LMFDB - Embedded newform 2112.4.a.bd.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,4,Mod(1,2112)] 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("2112.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,6,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$124.612033932$
Analytic rank:	$1$
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:	no (minimal twist has level 264)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.35235$ of defining polynomial
Character	$\chi$	$=$	2112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +14.7047 q^{5} -31.4094 q^{7} +9.00000 q^{9} -11.0000 q^{11} -22.7047 q^{13} -44.1141 q^{15} -46.1141 q^{17} +4.11410 q^{19} +94.2282 q^{21} +163.523 q^{23} +91.2282 q^{25} -27.0000 q^{27} +188.933 q^{29} +210.819 q^{31} +33.0000 q^{33} -461.866 q^{35} +76.5906 q^{37} +68.1141 q^{39} +396.436 q^{41} -501.980 q^{43} +132.342 q^{45} -54.7987 q^{47} +643.550 q^{49} +138.342 q^{51} -338.436 q^{53} -161.752 q^{55} -12.3423 q^{57} -565.409 q^{59} +820.758 q^{61} -282.685 q^{63} -333.866 q^{65} +134.362 q^{67} -490.570 q^{69} -299.027 q^{71} -875.329 q^{73} -273.685 q^{75} +345.503 q^{77} -175.906 q^{79} +81.0000 q^{81} -1465.79 q^{83} -678.094 q^{85} -566.799 q^{87} +377.544 q^{89} +713.141 q^{91} -632.456 q^{93} +60.4966 q^{95} +841.275 q^{97} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{5} - 16 q^{7} + 18 q^{9} - 22 q^{11} - 22 q^{13} - 18 q^{15} - 22 q^{17} - 62 q^{19} + 48 q^{21} + 210 q^{23} + 42 q^{25} - 54 q^{27} + 214 q^{29} + 328 q^{31} + 66 q^{33} - 596 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^5 - 16 * q^7 + 18 * q^9 - 22 * q^11 - 22 * q^13 - 18 * q^15 - 22 * q^17 - 62 * q^19 + 48 * q^21 + 210 * q^23 + 42 * q^25 - 54 * q^27 + 214 * q^29 + 328 * q^31 + 66 * q^33 - 596 * q^35 + 200 * q^37 + 66 * q^39 + 114 * q^41 - 606 * q^43 + 54 * q^45 + 382 * q^47 + 538 * q^49 + 66 * q^51 + 2 * q^53 - 66 * q^55 + 186 * q^57 - 1084 * q^59 + 354 * q^61 - 144 * q^63 - 340 * q^65 + 456 * q^67 - 630 * q^69 + 34 * q^71 - 112 * q^73 - 126 * q^75 + 176 * q^77 - 820 * q^79 + 162 * q^81 - 1012 * q^83 - 888 * q^85 - 642 * q^87 + 1036 * q^89 + 724 * q^91 - 984 * q^93 + 636 * q^95 + 1308 * 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
$5$	14.7047	1.31523	0.657614	−	0.753355i	$-0.271566\pi$
$5$	14.7047	1.31523	0.657614	+	0.753355i	$0.271566\pi$
$6$	0	0
$7$	−31.4094	−1.69595	−0.847974	−	0.530038i	$-0.822178\pi$
$7$	−31.4094	−1.69595	−0.847974	+	0.530038i	$0.822178\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−22.7047	−0.484396	−0.242198	−	0.970227i	$-0.577868\pi$
$13$	−22.7047	−0.484396	−0.242198	+	0.970227i	$0.577868\pi$
$14$	0	0
$15$	−44.1141	−0.759347
$16$	0	0
$17$	−46.1141	−0.657901	−0.328950	−	0.944347i	$-0.606695\pi$
$17$	−46.1141	−0.657901	−0.328950	+	0.944347i	$0.606695\pi$
$18$	0	0
$19$	4.11410	0.0496757	0.0248379	−	0.999691i	$-0.492093\pi$
$19$	4.11410	0.0496757	0.0248379	+	0.999691i	$0.492093\pi$
$20$	0	0
$21$	94.2282	0.979156
$22$	0	0
$23$	163.523	1.48248	0.741239	−	0.671241i	$-0.234238\pi$
$23$	163.523	1.48248	0.741239	+	0.671241i	$0.234238\pi$
$24$	0	0
$25$	91.2282	0.729826
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	188.933	1.20979	0.604896	−	0.796305i	$-0.293215\pi$
$29$	188.933	1.20979	0.604896	+	0.796305i	$0.293215\pi$
$30$	0	0
$31$	210.819	1.22142	0.610712	−	0.791852i	$-0.290883\pi$
$31$	210.819	1.22142	0.610712	+	0.791852i	$0.290883\pi$
$32$	0	0
$33$	33.0000	0.174078
$34$	0	0
$35$	−461.866	−2.23056
$36$	0	0
$37$	76.5906	0.340309	0.170154	−	0.985417i	$-0.445573\pi$
$37$	76.5906	0.340309	0.170154	+	0.985417i	$0.445573\pi$
$38$	0	0
$39$	68.1141	0.279666
$40$	0	0
$41$	396.436	1.51007	0.755036	−	0.655683i	$-0.227619\pi$
$41$	396.436	1.51007	0.755036	+	0.655683i	$0.227619\pi$
$42$	0	0
$43$	−501.980	−1.78026	−0.890130	−	0.455706i	$-0.849387\pi$
$43$	−501.980	−1.78026	−0.890130	+	0.455706i	$0.849387\pi$
$44$	0	0
$45$	132.342	0.438409
$46$	0	0
$47$	−54.7987	−0.170068	−0.0850342	−	0.996378i	$-0.527100\pi$
$47$	−54.7987	−0.170068	−0.0850342	+	0.996378i	$0.527100\pi$
$48$	0	0
$49$	643.550	1.87624
$50$	0	0
$51$	138.342	0.379839
$52$	0	0
$53$	−338.436	−0.877128	−0.438564	−	0.898700i	$-0.644513\pi$
$53$	−338.436	−0.877128	−0.438564	+	0.898700i	$0.644513\pi$
$54$	0	0
$55$	−161.752	−0.396556
$56$	0	0
$57$	−12.3423	−0.0286803
$58$	0	0
$59$	−565.409	−1.24763	−0.623814	−	0.781573i	$-0.714418\pi$
$59$	−565.409	−1.24763	−0.623814	+	0.781573i	$0.714418\pi$
$60$	0	0
$61$	820.758	1.72274	0.861372	−	0.507975i	$-0.169606\pi$
$61$	820.758	1.72274	0.861372	+	0.507975i	$0.169606\pi$
$62$	0	0
$63$	−282.685	−0.565316
$64$	0	0
$65$	−333.866	−0.637092
$66$	0	0
