LMFDB - Embedded newform 2112.4.a.ba.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,-16,0,2] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +3.48913 q^{5} -4.74456 q^{7} +9.00000 q^{9} -11.0000 q^{11} +15.0217 q^{13} -10.4674 q^{15} +73.1684 q^{17} +78.7011 q^{19} +14.2337 q^{21} +112.000 q^{23} -112.826 q^{25} -27.0000 q^{27} -243.125 q^{29} +278.717 q^{31} +33.0000 q^{33} -16.5544 q^{35} -102.380 q^{37} -45.0652 q^{39} -241.255 q^{41} +280.016 q^{43} +31.4021 q^{45} -169.870 q^{47} -320.489 q^{49} -219.505 q^{51} +409.652 q^{53} -38.3804 q^{55} -236.103 q^{57} -196.000 q^{59} +701.359 q^{61} -42.7011 q^{63} +52.4128 q^{65} -900.587 q^{67} -336.000 q^{69} +756.500 q^{71} -1019.81 q^{73} +338.478 q^{75} +52.1902 q^{77} -327.549 q^{79} +81.0000 q^{81} +756.619 q^{83} +255.294 q^{85} +729.375 q^{87} +508.978 q^{89} -71.2716 q^{91} -836.152 q^{93} +274.598 q^{95} +614.358 q^{97} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 16 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} + 76 q^{13} + 48 q^{15} - 26 q^{17} + 54 q^{19} - 6 q^{21} + 224 q^{23} + 142 q^{25} - 54 q^{27} - 222 q^{29} - 40 q^{31} + 66 q^{33} - 148 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 16 * q^5 + 2 * q^7 + 18 * q^9 - 22 * q^11 + 76 * q^13 + 48 * q^15 - 26 * q^17 + 54 * q^19 - 6 * q^21 + 224 * q^23 + 142 * q^25 - 54 * q^27 - 222 * q^29 - 40 * q^31 + 66 * q^33 - 148 * q^35 + 48 * q^37 - 228 * q^39 - 494 * q^41 + 66 * q^43 - 144 * q^45 - 64 * q^47 - 618 * q^49 + 78 * q^51 + 84 * q^53 + 176 * q^55 - 162 * q^57 - 392 * q^59 + 1104 * q^61 + 18 * q^63 - 1136 * q^65 - 928 * q^67 - 672 * q^69 + 456 * q^71 - 592 * q^73 - 426 * q^75 - 22 * q^77 - 230 * q^79 + 162 * q^81 - 348 * q^83 + 2188 * q^85 + 666 * q^87 + 972 * q^89 + 340 * q^91 + 120 * q^93 + 756 * q^95 - 1184 * 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
$5$	3.48913	0.312077	0.156038	−	0.987751i	$-0.450128\pi$
$5$	3.48913	0.312077	0.156038	+	0.987751i	$0.450128\pi$
$6$	0	0
$7$	−4.74456	−0.256182	−0.128091	−	0.991762i	$-0.540885\pi$
$7$	−4.74456	−0.256182	−0.128091	+	0.991762i	$0.540885\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	15.0217	0.320483	0.160242	−	0.987078i	$-0.448773\pi$
$13$	15.0217	0.320483	0.160242	+	0.987078i	$0.448773\pi$
$14$	0	0
$15$	−10.4674	−0.180178
$16$	0	0
$17$	73.1684	1.04388	0.521940	−	0.852982i	$-0.325209\pi$
$17$	73.1684	1.04388	0.521940	+	0.852982i	$0.325209\pi$
$18$	0	0
$19$	78.7011	0.950277	0.475138	−	0.879911i	$-0.342398\pi$
$19$	78.7011	0.950277	0.475138	+	0.879911i	$0.342398\pi$
$20$	0	0
$21$	14.2337	0.147907
$22$	0	0
$23$	112.000	1.01537	0.507687	−	0.861541i	$-0.330501\pi$
