LMFDB - Embedded newform 2112.4.a.bd.1.1

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			1.1
Root			$-5.35235$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -8.70470 q^{5} +15.4094 q^{7} +9.00000 q^{9} -11.0000 q^{11} +0.704700 q^{13} +26.1141 q^{15} +24.1141 q^{17} -66.1141 q^{19} -46.2282 q^{21} +46.4765 q^{23} -49.2282 q^{25} -27.0000 q^{27} +25.0671 q^{29} +117.181 q^{31} +33.0000 q^{33} -134.134 q^{35} +123.409 q^{37} -2.11410 q^{39} -282.436 q^{41} -104.020 q^{43} -78.3423 q^{45} +436.799 q^{47} -105.550 q^{49} -72.3423 q^{51} +340.436 q^{53} +95.7517 q^{55} +198.342 q^{57} -518.591 q^{59} -466.758 q^{61} +138.685 q^{63} -6.13420 q^{65} +321.638 q^{67} -139.430 q^{69} +333.027 q^{71} +763.329 q^{73} +147.685 q^{75} -169.503 q^{77} -644.094 q^{79} +81.0000 q^{81} +453.785 q^{83} -209.906 q^{85} -75.2013 q^{87} +658.456 q^{89} +10.8590 q^{91} -351.544 q^{93} +575.503 q^{95} +466.725 q^{97} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−8.70470	−0.778572	−0.389286	−	0.921117i	$-0.627278\pi$
$5$	−8.70470	−0.778572	−0.389286	+	0.921117i	$0.627278\pi$
$6$	0	0
$7$	15.4094	0.832029	0.416015	−	0.909358i	$-0.363426\pi$
$7$	15.4094	0.832029	0.416015	+	0.909358i	$0.363426\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	0.704700	0.0150345	0.00751725	−	0.999972i	$-0.497607\pi$
$13$	0.704700	0.0150345	0.00751725	+	0.999972i	$0.497607\pi$
$14$	0	0
$15$	26.1141	0.449509
$16$	0	0
$17$	24.1141	0.344031	0.172016	−	0.985094i	$-0.444972\pi$
$17$	24.1141	0.344031	0.172016	+	0.985094i	$0.444972\pi$
$18$	0	0
$19$	−66.1141	−0.798296	−0.399148	−	0.916887i	$-0.630694\pi$
$19$	−66.1141	−0.798296	−0.399148	+	0.916887i	$0.630694\pi$
$20$	0	0
$21$	−46.2282	−0.480372
$22$	0	0
$23$	46.4765	0.421349	0.210674	−	0.977556i	$-0.432434\pi$
$23$	46.4765	0.421349	0.210674	+	0.977556i	$0.432434\pi$
$24$	0	0
$25$	−49.2282	−0.393826
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	25.0671	0.160512	0.0802559	−	0.996774i	$-0.474426\pi$
$29$	25.0671	0.160512	0.0802559	+	0.996774i	$0.474426\pi$
$30$	0	0
$31$	117.181	0.678915	0.339457	−	0.940621i	$-0.389757\pi$
$31$	117.181	0.678915	0.339457	+	0.940621i	$0.389757\pi$
$32$	0	0
$33$	33.0000	0.174078
$34$	0	0
$35$	−134.134	−0.647795
$36$	0	0
$37$	123.409	0.548335	0.274167	−	0.961682i	$-0.411598\pi$
$37$	123.409	0.548335	0.274167	+	0.961682i	$0.411598\pi$
$38$	0	0
$39$	−2.11410	−0.00868018
$40$	0	0
$41$	−282.436	−1.07583	−0.537916	−	0.842998i	$-0.680788\pi$
$41$	−282.436	−1.07583	−0.537916	+	0.842998i	$0.680788\pi$
$42$	0	0
$43$	−104.020	−0.368905	−0.184453	−	0.982841i	$-0.559051\pi$
$43$	−104.020	−0.368905	−0.184453	+	0.982841i	$0.559051\pi$
$44$	0	0
$45$	−78.3423	−0.259524
$46$	0	0
$47$	436.799	1.35561	0.677805	−	0.735242i	$-0.262932\pi$
$47$	436.799	1.35561	0.677805	+	0.735242i	$0.262932\pi$
$48$	0	0
$49$	−105.550	−0.307727
$50$	0	0
$51$	−72.3423	−0.198626
$52$	0	0
$53$	340.436	0.882312	0.441156	−	0.897430i	$-0.354569\pi$
$53$	340.436	0.882312	0.441156	+	0.897430i	$0.354569\pi$
$54$	0	0
$55$	95.7517	0.234748
$56$	0	0
$57$	198.342	0.460896
$58$	0	0
$59$	−518.591	−1.14432	−0.572159	−	0.820143i	$-0.693894\pi$
$59$	−518.591	−1.14432	−0.572159	+	0.820143i	$0.693894\pi$
$60$	0	0
$61$	−466.758	−0.979710	−0.489855	−	0.871804i	$-0.662950\pi$
$61$	−466.758	−0.979710	−0.489855	+	0.871804i	$0.662950\pi$
$62$	0	0
$63$	138.685	0.277343
$64$	0	0
$65$	−6.13420	−0.0117054
$66$	0	0
$67$	321.638	0.586482	0.293241	−	0.956039i	$-0.405266\pi$
$67$	321.638	0.586482	0.293241	+	0.956039i	$0.405266\pi$
$68$	0	0
$69$	−139.430	−0.243266
$70$	0	0
$71$	333.027	0.556662	0.278331	−	0.960485i	$-0.410219\pi$
$71$	333.027	0.556662	0.278331	+	0.960485i	$0.410219\pi$
$72$	0	0
$73$	763.329	1.22385	0.611924	−	0.790917i	$-0.290396\pi$
$73$	763.329	1.22385	0.611924	+	0.790917i	$0.290396\pi$
$74$	0	0
$75$	147.685	0.227375
$76$	0	0
$77$	−169.503	−0.250866
$78$	0	0
$79$	−644.094	−0.917294	−0.458647	−	0.888619i	$-0.651666\pi$
$79$	−644.094	−0.917294	−0.458647	+	0.888619i	$0.651666\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	453.785	0.600113	0.300057	−	0.953921i	$-0.402994\pi$
$83$	453.785	0.600113	0.300057	+	0.953921i	$0.402994\pi$
$84$	0	0
$85$	−209.906	−0.267853
$86$	0	0
$87$	−75.2013	−0.0926716
$88$	0	0
$89$	658.456	0.784227	0.392114	−	0.919917i	$-0.371744\pi$
$89$	658.456	0.784227	0.392114	+	0.919917i	$0.371744\pi$
$90$	0	0
$91$	10.8590	0.0125092
$92$	0	0