$67$	134.362	0.245000	0.122500	−	0.992469i	$-0.460909\pi$
$67$	134.362	0.245000	0.122500	+	0.992469i	$0.460909\pi$
$68$	0	0
$69$	−490.570	−0.855909
$70$	0	0
$71$	−299.027	−0.499830	−0.249915	−	0.968268i	$-0.580403\pi$
$71$	−299.027	−0.499830	−0.249915	+	0.968268i	$0.580403\pi$
$72$	0	0
$73$	−875.329	−1.40342	−0.701709	−	0.712464i	$-0.747579\pi$
$73$	−875.329	−1.40342	−0.701709	+	0.712464i	$0.747579\pi$
$74$	0	0
$75$	−273.685	−0.421365
$76$	0	0
$77$	345.503	0.511348
$78$	0	0
$79$	−175.906	−0.250519	−0.125259	−	0.992124i	$-0.539976\pi$
$79$	−175.906	−0.250519	−0.125259	+	0.992124i	$0.539976\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1465.79	−1.93844	−0.969222	−	0.246188i	$-0.920822\pi$
$83$	−1465.79	−1.93844	−0.969222	+	0.246188i	$0.920822\pi$
$84$	0	0
$85$	−678.094	−0.865290
$86$	0	0
$87$	−566.799	−0.698474
$88$	0	0
$89$	377.544	0.449658	0.224829	−	0.974398i	$-0.427818\pi$
$89$	377.544	0.449658	0.224829	+	0.974398i	$0.427818\pi$
$90$	0	0
$91$	713.141	0.821511
$92$	0	0
$93$	−632.456	−0.705190
$94$	0	0
$95$	60.4966	0.0653350
$96$	0	0
$97$	841.275	0.880604	0.440302	−	0.897850i	$-0.354871\pi$
$97$	841.275	0.880604	0.440302	+	0.897850i	$0.354871\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−182.342	−0.179641	−0.0898205	−	0.995958i	$-0.528629\pi$
$101$	−182.342	−0.179641	−0.0898205	+	0.995958i	$0.528629\pi$
$102$	0	0
$103$	−861.826	−0.824449	−0.412224	−	0.911082i	$-0.635248\pi$
$103$	−861.826	−0.824449	−0.412224	+	0.911082i	$0.635248\pi$
$104$	0	0
$105$	1385.60	1.28781
$106$	0	0
$107$	189.597	0.171300	0.0856499	−	0.996325i	$-0.472703\pi$
$107$	189.597	0.171300	0.0856499	+	0.996325i	$0.472703\pi$
$108$	0	0
$109$	−440.074	−0.386710	−0.193355	−	0.981129i	$-0.561937\pi$
$109$	−440.074	−0.386710	−0.193355	+	0.981129i	$0.561937\pi$
$110$	0	0
$111$	−229.772	−0.196477
$112$	0	0
$113$	327.409	0.272567	0.136284	−	0.990670i	$-0.456484\pi$
$113$	327.409	0.272567	0.136284	+	0.990670i	$0.456484\pi$
$114$	0	0
$115$	2404.56	1.94980
$116$	0	0
$117$	−204.342	−0.161465
$118$	0	0
$119$	1448.42	1.11577
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1189.31	−0.871840
$124$	0	0
$125$	−496.604	−0.355341
$126$	0	0
$127$	163.289	0.114091	0.0570454	−	0.998372i	$-0.481832\pi$
$127$	163.289	0.114091	0.0570454	+	0.998372i	$0.481832\pi$
$128$	0	0
$129$	1505.94	1.02783
$130$	0	0
$131$	−2302.28	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$131$	−2302.28	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$132$	0	0
$133$	−129.221	−0.0842475
$134$	0	0
$135$	−397.027	−0.253116
$136$	0	0
$137$	−1964.12	−1.22486	−0.612431	−	0.790524i	$-0.709808\pi$
$137$	−1964.12	−1.22486	−0.612431	+	0.790524i	$0.709808\pi$
$138$	0	0
$139$	2383.23	1.45426	0.727132	−	0.686497i	$-0.240853\pi$
$139$	2383.23	1.45426	0.727132	+	0.686497i	$0.240853\pi$
$140$	0	0
$141$	164.396	0.0981890
$142$	0	0
$143$	249.752	0.146051
$144$	0	0
$145$	2778.20	1.59115
$146$	0	0
$147$	−1930.65	−1.08325
$148$	0	0
$149$	−842.262	−0.463093	−0.231546	−	0.972824i	$-0.574378\pi$
$149$	−842.262	−0.463093	−0.231546	+	0.972824i	$0.574378\pi$
$150$	0	0
$151$	1882.67	1.01463	0.507317	−	0.861760i	$-0.330637\pi$
$151$	1882.67	1.01463	0.507317	+	0.861760i	$0.330637\pi$
$152$	0	0
$153$	−415.027	−0.219300
$154$	0	0
$155$	3100.03	1.60645
$156$	0	0
$157$	1309.69	0.665763	0.332881	−	0.942969i	$-0.391979\pi$
$157$	1309.69	0.665763	0.332881	+	0.942969i	$0.391979\pi$
$158$	0	0
$159$	1015.31	0.506410
$160$	0	0
$161$	−5136.17	−2.51421
$162$	0	0
$163$	−4060.44	−1.95116	−0.975578	−	0.219653i	$-0.929508\pi$
$163$	−4060.44	−1.95116	−0.975578	+	0.219653i	$0.929508\pi$
$164$	0	0
$165$	485.255	0.228952
$166$	0	0
$167$	−2944.56	−1.36441	−0.682207	−	0.731159i	$-0.738980\pi$
$167$	−2944.56	−1.36441	−0.682207	+	0.731159i	$0.738980\pi$
$168$	0	0
$169$	−1681.50	−0.765360
$170$	0	0
$171$	37.0269	0.0165586
$172$	0	0
$173$	−2214.26	−0.973105	−0.486553	−	0.873651i	$-0.661746\pi$
$173$	−2214.26	−0.973105	−0.486553	+	0.873651i	$0.661746\pi$
$174$	0	0
$175$	−2865.42	−1.23775
$176$	0	0
$177$	1696.23	0.720318
$178$	0	0
$179$	1740.71	0.726854	0.363427	−	0.931623i	$-0.381607\pi$
$179$	1740.71	0.726854	0.363427	+	0.931623i	$0.381607\pi$
$180$	0	0
$181$	−2640.16	−1.08421	−0.542104	−	0.840311i	$-0.682372\pi$
$181$	−2640.16	−1.08421	−0.542104	+	0.840311i	$0.682372\pi$
$182$	0	0
$183$	−2462.28	−0.994626
$184$	0	0
$185$	1126.24	0.447583
$186$	0	0
$187$	507.255	0.198365
$188$	0	0
$189$	848.054	0.326385
$190$	0	0
$191$	4657.95	1.76460	0.882298	−	0.470692i	$-0.155996\pi$
$191$	4657.95	1.76460	0.882298	+	0.470692i	$0.155996\pi$
$192$	0	0
$193$	1392.81	0.519463	0.259731	−	0.965681i	$-0.416366\pi$
$193$	1392.81	0.519463	0.259731	+	0.965681i	$0.416366\pi$
$194$	0	0