$23$	112.000	1.01537	0.507687	+	0.861541i	$0.330501\pi$
$24$	0	0
$25$	−112.826	−0.902608
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−243.125	−1.55680	−0.778399	−	0.627769i	$-0.783968\pi$
$29$	−243.125	−1.55680	−0.778399	+	0.627769i	$0.783968\pi$
$30$	0	0
$31$	278.717	1.61481	0.807405	−	0.589998i	$-0.200871\pi$
$31$	278.717	1.61481	0.807405	+	0.589998i	$0.200871\pi$
$32$	0	0
$33$	33.0000	0.174078
$34$	0	0
$35$	−16.5544	−0.0799486
$36$	0	0
$37$	−102.380	−0.454898	−0.227449	−	0.973790i	$-0.573039\pi$
$37$	−102.380	−0.454898	−0.227449	+	0.973790i	$0.573039\pi$
$38$	0	0
$39$	−45.0652	−0.185031
$40$	0	0
$41$	−241.255	−0.918970	−0.459485	−	0.888186i	$-0.651966\pi$
$41$	−241.255	−0.918970	−0.459485	+	0.888186i	$0.651966\pi$
$42$	0	0
$43$	280.016	0.993071	0.496536	−	0.868016i	$-0.334605\pi$
$43$	280.016	0.993071	0.496536	+	0.868016i	$0.334605\pi$
$44$	0	0
$45$	31.4021	0.104026
$46$	0	0
$47$	−169.870	−0.527192	−0.263596	−	0.964633i	$-0.584909\pi$
$47$	−169.870	−0.527192	−0.263596	+	0.964633i	$0.584909\pi$
$48$	0	0
$49$	−320.489	−0.934371
$50$	0	0
$51$	−219.505	−0.602684
$52$	0	0
$53$	409.652	1.06170	0.530849	−	0.847466i	$-0.321873\pi$
$53$	409.652	1.06170	0.530849	+	0.847466i	$0.321873\pi$
$54$	0	0
$55$	−38.3804	−0.0940947
$56$	0	0
$57$	−236.103	−0.548643
$58$	0	0
$59$	−196.000	−0.432492	−0.216246	−	0.976339i	$-0.569381\pi$
$59$	−196.000	−0.432492	−0.216246	+	0.976339i	$0.569381\pi$
$60$	0	0
$61$	701.359	1.47213	0.736064	−	0.676912i	$-0.236682\pi$
$61$	701.359	1.47213	0.736064	+	0.676912i	$0.236682\pi$
$62$	0	0
$63$	−42.7011	−0.0853941
$64$	0	0
$65$	52.4128	0.100015
$66$	0	0
$67$	−900.587	−1.64215	−0.821076	−	0.570819i	$-0.806626\pi$
$67$	−900.587	−1.64215	−0.821076	+	0.570819i	$0.806626\pi$
$68$	0	0
$69$	−336.000	−0.586227
$70$	0	0
$71$	756.500	1.26451	0.632254	−	0.774762i	$-0.282130\pi$
$71$	756.500	1.26451	0.632254	+	0.774762i	$0.282130\pi$
$72$	0	0
$73$	−1019.81	−1.63507	−0.817536	−	0.575877i	$-0.804661\pi$
$73$	−1019.81	−1.63507	−0.817536	+	0.575877i	$0.804661\pi$
$74$	0	0
$75$	338.478	0.521121
$76$	0	0
$77$	52.1902	0.0772419
$78$	0	0
$79$	−327.549	−0.466483	−0.233241	−	0.972419i	$-0.574933\pi$
$79$	−327.549	−0.466483	−0.233241	+	0.972419i	$0.574933\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	756.619	1.00060	0.500300	−	0.865852i	$-0.333223\pi$
$83$	756.619	1.00060	0.500300	+	0.865852i	$0.333223\pi$
$84$	0	0
$85$	255.294	0.325771
$86$	0	0
$87$	729.375	0.898818
$88$	0	0
$89$	508.978	0.606198	0.303099	−	0.952959i	$-0.401979\pi$
$89$	508.978	0.606198	0.303099	+	0.952959i	$0.401979\pi$
$90$	0	0
$91$	−71.2716	−0.0821022
$92$	0	0
$93$	−836.152	−0.932311
$94$	0	0
$95$	274.598	0.296559
$96$	0	0