$93$	−351.544	−0.391972
$94$	0	0
$95$	575.503	0.621531
$96$	0	0
$97$	466.725	0.488544	0.244272	−	0.969707i	$-0.421451\pi$
$97$	466.725	0.488544	0.244272	+	0.969707i	$0.421451\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	28.3423	0.0279224	0.0139612	−	0.999903i	$-0.495556\pi$
$101$	28.3423	0.0279224	0.0139612	+	0.999903i	$0.495556\pi$
$102$	0	0
$103$	261.826	0.250470	0.125235	−	0.992127i	$-0.460031\pi$
$103$	261.826	0.250470	0.125235	+	0.992127i	$0.460031\pi$
$104$	0	0
$105$	402.403	0.374005
$106$	0	0
$107$	−793.597	−0.717009	−0.358504	−	0.933528i	$-0.616713\pi$
$107$	−793.597	−0.717009	−0.358504	+	0.933528i	$0.616713\pi$
$108$	0	0
$109$	426.074	0.374408	0.187204	−	0.982321i	$-0.440057\pi$
$109$	426.074	0.374408	0.187204	+	0.982321i	$0.440057\pi$
$110$	0	0
$111$	−370.228	−0.316581
$112$	0	0
$113$	280.591	0.233591	0.116795	−	0.993156i	$-0.462738\pi$
$113$	280.591	0.233591	0.116795	+	0.993156i	$0.462738\pi$
$114$	0	0
$115$	−404.564	−0.328050
$116$	0	0
$117$	6.34230	0.00501150
$118$	0	0
$119$	371.584	0.286244
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	847.309	0.621132
$124$	0	0
$125$	1516.60	1.08519
$126$	0	0
$127$	−2271.29	−1.58696	−0.793481	−	0.608594i	$-0.791734\pi$
$127$	−2271.29	−1.58696	−0.793481	+	0.608594i	$0.791734\pi$
$128$	0	0
$129$	312.060	0.212987
$130$	0	0
$131$	−897.718	−0.598733	−0.299366	−	0.954138i	$-0.596775\pi$
$131$	−897.718	−0.598733	−0.299366	+	0.954138i	$0.596775\pi$
$132$	0	0
$133$	−1018.78	−0.664205
$134$	0	0
$135$	235.027	0.149836
$136$	0	0
$137$	2624.12	1.63645	0.818226	−	0.574897i	$-0.194958\pi$
$137$	2624.12	1.63645	0.818226	+	0.574897i	$0.194958\pi$
$138$	0	0
$139$	−1245.23	−0.759848	−0.379924	−	0.925018i	$-0.624050\pi$
$139$	−1245.23	−0.759848	−0.379924	+	0.925018i	$0.624050\pi$
$140$	0	0
$141$	−1310.40	−0.782661
$142$	0	0
$143$	−7.75170	−0.00453307
$144$	0	0
$145$	−218.202	−0.124970
$146$	0	0
$147$	316.651	0.177666
$148$	0	0
$149$	960.262	0.527971	0.263986	−	0.964527i	$-0.414963\pi$
$149$	960.262	0.527971	0.263986	+	0.964527i	$0.414963\pi$
$150$	0	0
$151$	−3454.67	−1.86184	−0.930918	−	0.365229i	$-0.880991\pi$
$151$	−3454.67	−1.86184	−0.930918	+	0.365229i	$0.880991\pi$
$152$	0	0
$153$	217.027	0.114677
$154$	0	0
$155$	−1020.03	−0.528584
$156$	0	0
$157$	−141.691	−0.0720268	−0.0360134	−	0.999351i	$-0.511466\pi$
$157$	−141.691	−0.0720268	−0.0360134	+	0.999351i	$0.511466\pi$
$158$	0	0
$159$	−1021.31	−0.509403
$160$	0	0
$161$	716.175	0.350575
$162$	0	0
$163$	1136.44	0.546093	0.273046	−	0.962001i	$-0.411969\pi$
$163$	1136.44	0.546093	0.273046	+	0.962001i	$0.411969\pi$
$164$	0	0
$165$	−287.255	−0.135532
$166$	0	0
$167$	−135.436	−0.0627566	−0.0313783	−	0.999508i	$-0.509990\pi$
$167$	−135.436	−0.0627566	−0.0313783	+	0.999508i	$0.509990\pi$
$168$	0	0
$169$	−2196.50	−0.999774
$170$	0	0
$171$	−595.027	−0.266099
$172$	0	0
$173$	−411.738	−0.180947	−0.0904736	−	0.995899i	$-0.528838\pi$
$173$	−411.738	−0.180947	−0.0904736	+	0.995899i	$0.528838\pi$
$174$	0	0
$175$	−758.577	−0.327674
$176$	0	0
$177$	1555.77	0.660672
$178$	0	0
$179$	−2800.71	−1.16947	−0.584735	−	0.811225i	$-0.698801\pi$
$179$	−2800.71	−1.16947	−0.584735	+	0.811225i	$0.698801\pi$
$180$	0	0
$181$	1152.16	0.473146	0.236573	−	0.971614i	$-0.423976\pi$
$181$	1152.16	0.473146	0.236573	+	0.971614i	$0.423976\pi$
$182$	0	0
$183$	1400.28	0.565636
$184$	0	0
$185$	−1074.24	−0.426918
$186$	0	0
$187$	−265.255	−0.103729
$188$	0	0
$189$	−416.054	−0.160124
$190$	0	0
$191$	1404.05	0.531902	0.265951	−	0.963987i	$-0.414314\pi$
$191$	1404.05	0.531902	0.265951	+	0.963987i	$0.414314\pi$
$192$	0	0
$193$	−3616.81	−1.34893	−0.674465	−	0.738307i	$-0.735626\pi$
$193$	−3616.81	−1.34893	−0.674465	+	0.738307i	$0.735626\pi$
$194$	0	0
$195$	18.4026	0.00675814
$196$	0	0
$197$	3072.83	1.11132	0.555659	−	0.831410i	$-0.312466\pi$
$197$	3072.83	1.11132	0.555659	+	0.831410i	$0.312466\pi$
$198$	0	0
$199$	−3956.04	−1.40923	−0.704614	−	0.709591i	$-0.748880\pi$
$199$	−3956.04	−1.40923	−0.704614	+	0.709591i	$0.748880\pi$
$200$	0	0
$201$	−964.913	−0.338605
$202$	0	0
$203$	386.269	0.133551
$204$	0	0
$205$	2458.52	0.837613
$206$	0	0
$207$	418.289	0.140450
$208$	0	0
$209$	727.255	0.240695
$210$	0	0
$211$	−653.457	−0.213203	−0.106601	−	0.994302i	$-0.533997\pi$
$211$	−653.457	−0.213203	−0.106601	+	0.994302i	$0.533997\pi$
$212$	0	0
$213$	−999.081	−0.321389
$214$	0	0
$215$	905.464	0.287219
$216$	0	0
$217$	1805.69	0.564877