$195$	1001.60	0.367825
$196$	0	0
$197$	−1538.83	−0.556532	−0.278266	−	0.960504i	$-0.589760\pi$
$197$	−1538.83	−0.556532	−0.278266	+	0.960504i	$0.589760\pi$
$198$	0	0
$199$	−4751.96	−1.69275	−0.846376	−	0.532586i	$-0.821220\pi$
$199$	−4751.96	−1.69275	−0.846376	+	0.532586i	$0.821220\pi$
$200$	0	0
$201$	−403.087	−0.141451
$202$	0	0
$203$	−5934.27	−2.05174
$204$	0	0
$205$	5829.48	1.98609
$206$	0	0
$207$	1471.71	0.494160
$208$	0	0
$209$	−45.2551	−0.0149778
$210$	0	0
$211$	3115.46	1.01648	0.508239	−	0.861216i	$-0.330297\pi$
$211$	3115.46	1.01648	0.508239	+	0.861216i	$0.330297\pi$
$212$	0	0
$213$	897.081	0.288577
$214$	0	0
$215$	−7381.46	−2.34145
$216$	0	0
$217$	−6621.69	−2.07147
$218$	0	0
$219$	2625.99	0.810264
$220$	0	0
$221$	1047.01	0.318685
$222$	0	0
$223$	5390.75	1.61880	0.809398	−	0.587261i	$-0.199794\pi$
$223$	5390.75	1.61880	0.809398	+	0.587261i	$0.199794\pi$
$224$	0	0
$225$	821.054	0.243275
$226$	0	0
$227$	−4015.80	−1.17418	−0.587088	−	0.809523i	$-0.699726\pi$
$227$	−4015.80	−1.17418	−0.587088	+	0.809523i	$0.699726\pi$
$228$	0	0
$229$	3542.00	1.02210	0.511052	−	0.859550i	$-0.329256\pi$
$229$	3542.00	1.02210	0.511052	+	0.859550i	$0.329256\pi$
$230$	0	0
$231$	−1036.51	−0.295227
$232$	0	0
$233$	−1043.48	−0.293394	−0.146697	−	0.989181i	$-0.546864\pi$
$233$	−1043.48	−0.293394	−0.146697	+	0.989181i	$0.546864\pi$
$234$	0	0
$235$	−805.798	−0.223679
$236$	0	0
$237$	527.718	0.144637
$238$	0	0
$239$	1275.68	0.345258	0.172629	−	0.984987i	$-0.444774\pi$
$239$	1275.68	0.345258	0.172629	+	0.984987i	$0.444774\pi$
$240$	0	0
$241$	−5090.99	−1.36075	−0.680373	−	0.732866i	$-0.738182\pi$
$241$	−5090.99	−1.36075	−0.680373	+	0.732866i	$0.738182\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	9463.22	2.46768
$246$	0	0
$247$	−93.4094	−0.0240627
$248$	0	0
$249$	4397.36	1.11916
$250$	0	0
$251$	−761.919	−0.191601	−0.0958006	−	0.995401i	$-0.530541\pi$
$251$	−761.919	−0.191601	−0.0958006	+	0.995401i	$0.530541\pi$
$252$	0	0
$253$	−1798.76	−0.446984
$254$	0	0
$255$	2034.28	0.499575
$256$	0	0
$257$	2409.14	0.584740	0.292370	−	0.956305i	$-0.405556\pi$
$257$	2409.14	0.584740	0.292370	+	0.956305i	$0.405556\pi$
$258$	0	0
$259$	−2405.66	−0.577146
$260$	0	0
$261$	1700.40	0.403264
$262$	0	0
$263$	1820.47	0.426825	0.213413	−	0.976962i	$-0.431542\pi$
$263$	1820.47	0.426825	0.213413	+	0.976962i	$0.431542\pi$
$264$	0	0
$265$	−4976.60	−1.15362
$266$	0	0
$267$	−1132.63	−0.259610
$268$	0	0
$269$	1029.70	0.233390	0.116695	−	0.993168i	$-0.462770\pi$
$269$	1029.70	0.233390	0.116695	+	0.993168i	$0.462770\pi$
$270$	0	0
$271$	−3614.40	−0.810182	−0.405091	−	0.914276i	$-0.632760\pi$
$271$	−3614.40	−0.810182	−0.405091	+	0.914276i	$0.632760\pi$
$272$	0	0
$273$	−2139.42	−0.474300
$274$	0	0
$275$	−1003.51	−0.220051
$276$	0	0
$277$	−1360.57	−0.295122	−0.147561	−	0.989053i	$-0.547142\pi$
$277$	−1360.57	−0.295122	−0.147561	+	0.989053i	$0.547142\pi$
$278$	0	0
$279$	1897.37	0.407142
$280$	0	0
$281$	−3431.79	−0.728554	−0.364277	−	0.931291i	$-0.618684\pi$
$281$	−3431.79	−0.728554	−0.364277	+	0.931291i	$0.618684\pi$
$282$	0	0
$283$	7560.33	1.58804	0.794019	−	0.607893i	$-0.207985\pi$
$283$	7560.33	1.58804	0.794019	+	0.607893i	$0.207985\pi$
$284$	0	0
$285$	−181.490	−0.0377212
$286$	0	0
$287$	−12451.8	−2.56100
$288$	0	0
$289$	−2786.49	−0.567167
$290$	0	0
$291$	−2523.83	−0.508417
$292$	0	0
$293$	−6936.95	−1.38314	−0.691571	−	0.722308i	$-0.743081\pi$
$293$	−6936.95	−1.38314	−0.691571	+	0.722308i	$0.743081\pi$
$294$	0	0
$295$	−8314.18	−1.64092
$296$	0	0
$297$	297.000	0.0580259
$298$	0	0
$299$	−3712.75	−0.718107
$300$	0	0
$301$	15766.9	3.01923
$302$	0	0
$303$	547.027	0.103716
$304$	0	0
$305$	12069.0	2.26580
$306$	0	0
$307$	1201.00	0.223273	0.111636	−	0.993749i	$-0.464391\pi$
$307$	1201.00	0.223273	0.111636	+	0.993749i	$0.464391\pi$
$308$	0	0
$309$	2585.48	0.475996
$310$	0	0
$311$	−1764.93	−0.321801	−0.160901	−	0.986971i	$-0.551440\pi$
$311$	−1764.93	−0.321801	−0.160901	+	0.986971i	$0.551440\pi$
$312$	0	0
$313$	8384.67	1.51415	0.757076	−	0.653327i	$-0.226627\pi$
$313$	8384.67	1.51415	0.757076	+	0.653327i	$0.226627\pi$
$314$	0	0
$315$	−4156.79	−0.743520
$316$	0	0
$317$	−10218.7	−1.81053	−0.905265	−	0.424847i	$-0.860328\pi$
$317$	−10218.7	−1.81053	−0.905265	+	0.424847i	$0.860328\pi$
$318$	0	0
$319$	−2078.26	−0.364766
$320$	0	0
$321$	−568.792	−0.0989000
$322$	0	0
$323$	−189.718	−0.0326817
$324$	0	0
$325$	−2071.31	−0.353525
$326$	0	0
$327$	1320.22	0.223267
$328$	0	0
$329$	1721.19	0.288427
$330$	0	0
$331$	−5813.28	−0.965337	−0.482668	−	0.875803i	$-0.660332\pi$
$331$	−5813.28	−0.965337	−0.482668	+	0.875803i	$0.660332\pi$
$332$	0	0
$333$	689.315	0.113436
$334$	0	0
$335$	1975.76	0.322231
$336$	0	0