$97$	614.358	0.643079	0.321539	−	0.946896i	$-0.395800\pi$
$97$	614.358	0.643079	0.321539	+	0.946896i	$0.395800\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	1015.92	1.00087	0.500434	−	0.865775i	$-0.333174\pi$
$101$	1015.92	1.00087	0.500434	+	0.865775i	$0.333174\pi$
$102$	0	0
$103$	1102.16	1.05436	0.527181	−	0.849753i	$-0.323249\pi$
$103$	1102.16	1.05436	0.527181	+	0.849753i	$0.323249\pi$
$104$	0	0
$105$	49.6631	0.0461583
$106$	0	0
$107$	−1377.58	−1.24463	−0.622315	−	0.782767i	$-0.713808\pi$
$107$	−1377.58	−1.24463	−0.622315	+	0.782767i	$0.713808\pi$
$108$	0	0
$109$	−320.217	−0.281388	−0.140694	−	0.990053i	$-0.544933\pi$
$109$	−320.217	−0.281388	−0.140694	+	0.990053i	$0.544933\pi$
$110$	0	0
$111$	307.141	0.262636
$112$	0	0
$113$	−1629.45	−1.35651	−0.678254	−	0.734828i	$-0.737263\pi$
$113$	−1629.45	−1.35651	−0.678254	+	0.734828i	$0.737263\pi$
$114$	0	0
$115$	390.782	0.316875
$116$	0	0
$117$	135.196	0.106828
$118$	0	0
$119$	−347.152	−0.267423
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	723.766	0.530568
$124$	0	0
$125$	−829.805	−0.593760
$126$	0	0
$127$	2291.26	1.60091	0.800457	−	0.599390i	$-0.204590\pi$
$127$	2291.26	1.60091	0.800457	+	0.599390i	$0.204590\pi$
$128$	0	0
$129$	−840.049	−0.573350
$130$	0	0
$131$	1147.41	0.765267	0.382633	−	0.923900i	$-0.375017\pi$
$131$	1147.41	0.765267	0.382633	+	0.923900i	$0.375017\pi$
$132$	0	0
$133$	−373.402	−0.243444
$134$	0	0
$135$	−94.2064	−0.0600592
$136$	0	0
$137$	1268.60	0.791121	0.395561	−	0.918440i	$-0.370550\pi$
$137$	1268.60	0.791121	0.395561	+	0.918440i	$0.370550\pi$
$138$	0	0
$139$	486.288	0.296737	0.148368	−	0.988932i	$-0.452598\pi$
$139$	486.288	0.296737	0.148368	+	0.988932i	$0.452598\pi$
$140$	0	0
$141$	509.609	0.304374
$142$	0	0
$143$	−165.239	−0.0966294
$144$	0	0
$145$	−848.293	−0.485841
$146$	0	0
$147$	961.467	0.539459
$148$	0	0
$149$	−2354.11	−1.29434	−0.647169	−	0.762346i	$-0.724047\pi$
$149$	−2354.11	−1.29434	−0.647169	+	0.762346i	$0.724047\pi$
$150$	0	0
$151$	−570.070	−0.307229	−0.153615	−	0.988131i	$-0.549091\pi$
$151$	−570.070	−0.307229	−0.153615	+	0.988131i	$0.549091\pi$
$152$	0	0
$153$	658.516	0.347960
$154$	0	0
$155$	972.479	0.503945
$156$	0	0
$157$	2072.67	1.05361	0.526807	−	0.849985i	$-0.323389\pi$
$157$	2072.67	1.05361	0.526807	+	0.849985i	$0.323389\pi$
$158$	0	0
$159$	−1228.96	−0.612972
$160$	0	0
$161$	−531.391	−0.260121
$162$	0	0
$163$	−2676.51	−1.28614	−0.643069	−	0.765808i	$-0.722339\pi$
$163$	−2676.51	−1.28614	−0.643069	+	0.765808i	$0.722339\pi$
$164$	0	0
$165$	115.141	0.0543256
$166$	0	0
$167$	−1188.12	−0.550536	−0.275268	−	0.961368i	$-0.588767\pi$
$167$	−1188.12	−0.550536	−0.275268	+	0.961368i	$0.588767\pi$