$218$	0	0
$219$	−2289.99	−0.706589
$220$	0	0
$221$	16.9932	0.00517234
$222$	0	0
$223$	−5330.75	−1.60078	−0.800389	−	0.599481i	$-0.795374\pi$
$223$	−5330.75	−1.60078	−0.800389	+	0.599481i	$0.795374\pi$
$224$	0	0
$225$	−443.054	−0.131275
$226$	0	0
$227$	−36.2010	−0.0105848	−0.00529239	−	0.999986i	$-0.501685\pi$
$227$	−36.2010	−0.0105848	−0.00529239	+	0.999986i	$0.501685\pi$
$228$	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
$231$	508.510	0.144838
$232$	0	0
$233$	−130.517	−0.0366971	−0.0183486	−	0.999832i	$-0.505841\pi$
$233$	−130.517	−0.0366971	−0.0183486	+	0.999832i	$0.505841\pi$
$234$	0	0
$235$	−3802.20	−1.05544
$236$	0	0
$237$	1932.28	0.529600
$238$	0	0
$239$	−5091.68	−1.37805	−0.689024	−	0.724739i	$-0.741960\pi$
$239$	−5091.68	−1.37805	−0.689024	+	0.724739i	$0.741960\pi$
$240$	0	0
$241$	−6121.01	−1.63605	−0.818026	−	0.575181i	$-0.804932\pi$
$241$	−6121.01	−1.63605	−0.818026	+	0.575181i	$0.804932\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	918.785	0.239588
$246$	0	0
$247$	−46.5906	−0.0120020
$248$	0	0
$249$	−1361.36	−0.346476
$250$	0	0
$251$	−6146.08	−1.54557	−0.772783	−	0.634670i	$-0.781136\pi$
$251$	−6146.08	−1.54557	−0.772783	+	0.634670i	$0.781136\pi$
$252$	0	0
$253$	−511.242	−0.127041
$254$	0	0
$255$	629.718	0.154645
$256$	0	0
$257$	−5269.14	−1.27891	−0.639455	−	0.768828i	$-0.720840\pi$
$257$	−5269.14	−1.27891	−0.639455	+	0.768828i	$0.720840\pi$
$258$	0	0
$259$	1901.66	0.456231
$260$	0	0
$261$	225.604	0.0535040
$262$	0	0
$263$	−7496.47	−1.75761	−0.878806	−	0.477178i	$-0.841660\pi$
$263$	−7496.47	−1.75761	−0.878806	+	0.477178i	$0.841660\pi$
$264$	0	0
$265$	−2963.40	−0.686943
$266$	0	0
$267$	−1975.37	−0.452774
$268$	0	0
$269$	−4939.70	−1.11962	−0.559812	−	0.828620i	$-0.689127\pi$
$269$	−4939.70	−1.11962	−0.559812	+	0.828620i	$0.689127\pi$
$270$	0	0
$271$	2378.40	0.533128	0.266564	−	0.963817i	$-0.414112\pi$
$271$	2378.40	0.533128	0.266564	+	0.963817i	$0.414112\pi$
$272$	0	0
$273$	−32.5770	−0.00722216
$274$	0	0
$275$	541.510	0.118743
$276$	0	0
$277$	−1009.43	−0.218956	−0.109478	−	0.993989i	$-0.534918\pi$
$277$	−1009.43	−0.218956	−0.109478	+	0.993989i	$0.534918\pi$
$278$	0	0
$279$	1054.63	0.226305
$280$	0	0
$281$	−3970.21	−0.842857	−0.421428	−	0.906862i	$-0.638471\pi$
$281$	−3970.21	−0.842857	−0.421428	+	0.906862i	$0.638471\pi$
$282$	0	0
$283$	2433.67	0.511190	0.255595	−	0.966784i	$-0.417729\pi$
$283$	2433.67	0.511190	0.255595	+	0.966784i	$0.417729\pi$
$284$	0	0
$285$	−1726.51	−0.358841
$286$	0	0
$287$	−4352.17	−0.895124
$288$	0	0
$289$	−4331.51	−0.881643
$290$	0	0
$291$	−1400.17	−0.282061
$292$	0	0
$293$	2262.95	0.451204	0.225602	−	0.974220i	$-0.427565\pi$
$293$	2262.95	0.451204	0.225602	+	0.974220i	$0.427565\pi$
$294$	0	0
$295$	4514.18	0.890934
$296$	0	0
$297$	297.000	0.0580259
$298$	0	0
$299$	32.7520	0.00633477
$300$	0	0
$301$	−1602.89	−0.306940
$302$	0	0
$303$	−85.0269	−0.0161210
$304$	0	0
$305$	4062.99	0.762775
$306$	0	0
$307$	4689.00	0.871711	0.435856	−	0.900017i	$-0.356446\pi$
$307$	4689.00	0.871711	0.435856	+	0.900017i	$0.356446\pi$
$308$	0	0
$309$	−785.477	−0.144609
$310$	0	0
$311$	−1601.07	−0.291923	−0.145962	−	0.989290i	$-0.546628\pi$
$311$	−1601.07	−0.291923	−0.145962	+	0.989290i	$0.546628\pi$
$312$	0	0
$313$	−3928.67	−0.709462	−0.354731	−	0.934968i	$-0.615428\pi$
$313$	−3928.67	−0.709462	−0.354731	+	0.934968i	$0.615428\pi$
$314$	0	0
$315$	−1207.21	−0.215932
$316$	0	0
$317$	−363.321	−0.0643727	−0.0321864	−	0.999482i	$-0.510247\pi$
$317$	−363.321	−0.0643727	−0.0321864	+	0.999482i	$0.510247\pi$
$318$	0	0
$319$	−275.738	−0.0483961
$320$	0	0
$321$	2380.79	0.413965
$322$	0	0
$323$	−1594.28	−0.274638
$324$	0	0
$325$	−34.6911	−0.00592097
$326$	0	0
$327$	−1278.22	−0.216165
$328$	0	0
$329$	6730.81	1.12791
$330$	0	0
$331$	1537.28	0.255276	0.127638	−	0.991821i	$-0.459260\pi$
$331$	1537.28	0.255276	0.127638	+	0.991821i	$0.459260\pi$
$332$	0	0
$333$	1110.68	0.182778
$334$	0	0
$335$	−2799.76	−0.456618
$336$	0	0
$337$	−5196.05	−0.839902	−0.419951	−	0.907547i	$-0.637953\pi$
$337$	−5196.05	−0.839902	−0.419951	+	0.907547i	$0.637953\pi$
$338$	0	0
$339$	−841.772	−0.134864
$340$	0	0
$341$	−1288.99	−0.204701
$342$	0	0
$343$	−6911.89	−1.08807
$344$	0	0
$345$	1213.69	0.189400
$346$	0	0
$347$	2447.99	0.378717	0.189359	−	0.981908i	$-0.439359\pi$
$347$	2447.99	0.378717	0.189359	+	0.981908i	$0.439359\pi$
$348$	0	0
$349$	−9199.34	−1.41097	−0.705486	−	0.708724i	$-0.749271\pi$