$337$	−3931.95	−0.635569	−0.317785	−	0.948163i	$-0.602939\pi$
$337$	−3931.95	−0.635569	−0.317785	+	0.948163i	$0.602939\pi$
$338$	0	0
$339$	−982.228	−0.157367
$340$	0	0
$341$	−2319.01	−0.368273
$342$	0	0
$343$	−9440.11	−1.48606
$344$	0	0
$345$	−7213.69	−1.12572
$346$	0	0
$347$	−2467.99	−0.381811	−0.190906	−	0.981608i	$-0.561142\pi$
$347$	−2467.99	−0.381811	−0.190906	+	0.981608i	$0.561142\pi$
$348$	0	0
$349$	−10018.7	−1.53664	−0.768319	−	0.640067i	$-0.778907\pi$
$349$	−10018.7	−1.53664	−0.768319	+	0.640067i	$0.778907\pi$
$350$	0	0
$351$	613.027	0.0932221
$352$	0	0
$353$	9833.09	1.48261	0.741307	−	0.671167i	$-0.234206\pi$
$353$	9833.09	1.48261	0.741307	+	0.671167i	$0.234206\pi$
$354$	0	0
$355$	−4397.10	−0.657391
$356$	0	0
$357$	−4345.25	−0.644188
$358$	0	0
$359$	7551.32	1.11015	0.555074	−	0.831801i	$-0.312690\pi$
$359$	7551.32	1.11015	0.555074	+	0.831801i	$0.312690\pi$
$360$	0	0
$361$	−6842.07	−0.997532
$362$	0	0
$363$	−363.000	−0.0524864
$364$	0	0
$365$	−12871.5	−1.84582
$366$	0	0
$367$	−82.3494	−0.0117128	−0.00585641	−	0.999983i	$-0.501864\pi$
$367$	−82.3494	−0.0117128	−0.00585641	+	0.999983i	$0.501864\pi$
$368$	0	0
$369$	3567.93	0.503357
$370$	0	0
$371$	10630.1	1.48756
$372$	0	0
$373$	−5390.77	−0.748321	−0.374160	−	0.927364i	$-0.622069\pi$
$373$	−5390.77	−0.748321	−0.374160	+	0.927364i	$0.622069\pi$
$374$	0	0
$375$	1489.81	0.205156
$376$	0	0
$377$	−4289.66	−0.586019
$378$	0	0
$379$	−12296.7	−1.66660	−0.833298	−	0.552825i	$-0.813550\pi$
$379$	−12296.7	−1.66660	−0.833298	+	0.552825i	$0.813550\pi$
$380$	0	0
$381$	−489.866	−0.0658704
$382$	0	0
$383$	−11151.3	−1.48774	−0.743872	−	0.668322i	$-0.767013\pi$
$383$	−11151.3	−1.48774	−0.743872	+	0.668322i	$0.767013\pi$
$384$	0	0
$385$	5080.52	0.672539
$386$	0	0
$387$	−4517.82	−0.593420
$388$	0	0
$389$	−462.557	−0.0602894	−0.0301447	−	0.999546i	$-0.509597\pi$
$389$	−462.557	−0.0602894	−0.0301447	+	0.999546i	$0.509597\pi$
$390$	0	0
$391$	−7540.74	−0.975324
$392$	0	0
$393$	6906.85	0.886525
$394$	0	0
$395$	−2586.64	−0.329489
$396$	0	0
$397$	−5671.18	−0.716948	−0.358474	−	0.933540i	$-0.616703\pi$
$397$	−5671.18	−0.716948	−0.358474	+	0.933540i	$0.616703\pi$
$398$	0	0
$399$	387.664	0.0486403
$400$	0	0
$401$	1913.52	0.238295	0.119148	−	0.992877i	$-0.461984\pi$
$401$	1913.52	0.238295	0.119148	+	0.992877i	$0.461984\pi$
$402$	0	0
$403$	−4786.58	−0.591654
$404$	0	0
$405$	1191.08	0.146136
$406$	0	0
$407$	−842.497	−0.102607
$408$	0	0
$409$	−14523.4	−1.75583	−0.877915	−	0.478816i	$-0.841066\pi$
$409$	−14523.4	−1.75583	−0.877915	+	0.478816i	$0.841066\pi$
$410$	0	0
$411$	5892.36	0.707175
$412$	0	0
$413$	17759.2	2.11591
$414$	0	0
$415$	−21553.9	−2.54950
$416$	0	0
$417$	−7149.69	−0.839620
$418$	0	0
$419$	6357.41	0.741241	0.370620	−	0.928784i	$-0.379145\pi$
$419$	6357.41	0.741241	0.370620	+	0.928784i	$0.379145\pi$
$420$	0	0
$421$	−14025.7	−1.62369	−0.811844	−	0.583875i	$-0.801536\pi$
$421$	−14025.7	−1.62369	−0.811844	+	0.583875i	$0.801536\pi$
$422$	0	0
$423$	−493.188	−0.0566894
$424$	0	0
$425$	−4206.91	−0.480153
$426$	0	0
$427$	−25779.5	−2.92168
$428$	0	0
$429$	−749.255	−0.0843226
$430$	0	0
$431$	−591.826	−0.0661421	−0.0330711	−	0.999453i	$-0.510529\pi$
$431$	−591.826	−0.0661421	−0.0330711	+	0.999453i	$0.510529\pi$
$432$	0	0
$433$	10413.2	1.15572	0.577859	−	0.816137i	$-0.303888\pi$
$433$	10413.2	1.15572	0.577859	+	0.816137i	$0.303888\pi$
$434$	0	0
$435$	−8334.60	−0.918652
$436$	0	0
$437$	672.752	0.0736432
$438$	0	0
$439$	2161.15	0.234957	0.117479	−	0.993075i	$-0.462519\pi$
$439$	2161.15	0.234957	0.117479	+	0.993075i	$0.462519\pi$
$440$	0	0
$441$	5791.95	0.625413
$442$	0	0
$443$	2803.16	0.300636	0.150318	−	0.988638i	$-0.451970\pi$
$443$	2803.16	0.300636	0.150318	+	0.988638i	$0.451970\pi$
$444$	0	0
$445$	5551.67	0.591403
$446$	0	0
$447$	2526.79	0.267367
$448$	0	0
$449$	−4629.69	−0.486612	−0.243306	−	0.969950i	$-0.578232\pi$
$449$	−4629.69	−0.486612	−0.243306	+	0.969950i	$0.578232\pi$
$450$	0	0
$451$	−4360.80	−0.455304
$452$	0	0
$453$	−5648.01	−0.585799
$454$	0	0
$455$	10486.5	1.08047
$456$	0	0
$457$	−11001.8	−1.12614	−0.563068	−	0.826410i	$-0.690379\pi$
$457$	−11001.8	−1.12614	−0.563068	+	0.826410i	$0.690379\pi$
$458$	0	0
$459$	1245.08	0.126613
$460$	0	0
$461$	−13015.4	−1.31494	−0.657471	−	0.753480i	$-0.728374\pi$
$461$	−13015.4	−1.31494	−0.657471	+	0.753480i	$0.728374\pi$
$462$	0	0
$463$	8643.52	0.867599	0.433799	−	0.901009i	$-0.357173\pi$
$463$	8643.52	0.867599	0.433799	+	0.901009i	$0.357173\pi$
$464$	0	0
$465$	−9300.08	−0.927486
$466$	0	0
$467$	18045.9	1.78815	0.894073	−	0.447922i	$-0.147836\pi$
$467$	18045.9	1.78815	0.894073	+	0.447922i	$0.147836\pi$
$468$	0	0
$469$	−4220.24	−0.415507
$470$	0	0
$471$	−3929.07	−0.384378
$472$	0	0