$168$	0	0
$169$	−1971.35	−0.897290
$170$	0	0
$171$	708.310	0.316759
$172$	0	0
$173$	−807.147	−0.354718	−0.177359	−	0.984146i	$-0.556755\pi$
$173$	−807.147	−0.354718	−0.177359	+	0.984146i	$0.556755\pi$
$174$	0	0
$175$	535.310	0.231232
$176$	0	0
$177$	588.000	0.249699
$178$	0	0
$179$	1950.39	0.814408	0.407204	−	0.913337i	$-0.366504\pi$
$179$	1950.39	0.814408	0.407204	+	0.913337i	$0.366504\pi$
$180$	0	0
$181$	−1061.61	−0.435959	−0.217980	−	0.975953i	$-0.569947\pi$
$181$	−1061.61	−0.435959	−0.217980	+	0.975953i	$0.569947\pi$
$182$	0	0
$183$	−2104.08	−0.849933
$184$	0	0
$185$	−357.218	−0.141963
$186$	0	0
$187$	−804.853	−0.314742
$188$	0	0
$189$	128.103	0.0493023
$190$	0	0
$191$	2136.41	0.809348	0.404674	−	0.914461i	$-0.367385\pi$
$191$	2136.41	0.809348	0.404674	+	0.914461i	$0.367385\pi$
$192$	0	0
$193$	3947.76	1.47236	0.736181	−	0.676784i	$-0.236627\pi$
$193$	3947.76	1.47236	0.736181	+	0.676784i	$0.236627\pi$
$194$	0	0
$195$	−157.238	−0.0577439
$196$	0	0
$197$	−923.886	−0.334133	−0.167066	−	0.985946i	$-0.553429\pi$
$197$	−923.886	−0.334133	−0.167066	+	0.985946i	$0.553429\pi$
$198$	0	0
$199$	−476.152	−0.169616	−0.0848078	−	0.996397i	$-0.527028\pi$
$199$	−476.152	−0.169616	−0.0848078	+	0.996397i	$0.527028\pi$
$200$	0	0
$201$	2701.76	0.948097
$202$	0	0
$203$	1153.52	0.398824
$204$	0	0
$205$	−841.770	−0.286789
$206$	0	0
$207$	1008.00	0.338458
$208$	0	0
$209$	−865.712	−0.286519
$210$	0	0
$211$	4918.24	1.60467	0.802336	−	0.596872i	$-0.203590\pi$
$211$	4918.24	1.60467	0.802336	+	0.596872i	$0.203590\pi$
$212$	0	0
$213$	−2269.50	−0.730064
$214$	0	0
$215$	977.012	0.309915
$216$	0	0
$217$	−1322.39	−0.413686
$218$	0	0
$219$	3059.44	0.944010
$220$	0	0
$221$	1099.12	0.334546
$222$	0	0
$223$	2100.29	0.630700	0.315350	−	0.948975i	$-0.397878\pi$
$223$	2100.29	0.630700	0.315350	+	0.948975i	$0.397878\pi$
$224$	0	0
$225$	−1015.43	−0.300869
$226$	0	0
$227$	2257.16	0.659970	0.329985	−	0.943986i	$-0.392956\pi$
$227$	2257.16	0.659970	0.329985	+	0.943986i	$0.392956\pi$
$228$	0	0
$229$	5311.07	1.53260	0.766301	−	0.642482i	$-0.222095\pi$
$229$	5311.07	1.53260	0.766301	+	0.642482i	$0.222095\pi$
$230$	0	0
$231$	−156.571	−0.0445956
$232$	0	0
$233$	2466.27	0.693435	0.346718	−	0.937970i	$-0.387296\pi$
$233$	2466.27	0.693435	0.346718	+	0.937970i	$0.387296\pi$
$234$	0	0
$235$	−592.696	−0.164524
$236$	0	0
$237$	982.646	0.269324
$238$	0	0
$239$	1429.40	0.386863	0.193432	−	0.981114i	$-0.438038\pi$
$239$	1429.40	0.386863	0.193432	+	0.981114i	$0.438038\pi$
$240$	0	0
$241$	−978.989	−0.261669	−0.130835	−	0.991404i	$-0.541766\pi$
$241$	−978.989	−0.261669	−0.130835	+	0.991404i	$0.541766\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1118.23	−0.291595