$349$	−9199.34	−1.41097	−0.705486	+	0.708724i	$0.749271\pi$
$350$	0	0
$351$	−19.0269	−0.00289339
$352$	0	0
$353$	10394.9	1.56732	0.783662	−	0.621187i	$-0.213349\pi$
$353$	10394.9	1.56732	0.783662	+	0.621187i	$0.213349\pi$
$354$	0	0
$355$	−2898.90	−0.433402
$356$	0	0
$357$	−1114.75	−0.165263
$358$	0	0
$359$	−5979.32	−0.879043	−0.439521	−	0.898232i	$-0.644852\pi$
$359$	−5979.32	−0.879043	−0.439521	+	0.898232i	$0.644852\pi$
$360$	0	0
$361$	−2487.93	−0.362724
$362$	0	0
$363$	−363.000	−0.0524864
$364$	0	0
$365$	−6644.55	−0.952854
$366$	0	0
$367$	4646.35	0.660865	0.330433	−	0.943830i	$-0.392805\pi$
$367$	4646.35	0.660865	0.330433	+	0.943830i	$0.392805\pi$
$368$	0	0
$369$	−2541.93	−0.358611
$370$	0	0
$371$	5245.92	0.734109
$372$	0	0
$373$	−2043.23	−0.283631	−0.141815	−	0.989893i	$-0.545294\pi$
$373$	−2043.23	−0.283631	−0.141815	+	0.989893i	$0.545294\pi$
$374$	0	0
$375$	−4549.81	−0.626537
$376$	0	0
$377$	17.6648	0.00241322
$378$	0	0
$379$	−7755.29	−1.05109	−0.525544	−	0.850766i	$-0.676138\pi$
$379$	−7755.29	−1.05109	−0.525544	+	0.850766i	$0.676138\pi$
$380$	0	0
$381$	6813.87	0.916233
$382$	0	0
$383$	6897.32	0.920200	0.460100	−	0.887867i	$-0.347813\pi$
$383$	6897.32	0.920200	0.460100	+	0.887867i	$0.347813\pi$
$384$	0	0
$385$	1475.48	0.195317
$386$	0	0
$387$	−936.181	−0.122968
$388$	0	0
$389$	−2171.44	−0.283025	−0.141512	−	0.989937i	$-0.545196\pi$
$389$	−2171.44	−0.283025	−0.141512	+	0.989937i	$0.545196\pi$
$390$	0	0
$391$	1120.74	0.144957
$392$	0	0
$393$	2693.15	0.345678
$394$	0	0
$395$	5606.64	0.714180
$396$	0	0
$397$	−5764.82	−0.728786	−0.364393	−	0.931245i	$-0.618723\pi$
$397$	−5764.82	−0.728786	−0.364393	+	0.931245i	$0.618723\pi$
$398$	0	0
$399$	3056.34	0.383479
$400$	0	0
$401$	6314.48	0.786360	0.393180	−	0.919462i	$-0.371375\pi$
$401$	6314.48	0.786360	0.393180	+	0.919462i	$0.371375\pi$
$402$	0	0
$403$	82.5776	0.0102072
$404$	0	0
$405$	−705.081	−0.0865080
$406$	0	0
$407$	−1357.50	−0.165329
$408$	0	0
$409$	14223.4	1.71956	0.859781	−	0.510664i	$-0.170600\pi$
$409$	14223.4	1.71956	0.859781	+	0.510664i	$0.170600\pi$
$410$	0	0
$411$	−7872.36	−0.944805
$412$	0	0
$413$	−7991.17	−0.952106
$414$	0	0
$415$	−3950.07	−0.467232
$416$	0	0
$417$	3735.69	0.438699
$418$	0	0
$419$	6310.59	0.735782	0.367891	−	0.929869i	$-0.380080\pi$
$419$	6310.59	0.735782	0.367891	+	0.929869i	$0.380080\pi$
$420$	0	0
$421$	14533.7	1.68250	0.841248	−	0.540649i	$-0.181821\pi$
$421$	14533.7	1.68250	0.841248	+	0.540649i	$0.181821\pi$
$422$	0	0
$423$	3931.19	0.451870
$424$	0	0
$425$	−1187.09	−0.135488
$426$	0	0
$427$	−7192.47	−0.815147
$428$	0	0
$429$	23.2551	0.00261717
$430$	0	0
$431$	7507.83	0.839070	0.419535	−	0.907739i	$-0.362193\pi$
$431$	7507.83	0.839070	0.419535	+	0.907739i	$0.362193\pi$
$432$	0	0
$433$	3530.82	0.391871	0.195936	−	0.980617i	$-0.437226\pi$
$433$	3530.82	0.391871	0.195936	+	0.980617i	$0.437226\pi$
$434$	0	0
$435$	654.605	0.0721515
$436$	0	0
$437$	−3072.75	−0.336361
$438$	0	0
$439$	−601.155	−0.0653566	−0.0326783	−	0.999466i	$-0.510404\pi$
$439$	−601.155	−0.0653566	−0.0326783	+	0.999466i	$0.510404\pi$
$440$	0	0
$441$	−949.954	−0.102576
$442$	0	0
$443$	−13911.2	−1.49196	−0.745981	−	0.665967i	$-0.768019\pi$
$443$	−13911.2	−1.49196	−0.745981	+	0.665967i	$0.768019\pi$
$444$	0	0
$445$	−5731.67	−0.610578
$446$	0	0
$447$	−2880.79	−0.304824
$448$	0	0
$449$	10773.7	1.13239	0.566194	−	0.824272i	$-0.308415\pi$
$449$	10773.7	1.13239	0.566194	+	0.824272i	$0.308415\pi$
$450$	0	0
$451$	3106.80	0.324376
$452$	0	0
$453$	10364.0	1.07493
$454$	0	0
$455$	−94.5244	−0.00973928
$456$	0	0
$457$	−7818.16	−0.800259	−0.400129	−	0.916459i	$-0.631035\pi$
$457$	−7818.16	−0.800259	−0.400129	+	0.916459i	$0.631035\pi$
$458$	0	0
$459$	−651.081	−0.0662088
$460$	0	0
$461$	−1474.58	−0.148977	−0.0744883	−	0.997222i	$-0.523732\pi$
$461$	−1474.58	−0.148977	−0.0744883	+	0.997222i	$0.523732\pi$
$462$	0	0
$463$	−907.518	−0.0910927	−0.0455463	−	0.998962i	$-0.514503\pi$
$463$	−907.518	−0.0910927	−0.0455463	+	0.998962i	$0.514503\pi$
$464$	0	0
$465$	3060.08	0.305178
$466$	0	0
$467$	−19221.9	−1.90467	−0.952337	−	0.305048i	$-0.901327\pi$
$467$	−19221.9	−1.90467	−0.952337	+	0.305048i	$0.901327\pi$
$468$	0	0
$469$	4956.24	0.487970
$470$	0	0
$471$	425.074	0.0415847
$472$	0	0
$473$	1144.22	0.111229
$474$	0	0
$475$	3254.68	0.314389
$476$	0	0
$477$	3063.93	0.294104
$478$	0	0
$479$	4636.66	0.442285	0.221142	−	0.975242i	$-0.429021\pi$