$473$	5521.78	0.536769
$474$	0	0
$475$	375.322	0.0362546
$476$	0	0
$477$	−3045.93	−0.292376
$478$	0	0
$479$	−5616.66	−0.535766	−0.267883	−	0.963452i	$-0.586324\pi$
$479$	−5616.66	−0.535766	−0.267883	+	0.963452i	$0.586324\pi$
$480$	0	0
$481$	−1738.97	−0.164844
$482$	0	0
$483$	15408.5	1.45158
$484$	0	0
$485$	12370.7	1.15819
$486$	0	0
$487$	1213.03	0.112870	0.0564352	−	0.998406i	$-0.482027\pi$
$487$	1213.03	0.112870	0.0564352	+	0.998406i	$0.482027\pi$
$488$	0	0
$489$	12181.3	1.12650
$490$	0	0
$491$	−17453.7	−1.60423	−0.802113	−	0.597173i	$-0.796291\pi$
$491$	−17453.7	−1.60423	−0.802113	+	0.597173i	$0.796291\pi$
$492$	0	0
$493$	−8712.47	−0.795923
$494$	0	0
$495$	−1455.77	−0.132185
$496$	0	0
$497$	9392.26	0.847687
$498$	0	0
$499$	−9797.69	−0.878968	−0.439484	−	0.898250i	$-0.644839\pi$
$499$	−9797.69	−0.878968	−0.439484	+	0.898250i	$0.644839\pi$
$500$	0	0
$501$	8833.69	0.787745
$502$	0	0
$503$	−7796.11	−0.691076	−0.345538	−	0.938405i	$-0.612304\pi$
$503$	−7796.11	−0.691076	−0.345538	+	0.938405i	$0.612304\pi$
$504$	0	0
$505$	−2681.29	−0.236269
$506$	0	0
$507$	5044.49	0.441881
$508$	0	0
$509$	9180.73	0.799467	0.399734	−	0.916631i	$-0.369103\pi$
$509$	9180.73	0.799467	0.399734	+	0.916631i	$0.369103\pi$
$510$	0	0
$511$	27493.6	2.38012
$512$	0	0
$513$	−111.081	−0.00956010
$514$	0	0
$515$	−12672.9	−1.08434
$516$	0	0
$517$	602.786	0.0512775
$518$	0	0
$519$	6642.79	0.561823
$520$	0	0
$521$	9316.55	0.783427	0.391714	−	0.920087i	$-0.371882\pi$
$521$	9316.55	0.783427	0.391714	+	0.920087i	$0.371882\pi$
$522$	0	0
$523$	−6188.88	−0.517439	−0.258720	−	0.965952i	$-0.583301\pi$
$523$	−6188.88	−0.517439	−0.258720	+	0.965952i	$0.583301\pi$
$524$	0	0
$525$	8596.27	0.714613
$526$	0	0
$527$	−9721.72	−0.803576
$528$	0	0
$529$	14572.9	1.19774
$530$	0	0
$531$	−5088.68	−0.415876
$532$	0	0
$533$	−9000.97	−0.731473
$534$	0	0
$535$	2787.97	0.225298
$536$	0	0
$537$	−5222.14	−0.419649
$538$	0	0
$539$	−7079.05	−0.565708
$540$	0	0
$541$	−19682.8	−1.56420	−0.782098	−	0.623155i	$-0.785850\pi$
$541$	−19682.8	−1.56420	−0.782098	+	0.623155i	$0.785850\pi$
$542$	0	0
$543$	7920.48	0.625968
$544$	0	0
$545$	−6471.15	−0.508612
$546$	0	0
$547$	18294.1	1.42998	0.714989	−	0.699136i	$-0.246432\pi$
$547$	18294.1	1.42998	0.714989	+	0.699136i	$0.246432\pi$
$548$	0	0
$549$	7386.83	0.574248
$550$	0	0
$551$	777.289	0.0600973
$552$	0	0
$553$	5525.10	0.424867
$554$	0	0
$555$	−3378.73	−0.258412
$556$	0	0
$557$	6515.48	0.495637	0.247819	−	0.968806i	$-0.420286\pi$
$557$	6515.48	0.495637	0.247819	+	0.968806i	$0.420286\pi$
$558$	0	0
$559$	11397.3	0.862352
$560$	0	0
$561$	−1521.77	−0.114526
$562$	0	0
$563$	−2303.60	−0.172442	−0.0862212	−	0.996276i	$-0.527479\pi$
$563$	−2303.60	−0.172442	−0.0862212	+	0.996276i	$0.527479\pi$
$564$	0	0
$565$	4814.46	0.358488
$566$	0	0
$567$	−2544.16	−0.188439
$568$	0	0
$569$	−708.906	−0.0522300	−0.0261150	−	0.999659i	$-0.508314\pi$
$569$	−708.906	−0.0522300	−0.0261150	+	0.999659i	$0.508314\pi$
$570$	0	0
$571$	−12199.2	−0.894080	−0.447040	−	0.894514i	$-0.647522\pi$
$571$	−12199.2	−0.894080	−0.447040	+	0.894514i	$0.647522\pi$
$572$	0	0
$573$	−13973.9	−1.01879
$574$	0	0
$575$	14918.0	1.08195
$576$	0	0
$577$	−633.881	−0.0457345	−0.0228672	−	0.999739i	$-0.507280\pi$
$577$	−633.881	−0.0457345	−0.0228672	+	0.999739i	$0.507280\pi$
$578$	0	0
$579$	−4178.42	−0.299912
$580$	0	0
$581$	46039.4	3.28750
$582$	0	0
$583$	3722.80	0.264464
$584$	0	0
$585$	−3004.79	−0.212364
$586$	0	0
$587$	−5616.77	−0.394938	−0.197469	−	0.980309i	$-0.563272\pi$
$587$	−5616.77	−0.394938	−0.197469	+	0.980309i	$0.563272\pi$
$588$	0	0
$589$	867.330	0.0606752
$590$	0	0
$591$	4616.48	0.321314
$592$	0	0
$593$	6064.05	0.419934	0.209967	−	0.977709i	$-0.432664\pi$
$593$	6064.05	0.419934	0.209967	+	0.977709i	$0.432664\pi$
$594$	0	0
$595$	21298.5	1.46749
$596$	0	0
$597$	14255.9	0.977310
$598$	0	0
$599$	17217.8	1.17445	0.587227	−	0.809422i	$-0.300219\pi$
$599$	17217.8	1.17445	0.587227	+	0.809422i	$0.300219\pi$
$600$	0	0
$601$	−21218.2	−1.44011	−0.720056	−	0.693916i	$-0.755884\pi$
$601$	−21218.2	−1.44011	−0.720056	+	0.693916i	$0.755884\pi$
$602$	0	0
$603$	1209.26	0.0816666
$604$	0	0
$605$	1779.27	0.119566
$606$	0	0
$607$	8714.62	0.582727	0.291364	−	0.956612i	$-0.405891\pi$
$607$	8714.62	0.582727	0.291364	+	0.956612i	$0.405891\pi$
$608$	0	0
$609$	17802.8	1.18457
$610$	0	0
$611$	1244.19	0.0823805
$612$	0	0
$613$	2028.28	0.133640	0.0668200	−	0.997765i	$-0.478715\pi$
$613$	2028.28	0.133640	0.0668200	+	0.997765i	$0.478715\pi$
$614$	0	0
$615$	−17488.4	−1.14667
$616$	0	0
$617$	21101.3	1.37683	0.688416	−	0.725316i	$-0.258307\pi$
$617$	21101.3	1.37683	0.688416	+	0.725316i	$0.258307\pi$
$618$	0	0
$619$	16682.6	1.08325	0.541625	−	0.840620i	$-0.317809\pi$