$246$	0	0
$247$	1182.23	0.304548
$248$	0	0
$249$	−2269.86	−0.577696
$250$	0	0
$251$	6530.63	1.64227	0.821135	−	0.570734i	$-0.193341\pi$
$251$	6530.63	1.64227	0.821135	+	0.570734i	$0.193341\pi$
$252$	0	0
$253$	−1232.00	−0.306147
$254$	0	0
$255$	−765.882	−0.188084
$256$	0	0
$257$	8130.26	1.97335	0.986676	−	0.162696i	$-0.0520188\pi$
$257$	8130.26	1.97335	0.986676	+	0.162696i	$0.0520188\pi$
$258$	0	0
$259$	485.750	0.116537
$260$	0	0
$261$	−2188.12	−0.518933
$262$	0	0
$263$	−4549.42	−1.06665	−0.533326	−	0.845910i	$-0.679058\pi$
$263$	−4549.42	−1.06665	−0.533326	+	0.845910i	$0.679058\pi$
$264$	0	0
$265$	1429.33	0.331332
$266$	0	0
$267$	−1526.93	−0.349988
$268$	0	0
$269$	29.1522	0.00660760	0.00330380	−	0.999995i	$-0.498948\pi$
$269$	29.1522	0.00660760	0.00330380	+	0.999995i	$0.498948\pi$
$270$	0	0
$271$	7711.22	1.72850	0.864250	−	0.503063i	$-0.167794\pi$
$271$	7711.22	1.72850	0.864250	+	0.503063i	$0.167794\pi$
$272$	0	0
$273$	213.815	0.0474017
$274$	0	0
$275$	1241.09	0.272147
$276$	0	0
$277$	−1127.52	−0.244571	−0.122286	−	0.992495i	$-0.539022\pi$
$277$	−1127.52	−0.244571	−0.122286	+	0.992495i	$0.539022\pi$
$278$	0	0
$279$	2508.46	0.538270
$280$	0	0
$281$	−1872.47	−0.397517	−0.198758	−	0.980049i	$-0.563691\pi$
$281$	−1872.47	−0.397517	−0.198758	+	0.980049i	$0.563691\pi$
$282$	0	0
$283$	−2124.48	−0.446245	−0.223123	−	0.974790i	$-0.571625\pi$
$283$	−2124.48	−0.446245	−0.223123	+	0.974790i	$0.571625\pi$
$284$	0	0
$285$	−823.794	−0.171219
$286$	0	0
$287$	1144.65	0.235424
$288$	0	0
$289$	440.621	0.0896846
$290$	0	0
$291$	−1843.07	−0.371282
$292$	0	0
$293$	3324.19	0.662802	0.331401	−	0.943490i	$-0.392479\pi$
$293$	3324.19	0.662802	0.331401	+	0.943490i	$0.392479\pi$
$294$	0	0
$295$	−683.869	−0.134971
$296$	0	0
$297$	297.000	0.0580259
$298$	0	0
$299$	1682.44	0.325411
$300$	0	0
$301$	−1328.55	−0.254407
$302$	0	0
$303$	−3047.75	−0.577851
$304$	0	0
$305$	2447.13	0.459417
$306$	0	0
$307$	1698.94	0.315843	0.157921	−	0.987452i	$-0.449521\pi$
$307$	1698.94	0.315843	0.157921	+	0.987452i	$0.449521\pi$
$308$	0	0
$309$	−3306.49	−0.608736
$310$	0	0
$311$	6928.83	1.26334	0.631668	−	0.775239i	$-0.282370\pi$
$311$	6928.83	1.26334	0.631668	+	0.775239i	$0.282370\pi$
$312$	0	0
$313$	−3560.75	−0.643020	−0.321510	−	0.946906i	$-0.604190\pi$
$313$	−3560.75	−0.643020	−0.321510	+	0.946906i	$0.604190\pi$
$314$	0	0
$315$	−148.989	−0.0266495
$316$	0	0
$317$	−332.750	−0.0589561	−0.0294780	−	0.999565i	$-0.509385\pi$
$317$	−332.750	−0.0589561	−0.0294780	+	0.999565i	$0.509385\pi$
$318$	0	0
$319$	2674.37	0.469393
$320$	0	0
$321$	4132.73	0.718587
$322$	0	0
$323$	5758.43	0.991975
$324$	0	0
$325$	−1694.84	−0.289271
