884030	−0.442015	−	0.897008i	$-0.645736\pi$
$743$	−17904.0	−0.884030	−0.442015	+	0.897008i	$0.645736\pi$
$744$	0	0
$745$	−9370.00	−0.460792
$746$	0	0
$747$	−3924.00	−0.192198
$748$	0	0
$749$	5124.00	0.249969
$750$	0	0
$751$	−5408.00	−0.262771	−0.131385	−	0.991331i	$-0.541943\pi$
$751$	−5408.00	−0.262771	−0.131385	+	0.991331i	$0.541943\pi$
$752$	0	0
$753$	12636.0	0.611529
$754$	0	0
$755$	5480.00	0.264156
$756$	0	0
$757$	8318.00	0.399370	0.199685	−	0.979860i	$-0.436008\pi$
$757$	8318.00	0.399370	0.199685	+	0.979860i	$0.436008\pi$
$758$	0	0
$759$	4032.00	0.192823
$760$	0	0
$761$	6690.00	0.318676	0.159338	−	0.987224i	$-0.449064\pi$
$761$	6690.00	0.318676	0.159338	+	0.987224i	$0.449064\pi$
$762$	0	0
$763$	2646.00	0.125546
$764$	0	0
$765$	−6030.00	−0.284987
$766$	0	0
$767$	−11400.0	−0.536676
$768$	0	0
$769$	9266.00	0.434513	0.217257	−	0.976115i	$-0.430289\pi$
$769$	9266.00	0.434513	0.217257	+	0.976115i	$0.430289\pi$
$770$	0	0
$771$	15390.0	0.718881
$772$	0	0
$773$	9678.00	0.450315	0.225157	−	0.974322i	$-0.427710\pi$
$773$	9678.00	0.450315	0.225157	+	0.974322i	$0.427710\pi$
$774$	0	0
$775$	5600.00	0.259559
$776$	0	0
$777$	3066.00	0.141560
$778$	0	0
$779$	1656.00	0.0761648
$780$	0	0
$781$	−11520.0	−0.527808
$782$	0	0
$783$	−1566.00	−0.0714742
$784$	0	0
$785$	9590.00	0.436028
$786$	0	0
$787$	6860.00	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$787$	6860.00	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$788$	0	0
$789$	−2544.00	−0.114789
$790$	0	0
$791$	−10206.0	−0.458766
$792$	0	0
$793$	21540.0	0.964575
$794$	0	0
$795$	−11310.0	−0.504559
$796$	0	0
$797$	10950.0	0.486661	0.243331	−	0.969943i	$-0.421760\pi$
$797$	10950.0	0.486661	0.243331	+	0.969943i	$0.421760\pi$
$798$	0	0
$799$	27872.0	1.23409
$800$	0	0
$801$	−9342.00	−0.412089
$802$	0	0
$803$	−12792.0	−0.562167
$804$	0	0
$805$	3920.00	0.171630
$806$	0	0
$807$	−3822.00	−0.166717
$808$	0	0
$809$	26010.0	1.13036	0.565181	−	0.824967i	$-0.308806\pi$
$809$	26010.0	1.13036	0.565181	+	0.824967i	$0.308806\pi$
$810$	0	0
$811$	14628.0	0.633364	0.316682	−	0.948532i	$-0.397431\pi$
$811$	14628.0	0.633364	0.316682	+	0.948532i	$0.397431\pi$
$812$	0	0
$813$	−2592.00	−0.111815
$814$	0	0
$815$	−11580.0	−0.497705
$816$	0	0
$817$	−31280.0	−1.33947
$818$	0	0
$819$	−1890.00	−0.0806373
$820$	0	0
$821$	8718.00	0.370597	0.185299	−	0.982682i	$-0.440675\pi$
$821$	8718.00	0.370597	0.185299	+	0.982682i	$0.440675\pi$
$822$	0	0
$823$	7432.00	0.314779	0.157390	−	0.987537i	$-0.449692\pi$
$823$	7432.00	0.314779	0.157390	+	0.987537i	$0.449692\pi$
$824$	0	0
$825$	−900.000	−0.0379806
$826$	0	0
$827$	−17388.0	−0.731125	−0.365562	−	0.930787i	$-0.619123\pi$
$827$	−17388.0	−0.731125	−0.365562	+	0.930787i	$0.619123\pi$
$828$	0	0
$829$	7902.00	0.331059	0.165529	−	0.986205i	$-0.447067\pi$
$829$	7902.00	0.331059	0.165529	+	0.986205i	$0.447067\pi$
$830$	0	0
$831$	−25590.0	−1.06824
$832$	0	0
$833$	−6566.00	−0.273107