$479$	4636.66	0.442285	0.221142	+	0.975242i	$0.429021\pi$
$480$	0	0
$481$	86.9666	0.00824394
$482$	0	0
$483$	−2148.52	−0.202404
$484$	0	0
$485$	−4062.70	−0.380366
$486$	0	0
$487$	−10913.0	−1.01543	−0.507717	−	0.861524i	$-0.669511\pi$
$487$	−10913.0	−1.01543	−0.507717	+	0.861524i	$0.669511\pi$
$488$	0	0
$489$	−3409.33	−0.315287
$490$	0	0
$491$	9.70613	0.000892122 0	0.000446061	−	1.00000i	$-0.499858\pi$
$491$	9.70613	0.000892122 0	0.000446061		1.00000i	$0.499858\pi$
$492$	0	0
$493$	604.471	0.0552211
$494$	0	0
$495$	861.765	0.0782494
$496$	0	0
$497$	5131.74	0.463159
$498$	0	0
$499$	−15322.3	−1.37459	−0.687295	−	0.726378i	$-0.741202\pi$
$499$	−15322.3	−1.37459	−0.687295	+	0.726378i	$0.741202\pi$
$500$	0	0
$501$	406.308	0.0362325
$502$	0	0
$503$	−19219.9	−1.70372	−0.851862	−	0.523767i	$-0.824526\pi$
$503$	−19219.9	−1.70372	−0.851862	+	0.523767i	$0.824526\pi$
$504$	0	0
$505$	−246.711	−0.0217396
$506$	0	0
$507$	6589.51	0.577220
$508$	0	0
$509$	12013.3	1.04613	0.523064	−	0.852294i	$-0.324789\pi$
$509$	12013.3	1.04613	0.523064	+	0.852294i	$0.324789\pi$
$510$	0	0
$511$	11762.4	1.01828
$512$	0	0
$513$	1785.08	0.153632
$514$	0	0
$515$	−2279.11	−0.195009
$516$	0	0
$517$	−4804.79	−0.408732
$518$	0	0
$519$	1235.21	0.104470
$520$	0	0
$521$	−19336.6	−1.62601	−0.813004	−	0.582259i	$-0.802169\pi$
$521$	−19336.6	−1.62601	−0.813004	+	0.582259i	$0.802169\pi$
$522$	0	0
$523$	6662.88	0.557070	0.278535	−	0.960426i	$-0.410151\pi$
$523$	6662.88	0.557070	0.278535	+	0.960426i	$0.410151\pi$
$524$	0	0
$525$	2275.73	0.189183
$526$	0	0
$527$	2825.72	0.233568
$528$	0	0
$529$	−10006.9	−0.822465
$530$	0	0
$531$	−4667.32	−0.381439
$532$	0	0
$533$	−199.033	−0.0161746
$534$	0	0
$535$	6908.03	0.558243
$536$	0	0
$537$	8402.14	0.675193
$538$	0	0
$539$	1161.05	0.0927832
$540$	0	0
$541$	−17131.2	−1.36142	−0.680709	−	0.732554i	$-0.738328\pi$
$541$	−17131.2	−1.36142	−0.680709	+	0.732554i	$0.738328\pi$
$542$	0	0
$543$	−3456.48	−0.273171
$544$	0	0
$545$	−3708.85	−0.291504
$546$	0	0
$547$	5535.94	0.432723	0.216362	−	0.976313i	$-0.430581\pi$
$547$	5535.94	0.432723	0.216362	+	0.976313i	$0.430581\pi$
$548$	0	0
$549$	−4200.83	−0.326570
$550$	0	0
$551$	−1657.29	−0.128136
$552$	0	0
$553$	−9925.10	−0.763216
$554$	0	0
$555$	3222.73	0.246481
$556$	0	0
$557$	19554.5	1.48753	0.743763	−	0.668444i	$-0.233039\pi$
$557$	19554.5	1.48753	0.743763	+	0.668444i	$0.233039\pi$
$558$	0	0
$559$	−73.3030	−0.00554631
$560$	0	0
$561$	795.765	0.0598881
$562$	0	0
$563$	−8296.40	−0.621051	−0.310525	−	0.950565i	$-0.600505\pi$
$563$	−8296.40	−0.621051	−0.310525	+	0.950565i	$0.600505\pi$
$564$	0	0
$565$	−2442.46	−0.181867
$566$	0	0
$567$	1248.16	0.0924477
$568$	0	0
$569$	−4665.09	−0.343710	−0.171855	−	0.985122i	$-0.554976\pi$
$569$	−4665.09	−0.343710	−0.171855	+	0.985122i	$0.554976\pi$
$570$	0	0
$571$	−16810.8	−1.23207	−0.616034	−	0.787719i	$-0.711262\pi$
$571$	−16810.8	−1.23207	−0.616034	+	0.787719i	$0.711262\pi$
$572$	0	0
$573$	−4212.14	−0.307094
$574$	0	0
$575$	−2287.95	−0.165938
$576$	0	0
$577$	15705.9	1.13318	0.566590	−	0.824000i	$-0.308263\pi$
$577$	15705.9	1.13318	0.566590	+	0.824000i	$0.308263\pi$
$578$	0	0
$579$	10850.4	0.778805
$580$	0	0
$581$	6992.56	0.499312
$582$	0	0
$583$	−3744.80	−0.266027
$584$	0	0
$585$	−55.2078	−0.00390182
$586$	0	0
$587$	−6787.23	−0.477239	−0.238619	−	0.971113i	$-0.576695\pi$
$587$	−6787.23	−0.477239	−0.238619	+	0.971113i	$0.576695\pi$
$588$	0	0
$589$	−7747.33	−0.541975
$590$	0	0
$591$	−9218.48	−0.641620
$592$	0	0
$593$	2341.95	0.162180	0.0810898	−	0.996707i	$-0.474160\pi$
$593$	2341.95	0.162180	0.0810898	+	0.996707i	$0.474160\pi$
$594$	0	0
$595$	−3234.53	−0.222862
$596$	0	0
$597$	11868.1	0.813618
$598$	0	0
$599$	9984.25	0.681044	0.340522	−	0.940237i	$-0.389396\pi$
$599$	9984.25	0.681044	0.340522	+	0.940237i	$0.389396\pi$
$600$	0	0
$601$	−10449.8	−0.709248	−0.354624	−	0.935009i	$-0.615391\pi$
$601$	−10449.8	−0.709248	−0.354624	+	0.935009i	$0.615391\pi$
$602$	0	0
$603$	2894.74	0.195494
$604$	0	0
$605$	−1053.27	−0.0707793
$606$	0	0
$607$	11617.4	0.776829	0.388414	−	0.921485i	$-0.373023\pi$
$607$	11617.4	0.776829	0.388414	+	0.921485i	$0.373023\pi$
$608$	0	0
$609$	−1158.81	−0.0771055
$610$	0	0
$611$	307.812	0.0203809
$612$	0	0
$613$	−8810.28	−0.580495	−0.290248	−	0.956952i	$-0.593738\pi$
$613$	−8810.28	−0.580495	−0.290248	+	0.956952i	$0.593738\pi$
$614$	0	0
$615$	−7375.57	−0.483596
$616$	0	0