$326$	0	0
$327$	960.652	0.162459
$328$	0	0
$329$	805.957	0.135057
$330$	0	0
$331$	541.445	0.0899108	0.0449554	−	0.998989i	$-0.485685\pi$
$331$	541.445	0.0899108	0.0449554	+	0.998989i	$0.485685\pi$
$332$	0	0
$333$	−921.423	−0.151633
$334$	0	0
$335$	−3142.26	−0.512478
$336$	0	0
$337$	816.531	0.131986	0.0659930	−	0.997820i	$-0.478978\pi$
$337$	816.531	0.131986	0.0659930	+	0.997820i	$0.478978\pi$
$338$	0	0
$339$	4888.34	0.783180
$340$	0	0
$341$	−3065.89	−0.486883
$342$	0	0
$343$	3147.97	0.495552
$344$	0	0
$345$	−1172.35	−0.182948
$346$	0	0
$347$	−6260.53	−0.968539	−0.484269	−	0.874919i	$-0.660914\pi$
$347$	−6260.53	−0.968539	−0.484269	+	0.874919i	$0.660914\pi$
$348$	0	0
$349$	12768.5	1.95840	0.979198	−	0.202906i	$-0.0650386\pi$
$349$	12768.5	1.95840	0.979198	+	0.202906i	$0.0650386\pi$
$350$	0	0
$351$	−405.587	−0.0616771
$352$	0	0
$353$	−2649.28	−0.399453	−0.199727	−	0.979852i	$-0.564005\pi$
$353$	−2649.28	−0.399453	−0.199727	+	0.979852i	$0.564005\pi$
$354$	0	0
$355$	2639.52	0.394623
$356$	0	0
$357$	1041.46	0.154397
$358$	0	0
$359$	−3203.91	−0.471020	−0.235510	−	0.971872i	$-0.575676\pi$
$359$	−3203.91	−0.471020	−0.235510	+	0.971872i	$0.575676\pi$
$360$	0	0
$361$	−665.143	−0.0969737
$362$	0	0
$363$	−363.000	−0.0524864
$364$	0	0
$365$	−3558.26	−0.510268
$366$	0	0
$367$	−8429.40	−1.19894	−0.599470	−	0.800397i	$-0.704622\pi$
$367$	−8429.40	−1.19894	−0.599470	+	0.800397i	$0.704622\pi$
$368$	0	0
$369$	−2171.30	−0.306323
$370$	0	0
$371$	−1943.62	−0.271988
$372$	0	0
$373$	9388.53	1.30327	0.651635	−	0.758533i	$-0.274083\pi$
$373$	9388.53	1.30327	0.651635	+	0.758533i	$0.274083\pi$
$374$	0	0
$375$	2489.41	0.342807
$376$	0	0
$377$	−3652.16	−0.498928
$378$	0	0
$379$	14264.5	1.93329	0.966647	−	0.256112i	$-0.0824415\pi$
$379$	14264.5	1.93329	0.966647	+	0.256112i	$0.0824415\pi$
$380$	0	0
$381$	−6873.77	−0.924288
$382$	0	0
$383$	13462.2	1.79605	0.898026	−	0.439942i	$-0.145001\pi$
$383$	13462.2	1.79605	0.898026	+	0.439942i	$0.145001\pi$
$384$	0	0
$385$	182.098	0.0241054
$386$	0	0
$387$	2520.15	0.331024
$388$	0	0
$389$	941.881	0.122764	0.0613821	−	0.998114i	$-0.480449\pi$
$389$	941.881	0.122764	0.0613821	+	0.998114i	$0.480449\pi$
$390$	0	0
$391$	8194.87	1.05993
$392$	0	0
$393$	−3442.24	−0.441827
$394$	0	0
$395$	−1142.86	−0.145578
$396$	0	0
$397$	847.839	0.107183	0.0535917	−	0.998563i	$-0.482933\pi$
$397$	847.839	0.107183	0.0535917	+	0.998563i	$0.482933\pi$
$398$	0	0
$399$	1120.21	0.140553
$400$	0	0
$401$	12203.6	1.51975	0.759875	−	0.650069i	$-0.225260\pi$
$401$	12203.6	1.51975	0.759875	+	0.650069i	$0.225260\pi$
$402$	0	0
$403$	4186.82	0.517520
$404$	0	0
$405$	282.619	0.0346752
$406$	0	0
$407$	1126.18	0.137157