$619$	16682.6	1.08325	0.541625	+	0.840620i	$0.317809\pi$
$620$	0	0
$621$	−4415.13	−0.285303
$622$	0	0
$623$	−11858.4	−0.762596
$624$	0	0
$625$	−18705.9	−1.19718
$626$	0	0
$627$	135.765	0.00864744
$628$	0	0
$629$	−3531.91	−0.223889
$630$	0	0
$631$	13443.2	0.848122	0.424061	−	0.905634i	$-0.360604\pi$
$631$	13443.2	0.848122	0.424061	+	0.905634i	$0.360604\pi$
$632$	0	0
$633$	−9346.37	−0.586864
$634$	0	0
$635$	2401.11	0.150056
$636$	0	0
$637$	−14611.6	−0.908844
$638$	0	0
$639$	−2691.24	−0.166610
$640$	0	0
$641$	−15897.2	−0.979568	−0.489784	−	0.871844i	$-0.662924\pi$
$641$	−15897.2	−0.979568	−0.489784	+	0.871844i	$0.662924\pi$
$642$	0	0
$643$	8579.34	0.526184	0.263092	−	0.964771i	$-0.415258\pi$
$643$	8579.34	0.526184	0.263092	+	0.964771i	$0.415258\pi$
$644$	0	0
$645$	22144.4	1.35184
$646$	0	0
$647$	14732.8	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$647$	14732.8	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$648$	0	0
$649$	6219.50	0.376174
$650$	0	0
$651$	19865.1	1.19597
$652$	0	0
$653$	22322.3	1.33773	0.668865	−	0.743384i	$-0.266780\pi$
$653$	22322.3	1.33773	0.668865	+	0.743384i	$0.266780\pi$
$654$	0	0
$655$	−33854.4	−2.01954
$656$	0	0
$657$	−7877.96	−0.467806
$658$	0	0
$659$	9154.18	0.541117	0.270558	−	0.962704i	$-0.412792\pi$
$659$	9154.18	0.541117	0.270558	+	0.962704i	$0.412792\pi$
$660$	0	0
$661$	−4783.14	−0.281456	−0.140728	−	0.990048i	$-0.544944\pi$
$661$	−4783.14	−0.281456	−0.140728	+	0.990048i	$0.544944\pi$
$662$	0	0
$663$	−3141.02	−0.183993
$664$	0	0
$665$	−1900.16	−0.110805
$666$	0	0
$667$	30895.0	1.79349
$668$	0	0
$669$	−16172.3	−0.934612
$670$	0	0
$671$	−9028.34	−0.519427
$672$	0	0
$673$	22260.3	1.27500	0.637499	−	0.770451i	$-0.279969\pi$
$673$	22260.3	1.27500	0.637499	+	0.770451i	$0.279969\pi$
$674$	0	0
$675$	−2463.16	−0.140455
$676$	0	0
$677$	22534.2	1.27926	0.639631	−	0.768682i	$-0.279087\pi$
$677$	22534.2	1.27926	0.639631	+	0.768682i	$0.279087\pi$
$678$	0	0
$679$	−26423.9	−1.49346
$680$	0	0
$681$	12047.4	0.677911
$682$	0	0
$683$	−17007.2	−0.952799	−0.476400	−	0.879229i	$-0.658058\pi$
$683$	−17007.2	−0.952799	−0.476400	+	0.879229i	$0.658058\pi$
$684$	0	0
$685$	−28881.8	−1.61097
$686$	0	0
$687$	−10626.0	−0.590112
$688$	0	0
$689$	7684.09	0.424878
$690$	0	0
$691$	−13391.4	−0.737240	−0.368620	−	0.929580i	$-0.620170\pi$
$691$	−13391.4	−0.737240	−0.368620	+	0.929580i	$0.620170\pi$
$692$	0	0
$693$	3109.53	0.170449
$694$	0	0
$695$	35044.7	1.91269
$696$	0	0
$697$	−18281.3	−0.993477
$698$	0	0
$699$	3130.45	0.169391
$700$	0	0
$701$	22141.6	1.19298	0.596488	−	0.802622i	$-0.296562\pi$
$701$	22141.6	1.19298	0.596488	+	0.802622i	$0.296562\pi$
$702$	0	0
$703$	315.101	0.0169051
$704$	0	0
$705$	2417.40	0.129141
$706$	0	0
$707$	5727.26	0.304662
$708$	0	0
$709$	32076.4	1.69909	0.849545	−	0.527517i	$-0.176877\pi$
$709$	32076.4	1.69909	0.849545	+	0.527517i	$0.176877\pi$
$710$	0	0
$711$	−1583.15	−0.0835062
$712$	0	0
$713$	34473.8	1.81074
$714$	0	0
$715$	3672.52	0.192090
$716$	0	0
$717$	−3827.04	−0.199335
$718$	0	0
$719$	28097.5	1.45738	0.728692	−	0.684842i	$-0.240129\pi$
$719$	28097.5	1.45738	0.728692	+	0.684842i	$0.240129\pi$
$720$	0	0
$721$	27069.4	1.39822
$722$	0	0
$723$	15273.0	0.785627
$724$	0	0
$725$	17236.0	0.882937
$726$	0	0
$727$	38141.3	1.94578	0.972889	−	0.231272i	$-0.0742888\pi$
$727$	38141.3	1.94578	0.972889	+	0.231272i	$0.0742888\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	23148.4	1.17123
$732$	0	0
$733$	12575.8	0.633692	0.316846	−	0.948477i	$-0.397376\pi$
$733$	12575.8	0.633692	0.316846	+	0.948477i	$0.397376\pi$
$734$	0	0
$735$	−28389.6	−1.42472
$736$	0	0
$737$	−1477.99	−0.0738702
$738$	0	0
$739$	8569.75	0.426581	0.213291	−	0.976989i	$-0.431582\pi$
$739$	8569.75	0.426581	0.213291	+	0.976989i	$0.431582\pi$
$740$	0	0
$741$	280.228	0.0138926
$742$	0	0
$743$	−12941.1	−0.638983	−0.319492	−	0.947589i	$-0.603512\pi$
$743$	−12941.1	−0.638983	−0.319492	+	0.947589i	$0.603512\pi$
$744$	0	0
$745$	−12385.2	−0.609072
$746$	0	0
$747$	−13192.1	−0.646148
$748$	0	0
$749$	−5955.14	−0.290515
$750$	0	0
$751$	15218.5	0.739457	0.369728	−	0.929140i	$-0.379451\pi$
$751$	15218.5	0.739457	0.369728	+	0.929140i	$0.379451\pi$
$752$	0	0
$753$	2285.76	0.110621
$754$	0	0
$755$	27684.1	1.33447
$756$	0	0
$757$	−32516.9	−1.56122	−0.780612	−	0.625016i	$-0.785092\pi$
$757$	−32516.9	−1.56122	−0.780612	+	0.625016i	$0.785092\pi$
$758$	0	0
$759$	5396.28	0.258066
$760$	0	0
$761$	−13774.9	−0.656164	−0.328082	−	0.944649i	$-0.606402\pi$
$761$	−13774.9	−0.656164	−0.328082	+	0.944649i	$0.606402\pi$
$762$	0	0
$763$	13822.5	0.655841
$764$	0	0
$765$	−6102.85	−0.288430
$766$	0	0
$767$	12837.5	0.604346
$768$	0	0
$769$	−19291.0	−0.904620	−0.452310	−	0.891861i	$-0.649400\pi$