$617$	−12093.3	−0.789070	−0.394535	−	0.918881i	$-0.629094\pi$
$617$	−12093.3	−0.789070	−0.394535	+	0.918881i	$0.629094\pi$
$618$	0	0
$619$	−5462.65	−0.354705	−0.177352	−	0.984147i	$-0.556753\pi$
$619$	−5462.65	−0.354705	−0.177352	+	0.984147i	$0.556753\pi$
$620$	0	0
$621$	−1254.87	−0.0810886
$622$	0	0
$623$	10146.4	0.652500
$624$	0	0
$625$	−7048.06	−0.451076
$626$	0	0
$627$	−2181.77	−0.138965
$628$	0	0
$629$	2975.91	0.188644
$630$	0	0
$631$	11476.8	0.724064	0.362032	−	0.932166i	$-0.382083\pi$
$631$	11476.8	0.724064	0.362032	+	0.932166i	$0.382083\pi$
$632$	0	0
$633$	1960.37	0.123093
$634$	0	0
$635$	19770.9	1.23556
$636$	0	0
$637$	−74.3814	−0.00462653
$638$	0	0
$639$	2997.24	0.185554
$640$	0	0
$641$	−14726.8	−0.907445	−0.453723	−	0.891143i	$-0.649904\pi$
$641$	−14726.8	−0.907445	−0.453723	+	0.891143i	$0.649904\pi$
$642$	0	0
$643$	−9071.34	−0.556359	−0.278180	−	0.960529i	$-0.589731\pi$
$643$	−9071.34	−0.556359	−0.278180	+	0.960529i	$0.589731\pi$
$644$	0	0
$645$	−2716.39	−0.165826
$646$	0	0
$647$	−7482.76	−0.454680	−0.227340	−	0.973816i	$-0.573003\pi$
$647$	−7482.76	−0.454680	−0.227340	+	0.973816i	$0.573003\pi$
$648$	0	0
$649$	5704.50	0.345025
$650$	0	0
$651$	−5417.08	−0.326132
$652$	0	0
$653$	4507.72	0.270139	0.135070	−	0.990836i	$-0.456874\pi$
$653$	4507.72	0.270139	0.135070	+	0.990836i	$0.456874\pi$
$654$	0	0
$655$	7814.37	0.466157
$656$	0	0
$657$	6869.96	0.407949
$658$	0	0
$659$	−17626.2	−1.04191	−0.520955	−	0.853584i	$-0.674424\pi$
$659$	−17626.2	−1.04191	−0.520955	+	0.853584i	$0.674424\pi$
$660$	0	0
$661$	9871.14	0.580852	0.290426	−	0.956898i	$-0.406203\pi$
$661$	9871.14	0.580852	0.290426	+	0.956898i	$0.406203\pi$
$662$	0	0
$663$	−50.9796	−0.00298625
$664$	0	0
$665$	8868.16	0.517132
$666$	0	0
$667$	1165.03	0.0676315
$668$	0	0
$669$	15992.3	0.924210
$670$	0	0
$671$	5134.34	0.295394
$672$	0	0
$673$	31483.7	1.80328	0.901639	−	0.432490i	$-0.142365\pi$
$673$	31483.7	1.80328	0.901639	+	0.432490i	$0.142365\pi$
$674$	0	0
$675$	1329.16	0.0757918
$676$	0	0
$677$	19935.8	1.13175	0.565875	−	0.824491i	$-0.308539\pi$
$677$	19935.8	1.13175	0.565875	+	0.824491i	$0.308539\pi$
$678$	0	0
$679$	7191.95	0.406483
$680$	0	0
$681$	108.603	0.00611113
$682$	0	0
$683$	31731.2	1.77769	0.888843	−	0.458211i	$-0.151510\pi$
$683$	31731.2	1.77769	0.888843	+	0.458211i	$0.151510\pi$
$684$	0	0
$685$	−22842.2	−1.27410
$686$	0	0
$687$	−10626.0	−0.590112
$688$	0	0
$689$	239.905	0.0132651
$690$	0	0
$691$	−1452.60	−0.0799705	−0.0399853	−	0.999200i	$-0.512731\pi$
$691$	−1452.60	−0.0799705	−0.0399853	+	0.999200i	$0.512731\pi$
$692$	0	0
$693$	−1525.53	−0.0836221
$694$	0	0
$695$	10839.3	0.591597
$696$	0	0
$697$	−6810.70	−0.370120
$698$	0	0
$699$	391.550	0.0211871
$700$	0	0
$701$	20760.4	1.11856	0.559280	−	0.828979i	$-0.311078\pi$
$701$	20760.4	1.11856	0.559280	+	0.828979i	$0.311078\pi$
$702$	0	0
$703$	−8159.10	−0.437733
$704$	0	0
$705$	11406.6	0.609358
$706$	0	0
$707$	436.738	0.0232323
$708$	0	0
$709$	−27664.4	−1.46539	−0.732693	−	0.680560i	$-0.761737\pi$
$709$	−27664.4	−1.46539	−0.732693	+	0.680560i	$0.761737\pi$
$710$	0	0
$711$	−5796.85	−0.305765
$712$	0	0
$713$	5446.17	0.286060
$714$	0	0
$715$	67.4762	0.00352932
$716$	0	0
$717$	15275.0	0.795616
$718$	0	0
$719$	−3575.46	−0.185455	−0.0927275	−	0.995692i	$-0.529559\pi$
$719$	−3575.46	−0.185455	−0.0927275	+	0.995692i	$0.529559\pi$
$720$	0	0
$721$	4034.58	0.208399
$722$	0	0
$723$	18363.0	0.944575
$724$	0	0
$725$	−1234.01	−0.0632137
$726$	0	0
$727$	−6945.25	−0.354312	−0.177156	−	0.984183i	$-0.556690\pi$
$727$	−6945.25	−0.354312	−0.177156	+	0.984183i	$0.556690\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2508.35	−0.126915
$732$	0	0
$733$	−17645.8	−0.889170	−0.444585	−	0.895737i	$-0.646649\pi$
$733$	−17645.8	−0.889170	−0.444585	+	0.895737i	$0.646649\pi$
$734$	0	0
$735$	−2756.35	−0.138326
$736$	0	0
$737$	−3538.01	−0.176831
$738$	0	0
$739$	22264.2	1.10826	0.554130	−	0.832430i	$-0.313051\pi$
$739$	22264.2	1.10826	0.554130	+	0.832430i	$0.313051\pi$
$740$	0	0
$741$	139.772	0.00692935
$742$	0	0
$743$	−12238.9	−0.604307	−0.302154	−	0.953259i	$-0.597706\pi$
$743$	−12238.9	−0.604307	−0.302154	+	0.953259i	$0.597706\pi$
$744$	0	0
$745$	−8358.79	−0.411064
$746$	0	0
$747$	4084.07	0.200038
$748$	0	0
$749$	−12228.9	−0.596573
$750$	0	0
$751$	−37218.5	−1.80842	−0.904210	−	0.427088i	$-0.859539\pi$
$751$	−37218.5	−1.80842	−0.904210	+	0.427088i	$0.859539\pi$
$752$	0	0
$753$	18438.2	0.892333
$754$	0	0