$834$	0	0
$835$	8680.00	0.359741
$836$	0	0
$837$	6048.00	0.249760
$838$	0	0
$839$	31848.0	1.31051	0.655253	−	0.755409i	$-0.272562\pi$
$839$	31848.0	1.31051	0.655253	+	0.755409i	$0.272562\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	0	0
$843$	−16146.0	−0.659665
$844$	0	0
$845$	−6485.00	−0.264013
$846$	0	0
$847$	8309.00	0.337073
$848$	0	0
$849$	−18708.0	−0.756251
$850$	0	0
$851$	16352.0	0.658683
$852$	0	0
$853$	30150.0	1.21022	0.605109	−	0.796142i	$-0.293129\pi$
$853$	30150.0	1.21022	0.605109	+	0.796142i	$0.293129\pi$
$854$	0	0
$855$	4140.00	0.165597
$856$	0	0
$857$	−4350.00	−0.173388	−0.0866938	−	0.996235i	$-0.527630\pi$
$857$	−4350.00	−0.173388	−0.0866938	+	0.996235i	$0.527630\pi$
$858$	0	0
$859$	30676.0	1.21845	0.609227	−	0.792996i	$-0.291480\pi$
$859$	30676.0	1.21845	0.609227	+	0.792996i	$0.291480\pi$
$860$	0	0
$861$	−378.000	−0.0149619
$862$	0	0
$863$	23688.0	0.934356	0.467178	−	0.884163i	$-0.345271\pi$
$863$	23688.0	0.934356	0.467178	+	0.884163i	$0.345271\pi$
$864$	0	0
$865$	−12210.0	−0.479945
$866$	0	0
$867$	39129.0	1.53275
$868$	0	0
$869$	10752.0	0.419720
$870$	0	0
$871$	−12360.0	−0.480830
$872$	0	0
$873$	−6318.00	−0.244939
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	31910.0	1.22865	0.614324	−	0.789054i	$-0.289429\pi$
$877$	31910.0	1.22865	0.614324	+	0.789054i	$0.289429\pi$
$878$	0	0
$879$	−2454.00	−0.0941654
$880$	0	0
$881$	50250.0	1.92164	0.960820	−	0.277172i	$-0.0893971\pi$
$881$	50250.0	1.92164	0.960820	+	0.277172i	$0.0893971\pi$
$882$	0	0
$883$	−5980.00	−0.227908	−0.113954	−	0.993486i	$-0.536352\pi$
$883$	−5980.00	−0.227908	−0.113954	+	0.993486i	$0.536352\pi$
$884$	0	0
$885$	−5700.00	−0.216501
$886$	0	0
$887$	24568.0	0.930003	0.465002	−	0.885310i	$-0.346054\pi$
$887$	24568.0	0.930003	0.465002	+	0.885310i	$0.346054\pi$
$888$	0	0
$889$	4256.00	0.160564
$890$	0	0
$891$	−972.000	−0.0365468
$892$	0	0
$893$	−19136.0	−0.717091
$894$	0	0
$895$	20460.0	0.764137
$896$	0	0
$897$	−10080.0	−0.375208
$898$	0	0
$899$	−12992.0	−0.481988
$900$	0	0
$901$	101036.	3.73585
$902$	0	0
$903$	7140.00	0.263128
$904$	0	0
$905$	6350.00	0.233239
$906$	0	0
$907$	−13252.0	−0.485144	−0.242572	−	0.970133i	$-0.577991\pi$
$907$	−13252.0	−0.485144	−0.242572	+	0.970133i	$0.577991\pi$
$908$	0	0
$909$	414.000	0.0151062
$910$	0	0
$911$	6744.00	0.245267	0.122634	−	0.992452i	$-0.460866\pi$
$911$	6744.00	0.245267	0.122634	+	0.992452i	$0.460866\pi$
$912$	0	0
$913$	5232.00	0.189654
$914$	0	0
$915$	10770.0	0.389120
$916$	0	0
$917$	−6692.00	−0.240992
$918$	0	0
$919$	45336.0	1.62731	0.813654	−	0.581349i	$-0.197475\pi$
$919$	45336.0	1.62731	0.813654	+	0.581349i	$0.197475\pi$
$920$	0	0
$921$	6804.00	0.243430
$922$	0	0
$923$	28800.0	1.02705
$924$	0	0
$925$	−3650.00	−0.129742
$926$	0	0
$927$	−16920.0	−0.599488
$928$	0	0
$929$	30074.0	1.06211	0.531053	−	0.847339i	$-0.321797\pi$
$929$	30074.0	1.06211	0.531053	+	0.847339i	$0.321797\pi$
$930$	0	0