$408$	0	0
$409$	8759.53	1.05900	0.529500	−	0.848310i	$-0.322380\pi$
$409$	8759.53	1.05900	0.529500	+	0.848310i	$0.322380\pi$
$410$	0	0
$411$	−3805.79	−0.456754
$412$	0	0
$413$	929.934	0.110797
$414$	0	0
$415$	2639.94	0.312264
$416$	0	0
$417$	−1458.86	−0.171321
$418$	0	0
$419$	11188.4	1.30451	0.652256	−	0.757999i	$-0.273823\pi$
$419$	11188.4	1.30451	0.652256	+	0.757999i	$0.273823\pi$
$420$	0	0
$421$	14082.3	1.63023	0.815116	−	0.579298i	$-0.196673\pi$
$421$	14082.3	1.63023	0.815116	+	0.579298i	$0.196673\pi$
$422$	0	0
$423$	−1528.83	−0.175731
$424$	0	0
$425$	−8255.30	−0.942214
$426$	0	0
$427$	−3327.64	−0.377133
$428$	0	0
$429$	495.718	0.0557890
$430$	0	0
$431$	−5616.05	−0.627647	−0.313823	−	0.949481i	$-0.601610\pi$
$431$	−5616.05	−0.627647	−0.313823	+	0.949481i	$0.601610\pi$
$432$	0	0
$433$	7195.75	0.798627	0.399314	−	0.916814i	$-0.369248\pi$
$433$	7195.75	0.798627	0.399314	+	0.916814i	$0.369248\pi$
$434$	0	0
$435$	2544.88	0.280500
$436$	0	0
$437$	8814.52	0.964887
$438$	0	0
$439$	101.959	0.0110848	0.00554240	−	0.999985i	$-0.498236\pi$
$439$	101.959	0.0110848	0.00554240	+	0.999985i	$0.498236\pi$
$440$	0	0
$441$	−2884.40	−0.311457
$442$	0	0
$443$	−4953.74	−0.531285	−0.265642	−	0.964072i	$-0.585584\pi$
$443$	−4953.74	−0.531285	−0.265642	+	0.964072i	$0.585584\pi$
$444$	0	0
$445$	1775.89	0.189180
$446$	0	0
$447$	7062.34	0.747287
$448$	0	0
$449$	−11602.0	−1.21945	−0.609723	−	0.792615i	$-0.708719\pi$
$449$	−11602.0	−1.21945	−0.609723	+	0.792615i	$0.708719\pi$
$450$	0	0
$451$	2653.81	0.277080
$452$	0	0
$453$	1710.21	0.177379
$454$	0	0
$455$	−248.676	−0.0256222
$456$	0	0
$457$	−3530.68	−0.361397	−0.180698	−	0.983539i	$-0.557836\pi$
$457$	−3530.68	−0.361397	−0.180698	+	0.983539i	$0.557836\pi$
$458$	0	0
$459$	−1975.55	−0.200895
$460$	0	0
$461$	−11566.3	−1.16854	−0.584271	−	0.811559i	$-0.698619\pi$
$461$	−11566.3	−1.16854	−0.584271	+	0.811559i	$0.698619\pi$
$462$	0	0
$463$	10888.5	1.09294	0.546470	−	0.837479i	$-0.315971\pi$
$463$	10888.5	1.09294	0.546470	+	0.837479i	$0.315971\pi$
$464$	0	0
$465$	−2917.44	−0.290953
$466$	0	0
$467$	−10688.0	−1.05906	−0.529529	−	0.848292i	$-0.677631\pi$
$467$	−10688.0	−1.05906	−0.529529	+	0.848292i	$0.677631\pi$
$468$	0	0
$469$	4272.89	0.420690
$470$	0	0
$471$	−6218.02	−0.608304
$472$	0	0
$473$	−3080.18	−0.299422
$474$	0	0
$475$	−8879.53	−0.857728
$476$	0	0
$477$	3686.87	0.353900
$478$	0	0
$479$	2341.90	0.223391	0.111696	−	0.993742i	$-0.464372\pi$
$479$	2341.90	0.223391	0.111696	+	0.993742i	$0.464372\pi$
$480$	0	0
$481$	−1537.93	−0.145787
$482$	0	0
$483$	1594.17	0.150181
$484$	0	0
$485$	2143.57	0.200690
$486$	0	0
$487$	6748.91	0.627972	0.313986	−	0.949428i	$-0.398335\pi$