$755$	30071.9	1.44957
$756$	0	0
$757$	17672.9	0.848523	0.424261	−	0.905540i	$-0.360534\pi$
$757$	17672.9	0.848523	0.424261	+	0.905540i	$0.360534\pi$
$758$	0	0
$759$	1533.72	0.0733474
$760$	0	0
$761$	−1719.08	−0.0818878	−0.0409439	−	0.999161i	$-0.513036\pi$
$761$	−1719.08	−0.0818878	−0.0409439	+	0.999161i	$0.513036\pi$
$762$	0	0
$763$	6565.54	0.311519
$764$	0	0
$765$	−1889.15	−0.0892843
$766$	0	0
$767$	−365.451	−0.0172043
$768$	0	0
$769$	−7164.97	−0.335989	−0.167994	−	0.985788i	$-0.553729\pi$
$769$	−7164.97	−0.335989	−0.167994	+	0.985788i	$0.553729\pi$
$770$	0	0
$771$	15807.4	0.738379
$772$	0	0
$773$	5441.59	0.253196	0.126598	−	0.991954i	$-0.459594\pi$
$773$	5441.59	0.253196	0.126598	+	0.991954i	$0.459594\pi$
$774$	0	0
$775$	−5768.62	−0.267374
$776$	0	0
$777$	−5704.99	−0.263405
$778$	0	0
$779$	18673.0	0.858832
$780$	0	0
$781$	−3663.30	−0.167840
$782$	0	0
$783$	−676.812	−0.0308905
$784$	0	0
$785$	1233.38	0.0560780
$786$	0	0
$787$	20941.9	0.948535	0.474267	−	0.880381i	$-0.342713\pi$
$787$	20941.9	0.948535	0.474267	+	0.880381i	$0.342713\pi$
$788$	0	0
$789$	22489.4	1.01476
$790$	0	0
$791$	4323.73	0.194354
$792$	0	0
$793$	−328.925	−0.0147295
$794$	0	0
$795$	8890.19	0.396607
$796$	0	0
$797$	−2765.45	−0.122908	−0.0614538	−	0.998110i	$-0.519574\pi$
$797$	−2765.45	−0.122908	−0.0614538	+	0.998110i	$0.519574\pi$
$798$	0	0
$799$	10533.0	0.466372
$800$	0	0
$801$	5926.11	0.261409
$802$	0	0
$803$	−8396.62	−0.369004
$804$	0	0
$805$	−6234.09	−0.272948
$806$	0	0
$807$	14819.1	0.646415
$808$	0	0
$809$	−44402.3	−1.92967	−0.964834	−	0.262860i	$-0.915334\pi$
$809$	−44402.3	−1.92967	−0.964834	+	0.262860i	$0.915334\pi$
$810$	0	0
$811$	42366.7	1.83440	0.917198	−	0.398431i	$-0.130445\pi$
$811$	42366.7	1.83440	0.917198	+	0.398431i	$0.130445\pi$
$812$	0	0
$813$	−7135.21	−0.307802
$814$	0	0
$815$	−9892.40	−0.425173
$816$	0	0
$817$	6877.20	0.294495
$818$	0	0
$819$	97.7310	0.00416972
$820$	0	0
$821$	36258.9	1.54135	0.770673	−	0.637231i	$-0.219920\pi$
$821$	36258.9	1.54135	0.770673	+	0.637231i	$0.219920\pi$
$822$	0	0
$823$	13724.9	0.581313	0.290657	−	0.956827i	$-0.406126\pi$
$823$	13724.9	0.581313	0.290657	+	0.956827i	$0.406126\pi$
$824$	0	0
$825$	−1624.53	−0.0685562
$826$	0	0
$827$	−5520.55	−0.232126	−0.116063	−	0.993242i	$-0.537027\pi$
$827$	−5520.55	−0.232126	−0.116063	+	0.993242i	$0.537027\pi$
$828$	0	0
$829$	−39751.1	−1.66539	−0.832697	−	0.553728i	$-0.813205\pi$
$829$	−39751.1	−1.66539	−0.832697	+	0.553728i	$0.813205\pi$
$830$	0	0
$831$	3028.29	0.126414
$832$	0	0
$833$	−2545.25	−0.105868
$834$	0	0
$835$	1178.93	0.0488605
$836$	0	0
$837$	−3163.89	−0.130657
$838$	0	0
$839$	−4446.07	−0.182950	−0.0914752	−	0.995807i	$-0.529158\pi$
$839$	−4446.07	−0.182950	−0.0914752	+	0.995807i	$0.529158\pi$
$840$	0	0
$841$	−23760.6	−0.974236
$842$	0	0
$843$	11910.6	0.486624
$844$	0	0
$845$	19119.9	0.778396
$846$	0	0
$847$	1864.54	0.0756390
$848$	0	0
$849$	−7301.01	−0.295136
$850$	0	0
$851$	5735.64	0.231040
$852$	0	0
$853$	−30413.1	−1.22078	−0.610389	−	0.792102i	$-0.708987\pi$
$853$	−30413.1	−1.22078	−0.610389	+	0.792102i	$0.708987\pi$
$854$	0	0
$855$	5179.53	0.207177
$856$	0	0
$857$	13768.4	0.548799	0.274399	−	0.961616i	$-0.411521\pi$
$857$	13768.4	0.548799	0.274399	+	0.961616i	$0.411521\pi$
$858$	0	0
$859$	−47420.4	−1.88354	−0.941772	−	0.336251i	$-0.890841\pi$
$859$	−47420.4	−1.88354	−0.941772	+	0.336251i	$0.890841\pi$
$860$	0	0
$861$	13056.5	0.516800
$862$	0	0
$863$	15451.5	0.609471	0.304736	−	0.952437i	$-0.401432\pi$
$863$	15451.5	0.609471	0.304736	+	0.952437i	$0.401432\pi$
$864$	0	0
$865$	3584.06	0.140880
$866$	0	0
$867$	12994.5	0.509017
$868$	0	0
$869$	7085.03	0.276575
$870$	0	0
$871$	226.658	0.00881747
$872$	0	0
$873$	4200.52	0.162848
$874$	0	0
$875$	23370.0	0.902913
$876$	0	0
$877$	40774.6	1.56997	0.784983	−	0.619517i	$-0.212672\pi$
$877$	40774.6	1.56997	0.784983	+	0.619517i	$0.212672\pi$
$878$	0	0
$879$	−6788.84	−0.260503
$880$	0	0
$881$	14320.0	0.547619	0.273809	−	0.961784i	$-0.411716\pi$
$881$	14320.0	0.547619	0.273809	+	0.961784i	$0.411716\pi$
$882$	0	0
$883$	44446.1	1.69392	0.846959	−	0.531658i	$-0.178431\pi$
$883$	44446.1	1.69392	0.846959	+	0.531658i	$0.178431\pi$
$884$	0	0
$885$	−13542.5	−0.514381
$886$	0	0
$887$	45117.2	1.70788	0.853939	−	0.520373i	$-0.174207\pi$
$887$	45117.2	1.70788	0.853939	+	0.520373i	$0.174207\pi$
$888$	0	0
$889$	−34999.2	−1.32040
$890$	0	0
$891$	−891.000	−0.0335013