$769$	−19291.0	−0.904620	−0.452310	+	0.891861i	$0.649400\pi$
$770$	0	0
$771$	−7227.42	−0.337600
$772$	0	0
$773$	22928.4	1.06685	0.533427	−	0.845846i	$-0.320904\pi$
$773$	22928.4	1.06685	0.533427	+	0.845846i	$0.320904\pi$
$774$	0	0
$775$	19232.6	0.891427
$776$	0	0
$777$	7216.99	0.333215
$778$	0	0
$779$	1630.98	0.0750140
$780$	0	0
$781$	3289.30	0.150705
$782$	0	0
$783$	−5101.19	−0.232825
$784$	0	0
$785$	19258.6	0.875630
$786$	0	0
$787$	−2771.86	−0.125548	−0.0627739	−	0.998028i	$-0.519995\pi$
$787$	−2771.86	−0.125548	−0.0627739	+	0.998028i	$0.519995\pi$
$788$	0	0
$789$	−5461.41	−0.246428
$790$	0	0
$791$	−10283.7	−0.462260
$792$	0	0
$793$	−18635.1	−0.834490
$794$	0	0
$795$	14929.8	0.666045
$796$	0	0
$797$	−40852.5	−1.81565	−0.907824	−	0.419351i	$-0.862258\pi$
$797$	−40852.5	−1.81565	−0.907824	+	0.419351i	$0.862258\pi$
$798$	0	0
$799$	2526.99	0.111888
$800$	0	0
$801$	3397.89	0.149886
$802$	0	0
$803$	9628.62	0.423146
$804$	0	0
$805$	−75525.9	−3.30676
$806$	0	0
$807$	−3089.10	−0.134748
$808$	0	0
$809$	24304.3	1.05623	0.528117	−	0.849172i	$-0.322898\pi$
$809$	24304.3	1.05623	0.528117	+	0.849172i	$0.322898\pi$
$810$	0	0
$811$	25535.3	1.10563	0.552815	−	0.833304i	$-0.313554\pi$
$811$	25535.3	1.10563	0.552815	+	0.833304i	$0.313554\pi$
$812$	0	0
$813$	10843.2	0.467759
$814$	0	0
$815$	−59707.6	−2.56622
$816$	0	0
$817$	−2065.20	−0.0884358
$818$	0	0
$819$	6418.27	0.273837
$820$	0	0
$821$	−10676.9	−0.453870	−0.226935	−	0.973910i	$-0.572870\pi$
$821$	−10676.9	−0.453870	−0.226935	+	0.973910i	$0.572870\pi$
$822$	0	0
$823$	−9824.93	−0.416131	−0.208065	−	0.978115i	$-0.566717\pi$
$823$	−9824.93	−0.416131	−0.208065	+	0.978115i	$0.566717\pi$
$824$	0	0
$825$	3010.53	0.127046
$826$	0	0
$827$	−39651.5	−1.66725	−0.833625	−	0.552331i	$-0.813739\pi$
$827$	−39651.5	−1.66725	−0.833625	+	0.552331i	$0.813739\pi$
$828$	0	0
$829$	−33336.9	−1.39667	−0.698335	−	0.715771i	$-0.746075\pi$
$829$	−33336.9	−1.39667	−0.698335	+	0.715771i	$0.746075\pi$
$830$	0	0
$831$	4081.71	0.170389
$832$	0	0
$833$	−29676.7	−1.23438
$834$	0	0
$835$	−43298.9	−1.79452
$836$	0	0
$837$	−5692.11	−0.235063
$838$	0	0
$839$	−24507.9	−1.00847	−0.504236	−	0.863566i	$-0.668226\pi$
$839$	−24507.9	−1.00847	−0.504236	+	0.863566i	$0.668226\pi$
$840$	0	0
$841$	11306.6	0.463596
$842$	0	0
$843$	10295.4	0.420631
$844$	0	0
$845$	−24725.9	−1.00662
$846$	0	0
$847$	−3800.54	−0.154177
$848$	0	0
$849$	−22681.0	−0.916855
$850$	0	0
$851$	12524.4	0.504500
$852$	0	0
$853$	−9648.93	−0.387307	−0.193654	−	0.981070i	$-0.562034\pi$
$853$	−9648.93	−0.387307	−0.193654	+	0.981070i	$0.562034\pi$
$854$	0	0
$855$	544.469	0.0217783
$856$	0	0
$857$	−40658.4	−1.62061	−0.810307	−	0.586006i	$-0.800699\pi$
$857$	−40658.4	−1.62061	−0.810307	+	0.586006i	$0.800699\pi$
$858$	0	0
$859$	−367.553	−0.0145992	−0.00729962	−	0.999973i	$-0.502324\pi$
$859$	−367.553	−0.0145992	−0.00729962	+	0.999973i	$0.502324\pi$
$860$	0	0
$861$	37355.5	1.47860
$862$	0	0
$863$	−30173.5	−1.19017	−0.595085	−	0.803663i	$-0.702882\pi$
$863$	−30173.5	−1.19017	−0.595085	+	0.803663i	$0.702882\pi$
$864$	0	0
$865$	−32560.1	−1.27986
$866$	0	0
$867$	8359.47	0.327454
$868$	0	0
$869$	1934.97	0.0755342
$870$	0	0
$871$	−3050.66	−0.118677
$872$	0	0
$873$	7571.48	0.293535
$874$	0	0
$875$	15598.0	0.602640
$876$	0	0
$877$	8399.40	0.323407	0.161703	−	0.986839i	$-0.448301\pi$
$877$	8399.40	0.323407	0.161703	+	0.986839i	$0.448301\pi$
$878$	0	0
$879$	20810.8	0.798558
$880$	0	0
$881$	−44344.0	−1.69578	−0.847892	−	0.530168i	$-0.822129\pi$
$881$	−44344.0	−1.69578	−0.847892	+	0.530168i	$0.822129\pi$
$882$	0	0
$883$	20193.9	0.769626	0.384813	−	0.922995i	$-0.374266\pi$
$883$	20193.9	0.769626	0.384813	+	0.922995i	$0.374266\pi$
$884$	0	0
$885$	24942.5	0.947383
$886$	0	0
$887$	−23753.2	−0.899160	−0.449580	−	0.893240i	$-0.648426\pi$
$887$	−23753.2	−0.899160	−0.449580	+	0.893240i	$0.648426\pi$
$888$	0	0
$889$	−5128.80	−0.193492
$890$	0	0
$891$	−891.000	−0.0335013
$892$	0	0
$893$	−225.447	−0.00844827
$894$	0	0
$895$	25596.6	0.955979
$896$	0	0
$897$	11138.3	0.414599
$898$	0	0
$899$	39830.6	1.47767
$900$	0	0
$901$	15606.7	0.577063
$902$	0	0
$903$	−47300.7	−1.74315
$904$	0	0
$905$	−38822.8	−1.42598
$906$	0	0
$907$	−36738.9	−1.34498	−0.672489	−	0.740107i	$-0.734775\pi$
$907$	−36738.9	−1.34498	−0.672489	+	0.740107i	$0.734775\pi$
$908$	0	0
$909$	−1641.08	−0.0598803
$910$	0	0
$911$	3191.71	0.116077	0.0580385	−	0.998314i	$-0.481515\pi$
$911$	3191.71	0.116077	0.0580385	+	0.998314i	$0.481515\pi$
$912$	0	0
$913$	16123.6	0.584463
$914$	0	0
$915$	−36207.0	−1.30816
$916$	0	0
$917$	72313.3	2.60414
$918$	0	0
$919$	−9390.54	−0.337068	−0.168534	−	0.985696i	$-0.553903\pi$
$919$	−9390.54	−0.337068	−0.168534	+	0.985696i	$0.553903\pi$