$931$	4508.00	0.158694
$932$	0	0
$933$	−19944.0	−0.699826
$934$	0	0
$935$	8040.00	0.281215
$936$	0	0
$937$	21754.0	0.758455	0.379227	−	0.925303i	$-0.376190\pi$
$937$	21754.0	0.758455	0.379227	+	0.925303i	$0.376190\pi$
$938$	0	0
$939$	29454.0	1.02364
$940$	0	0
$941$	14550.0	0.504056	0.252028	−	0.967720i	$-0.418903\pi$
$941$	14550.0	0.504056	0.252028	+	0.967720i	$0.418903\pi$
$942$	0	0
$943$	−2016.00	−0.0696182
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−46660.0	−1.60110	−0.800552	−	0.599263i	$-0.795460\pi$
$947$	−46660.0	−1.60110	−0.800552	+	0.599263i	$0.795460\pi$
$948$	0	0
$949$	31980.0	1.09390
$950$	0	0
$951$	2802.00	0.0955427
$952$	0	0
$953$	20810.0	0.707347	0.353674	−	0.935369i	$-0.384932\pi$
$953$	20810.0	0.707347	0.353674	+	0.935369i	$0.384932\pi$
$954$	0	0
$955$	−24520.0	−0.830836
$956$	0	0
$957$	2088.00	0.0705282
$958$	0	0
$959$	2618.00	0.0881539
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	−6588.00	−0.220452
$964$	0	0
$965$	10890.0	0.363276
$966$	0	0
$967$	−2776.00	−0.0923166	−0.0461583	−	0.998934i	$-0.514698\pi$
$967$	−2776.00	−0.0923166	−0.0461583	+	0.998934i	$0.514698\pi$
$968$	0	0
$969$	−36984.0	−1.22611
$970$	0	0
$971$	−27292.0	−0.902000	−0.451000	−	0.892524i	$-0.648933\pi$
$971$	−27292.0	−0.902000	−0.451000	+	0.892524i	$0.648933\pi$
$972$	0	0
$973$	2772.00	0.0913322
$974$	0	0
$975$	2250.00	0.0739053
$976$	0	0
$977$	−62.0000	−0.00203025	−0.00101513	−	0.999999i	$-0.500323\pi$
$977$	−62.0000	−0.00203025	−0.00101513	+	0.999999i	$0.500323\pi$
$978$	0	0
$979$	12456.0	0.406635
$980$	0	0
$981$	−3402.00	−0.110721
$982$	0	0
$983$	−37912.0	−1.23012	−0.615058	−	0.788481i	$-0.710868\pi$
$983$	−37912.0	−1.23012	−0.615058	+	0.788481i	$0.710868\pi$
$984$	0	0
$985$	−14250.0	−0.460957
$986$	0	0
$987$	4368.00	0.140866
$988$	0	0
$989$	38080.0	1.22434
$990$	0	0
$991$	−10656.0	−0.341573	−0.170787	−	0.985308i	$-0.554631\pi$
$991$	−10656.0	−0.341573	−0.170787	+	0.985308i	$0.554631\pi$
$992$	0	0
$993$	−6876.00	−0.219741
$994$	0	0
$995$	5720.00	0.182247
$996$	0	0
$997$	−29434.0	−0.934989	−0.467495	−	0.883996i	$-0.654843\pi$
$997$	−29434.0	−0.934989	−0.467495	+	0.883996i	$0.654843\pi$
$998$	0	0
$999$	−3942.00	−0.124844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.u.1.1		1
4.3	odd	2		105.4.a.b.1.1	✓	1
12.11	even	2		315.4.a.a.1.1		1
20.3	even	4		525.4.d.a.274.1		2
20.7	even	4		525.4.d.a.274.2		2
20.19	odd	2		525.4.a.a.1.1		1
28.27	even	2		735.4.a.j.1.1		1
60.59	even	2		1575.4.a.l.1.1		1
84.83	odd	2		2205.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.b.1.1	✓	1	4.3	odd	2
315.4.a.a.1.1		1	12.11	even	2
525.4.a.a.1.1		1	20.19	odd	2
525.4.d.a.274.1		2	20.3	even	4
525.4.d.a.274.2		2	20.7	even	4
735.4.a.j.1.1		1	28.27	even	2
1575.4.a.l.1.1		1	60.59	even	2
1680.4.a.u.1.1		1	1.1	even	1	trivial
2205.4.a.b.1.1		1	84.83	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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