$892$	0	0
$893$	−28878.6	−1.08218
$894$	0	0
$895$	24379.4	0.910516
$896$	0	0
$897$	−98.2560	−0.00365738
$898$	0	0
$899$	2937.39	0.108974
$900$	0	0
$901$	8209.31	0.303543
$902$	0	0
$903$	4808.66	0.177212
$904$	0	0
$905$	−10029.2	−0.368378
$906$	0	0
$907$	−29201.1	−1.06903	−0.534513	−	0.845160i	$-0.679505\pi$
$907$	−29201.1	−1.06903	−0.534513	+	0.845160i	$0.679505\pi$
$908$	0	0
$909$	255.081	0.00930747
$910$	0	0
$911$	2138.29	0.0777658	0.0388829	−	0.999244i	$-0.487620\pi$
$911$	2138.29	0.0777658	0.0388829	+	0.999244i	$0.487620\pi$
$912$	0	0
$913$	−4991.64	−0.180941
$914$	0	0
$915$	−12189.0	−0.440388
$916$	0	0
$917$	−13833.3	−0.498163
$918$	0	0
$919$	3250.54	0.116676	0.0583381	−	0.998297i	$-0.481420\pi$
$919$	3250.54	0.116676	0.0583381	+	0.998297i	$0.481420\pi$
$920$	0	0
$921$	−14067.0	−0.503283
$922$	0	0
$923$	234.684	0.00836914
$924$	0	0
$925$	−6075.22	−0.215948
$926$	0	0
$927$	2356.43	0.0834901
$928$	0	0
$929$	44175.2	1.56011	0.780055	−	0.625711i	$-0.215191\pi$
$929$	44175.2	1.56011	0.780055	+	0.625711i	$0.215191\pi$
$930$	0	0
$931$	6978.37	0.245657
$932$	0	0
$933$	4803.20	0.168542
$934$	0	0
$935$	2308.97	0.0807607
$936$	0	0
$937$	21311.9	0.743039	0.371520	−	0.928425i	$-0.378837\pi$
$937$	21311.9	0.743039	0.371520	+	0.928425i	$0.378837\pi$
$938$	0	0
$939$	11786.0	0.409608
$940$	0	0
$941$	5671.45	0.196476	0.0982381	−	0.995163i	$-0.468679\pi$
$941$	5671.45	0.196476	0.0982381	+	0.995163i	$0.468679\pi$
$942$	0	0
$943$	−13126.7	−0.453301
$944$	0	0
$945$	3621.62	0.124668
$946$	0	0
$947$	−7820.51	−0.268355	−0.134178	−	0.990957i	$-0.542839\pi$
$947$	−7820.51	−0.268355	−0.134178	+	0.990957i	$0.542839\pi$
$948$	0	0
$949$	537.918	0.0184000
$950$	0	0
$951$	1089.96	0.0371656
$952$	0	0
$953$	27278.2	0.927205	0.463603	−	0.886043i	$-0.346557\pi$
$953$	27278.2	0.927205	0.463603	+	0.886043i	$0.346557\pi$
$954$	0	0
$955$	−12221.8	−0.414124
$956$	0	0
$957$	827.214	0.0279415
$958$	0	0
$959$	40436.1	1.36158
$960$	0	0
$961$	−16059.6	−0.539074
$962$	0	0
$963$	−7142.38	−0.239003
$964$	0	0
$965$	31483.2	1.05024
$966$	0	0
$967$	46414.3	1.54352	0.771760	−	0.635914i	$-0.219377\pi$
$967$	46414.3	1.54352	0.771760	+	0.635914i	$0.219377\pi$
$968$	0	0
$969$	4782.85	0.158563
$970$	0	0
$971$	10453.6	0.345490	0.172745	−	0.984967i	$-0.444736\pi$
$971$	10453.6	0.345490	0.172745	+	0.984967i	$0.444736\pi$
$972$	0	0
$973$	−19188.2	−0.632216
$974$	0	0
$975$	104.073	0.00341848
$976$	0	0
$977$	34305.5	1.12337	0.561684	−	0.827352i	$-0.310154\pi$
$977$	34305.5	1.12337	0.561684	+	0.827352i	$0.310154\pi$
$978$	0	0
$979$	−7243.02	−0.236453
$980$	0	0
$981$	3834.67	0.124803
$982$	0	0
$983$	6251.79	0.202850	0.101425	−	0.994843i	$-0.467660\pi$
$983$	6251.79	0.202850	0.101425	+	0.994843i	$0.467660\pi$
$984$	0	0
$985$	−26748.0	−0.865242
$986$	0	0
$987$	−20192.4	−0.651197
$988$	0	0
$989$	−4834.49	−0.155438
$990$	0	0
$991$	36675.1	1.17560	0.587801	−	0.809005i	$-0.299994\pi$
$991$	36675.1	1.17560	0.587801	+	0.809005i	$0.299994\pi$
$992$	0	0
$993$	−4611.83	−0.147384
$994$	0	0
$995$	34436.1	1.09719
$996$	0	0
$997$	−3203.74	−0.101769	−0.0508844	−	0.998705i	$-0.516204\pi$
$997$	−3203.74	−0.101769	−0.0508844	+	0.998705i	$0.516204\pi$
$998$	0	0
$999$	−3332.05	−0.105527

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -8.70470 q^{5} +15.4094 q^{7} +9.00000 q^{9} -11.0000 q^{11} +0.704700 q^{13} +26.1141 q^{15} +24.1141 q^{17} -66.1141 q^{19} -46.2282 q^{21} +46.4765 q^{23} -49.2282 q^{25} -27.0000 q^{27} +25.0671 q^{29} +117.181 q^{31} +33.0000 q^{33} -134.134 q^{35} +123.409 q^{37} -2.11410 q^{39} -282.436 q^{41} -104.020 q^{43} -78.3423 q^{45} +436.799 q^{47} -105.550 q^{49} -72.3423 q^{51} +340.436 q^{53} +95.7517 q^{55} +198.342 q^{57} -518.591 q^{59} -466.758 q^{61} +138.685 q^{63} -6.13420 q^{65} +321.638 q^{67} -139.430 q^{69} +333.027 q^{71} +763.329 q^{73} +147.685 q^{75} -169.503 q^{77} -644.094 q^{79} +81.0000 q^{81} +453.785 q^{83} -209.906 q^{85} -75.2013 q^{87} +658.456 q^{89} +10.8590 q^{91} -351.544 q^{93} +575.503 q^{95} +466.725 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} - 16 q^{7} + 18 q^{9} - 22 q^{11} - 22 q^{13} - 18 q^{15} - 22 q^{17} - 62 q^{19} + 48 q^{21} + 210 q^{23} + 42 q^{25} - 54 q^{27} + 214 q^{29} + 328 q^{31} + 66 q^{33} - 596 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 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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