$920$	0	0
$921$	−3603.00	−0.128906
$922$	0	0
$923$	6789.32	0.242116
$924$	0	0
$925$	6987.22	0.248366
$926$	0	0
$927$	−7756.43	−0.274816
$928$	0	0
$929$	22076.8	0.779672	0.389836	−	0.920884i	$-0.372532\pi$
$929$	22076.8	0.779672	0.389836	+	0.920884i	$0.372532\pi$
$930$	0	0
$931$	2647.63	0.0932036
$932$	0	0
$933$	5294.80	0.185792
$934$	0	0
$935$	7459.03	0.260895
$936$	0	0
$937$	−11835.9	−0.412658	−0.206329	−	0.978483i	$-0.566152\pi$
$937$	−11835.9	−0.412658	−0.206329	+	0.978483i	$0.566152\pi$
$938$	0	0
$939$	−25154.0	−0.874196
$940$	0	0
$941$	29806.5	1.03259	0.516294	−	0.856411i	$-0.327311\pi$
$941$	29806.5	1.03259	0.516294	+	0.856411i	$0.327311\pi$
$942$	0	0
$943$	64826.7	2.23865
$944$	0	0
$945$	12470.4	0.429271
$946$	0	0
$947$	−27203.5	−0.933469	−0.466734	−	0.884398i	$-0.654570\pi$
$947$	−27203.5	−0.933469	−0.466734	+	0.884398i	$0.654570\pi$
$948$	0	0
$949$	19874.1	0.679810
$950$	0	0
$951$	30656.0	1.04531
$952$	0	0
$953$	−43816.2	−1.48934	−0.744672	−	0.667430i	$-0.767394\pi$
$953$	−43816.2	−1.48934	−0.744672	+	0.667430i	$0.767394\pi$
$954$	0	0
$955$	68493.8	2.32085
$956$	0	0
$957$	6234.79	0.210598
$958$	0	0
$959$	61691.9	2.07730
$960$	0	0
$961$	14653.6	0.491879
$962$	0	0
$963$	1706.38	0.0570999
$964$	0	0
$965$	20480.8	0.683212
$966$	0	0
$967$	38829.7	1.29129	0.645645	−	0.763637i	$-0.276588\pi$
$967$	38829.7	1.29129	0.645645	+	0.763637i	$0.276588\pi$
$968$	0	0
$969$	569.154	0.0188688
$970$	0	0
$971$	−28265.6	−0.934177	−0.467088	−	0.884211i	$-0.654697\pi$
$971$	−28265.6	−0.934177	−0.467088	+	0.884211i	$0.654697\pi$
$972$	0	0
$973$	−74855.8	−2.46636
$974$	0	0
$975$	6213.93	0.204108
$976$	0	0
$977$	10802.5	0.353738	0.176869	−	0.984234i	$-0.443403\pi$
$977$	10802.5	0.353738	0.176869	+	0.984234i	$0.443403\pi$
$978$	0	0
$979$	−4152.98	−0.135577
$980$	0	0
$981$	−3960.67	−0.128903
$982$	0	0
$983$	−185.792	−0.00602834	−0.00301417	−	0.999995i	$-0.500959\pi$
$983$	−185.792	−0.00602834	−0.00301417	+	0.999995i	$0.500959\pi$
$984$	0	0
$985$	−22628.0	−0.731967
$986$	0	0
$987$	−5163.58	−0.166523
$988$	0	0
$989$	−82085.5	−2.63920
$990$	0	0
$991$	51188.9	1.64084	0.820418	−	0.571764i	$-0.193741\pi$
$991$	51188.9	1.64084	0.820418	+	0.571764i	$0.193741\pi$
$992$	0	0
$993$	17439.8	0.557338
$994$	0	0
$995$	−69876.1	−2.22635
$996$	0	0
$997$	1969.74	0.0625700	0.0312850	−	0.999511i	$-0.490040\pi$
$997$	1969.74	0.0625700	0.0312850	+	0.999511i	$0.490040\pi$
$998$	0	0
$999$	−2067.95	−0.0654924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.bd.1.2		2
4.3	odd	2		2112.4.a.bm.1.2		2
8.3	odd	2		264.4.a.f.1.1	✓	2
8.5	even	2		528.4.a.q.1.1		2
24.5	odd	2		1584.4.a.be.1.2		2
24.11	even	2		792.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.4.a.f.1.1	✓	2	8.3	odd	2
528.4.a.q.1.1		2	8.5	even	2
792.4.a.i.1.2		2	24.11	even	2
1584.4.a.be.1.2		2	24.5	odd	2
2112.4.a.bd.1.2		2	1.1	even	1	trivial
2112.4.a.bm.1.2		2	4.3	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +14.7047 q^{5} -31.4094 q^{7} +9.00000 q^{9} -11.0000 q^{11} -22.7047 q^{13} -44.1141 q^{15} -46.1141 q^{17} +4.11410 q^{19} +94.2282 q^{21} +163.523 q^{23} +91.2282 q^{25} -27.0000 q^{27} +188.933 q^{29} +210.819 q^{31} +33.0000 q^{33} -461.866 q^{35} +76.5906 q^{37} +68.1141 q^{39} +396.436 q^{41} -501.980 q^{43} +132.342 q^{45} -54.7987 q^{47} +643.550 q^{49} +138.342 q^{51} -338.436 q^{53} -161.752 q^{55} -12.3423 q^{57} -565.409 q^{59} +820.758 q^{61} -282.685 q^{63} -333.866 q^{65} +134.362 q^{67} -490.570 q^{69} -299.027 q^{71} -875.329 q^{73} -273.685 q^{75} +345.503 q^{77} -175.906 q^{79} +81.0000 q^{81} -1465.79 q^{83} -678.094 q^{85} -566.799 q^{87} +377.544 q^{89} +713.141 q^{91} -632.456 q^{93} +60.4966 q^{95} +841.275 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} - 16 q^{7} + 18 q^{9} - 22 q^{11} - 22 q^{13} - 18 q^{15} - 22 q^{17} - 62 q^{19} + 48 q^{21} + 210 q^{23} + 42 q^{25} - 54 q^{27} + 214 q^{29} + 328 q^{31} + 66 q^{33} - 596 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^5 - 16 * q^7 + 18 * q^9 - 22 * q^11 - 22 * q^13 - 18 * q^15 - 22 * q^17 - 62 * q^19 + 48 * q^21 + 210 * q^23 + 42 * q^25 - 54 * q^27 + 214 * q^29 + 328 * q^31 + 66 * q^33 - 596 * q^35 + 200 * q^37 + 66 * q^39 + 114 * q^41 - 606 * q^43 + 54 * q^45 + 382 * q^47 + 538 * q^49 + 66 * q^51 + 2 * q^53 - 66 * q^55 + 186 * q^57 - 1084 * q^59 + 354 * q^61 - 144 * q^63 - 340 * q^65 + 456 * q^67 - 630 * q^69 + 34 * q^71 - 112 * q^73 - 126 * q^75 + 176 * q^77 - 820 * q^79 + 162 * q^81 - 1012 * q^83 - 888 * q^85 - 642 * q^87 + 1036 * q^89 + 724 * q^91 - 984 * q^93 + 636 * q^95 + 1308 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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