$487$	6748.91	0.627972	0.313986	+	0.949428i	$0.398335\pi$
$488$	0	0
$489$	8029.53	0.742552
$490$	0	0
$491$	−7361.40	−0.676609	−0.338305	−	0.941037i	$-0.609853\pi$
$491$	−7361.40	−0.676609	−0.338305	+	0.941037i	$0.609853\pi$
$492$	0	0
$493$	−17789.1	−1.62511
$494$	0	0
$495$	−345.423	−0.0313649
$496$	0	0
$497$	−3589.26	−0.323944
$498$	0	0
$499$	−10381.7	−0.931359	−0.465680	−	0.884953i	$-0.654190\pi$
$499$	−10381.7	−0.931359	−0.465680	+	0.884953i	$0.654190\pi$
$500$	0	0
$501$	3564.36	0.317852
$502$	0	0
$503$	19149.0	1.69744	0.848721	−	0.528840i	$-0.177373\pi$
$503$	19149.0	1.69744	0.848721	+	0.528840i	$0.177373\pi$
$504$	0	0
$505$	3544.67	0.312348
$506$	0	0
$507$	5914.04	0.518051
$508$	0	0
$509$	−16073.2	−1.39967	−0.699836	−	0.714303i	$-0.746744\pi$
$509$	−16073.2	−1.39967	−0.699836	+	0.714303i	$0.746744\pi$
$510$	0	0
$511$	4838.58	0.418877
$512$	0	0
$513$	−2124.93	−0.182881
$514$	0	0
$515$	3845.58	0.329042
$516$	0	0
$517$	1868.56	0.158954
$518$	0	0
$519$	2421.44	0.204797
$520$	0	0
$521$	−18955.3	−1.59395	−0.796975	−	0.604012i	$-0.793568\pi$
$521$	−18955.3	−1.59395	−0.796975	+	0.604012i	$0.793568\pi$
$522$	0	0
$523$	4442.19	0.371402	0.185701	−	0.982606i	$-0.440544\pi$
$523$	4442.19	0.371402	0.185701	+	0.982606i	$0.440544\pi$
$524$	0	0
$525$	−1605.93	−0.133502
$526$	0	0
$527$	20393.3	1.68567
$528$	0	0
$529$	377.000	0.0309855
$530$	0	0
$531$	−1764.00	−0.144164
$532$	0	0
$533$	−3624.08	−0.294515
$534$	0	0
$535$	−4806.54	−0.388420
$536$	0	0
$537$	−5851.17	−0.470199
$538$	0	0
$539$	3525.38	0.281723
$540$	0	0
$541$	−2180.90	−0.173316	−0.0866580	−	0.996238i	$-0.527619\pi$
$541$	−2180.90	−0.173316	−0.0866580	+	0.996238i	$0.527619\pi$
$542$	0	0
$543$	3184.82	0.251701
$544$	0	0
$545$	−1117.28	−0.0878146
$546$	0	0
$547$	−8225.04	−0.642920	−0.321460	−	0.946923i	$-0.604174\pi$
$547$	−8225.04	−0.642920	−0.321460	+	0.946923i	$0.604174\pi$
$548$	0	0
$549$	6312.23	0.490709
$550$	0	0
$551$	−19134.2	−1.47939
$552$	0	0
$553$	1554.08	0.119505
$554$	0	0
$555$	1071.65	0.0819625
$556$	0	0
$557$	25181.9	1.91561	0.957804	−	0.287423i	$-0.0927986\pi$
$557$	25181.9	1.91561	0.957804	+	0.287423i	$0.0927986\pi$
$558$	0	0
$559$	4206.33	0.318263
$560$	0	0
$561$	2414.56	0.181716
$562$	0	0
$563$	4504.50	0.337197	0.168599	−	0.985685i	$-0.446076\pi$
$563$	4504.50	0.337197	0.168599	+	0.985685i	$0.446076\pi$
$564$	0	0
$565$	−5685.34	−0.423335
$566$	0	0
$567$	−384.310	−0.0284647
$568$	0	0
$569$	−13447.0	−0.990732	−0.495366	−	0.868684i	$-0.664966\pi$
$569$	−13447.0	−0.990732	−0.495366	+	0.868684i	$0.664966\pi$
$570$	0	0
$571$	2605.52	0.190959	0.0954795	−	0.995431i	$-0.469562\pi$
$571$	2605.52	0.190959	0.0954795	+	0.995431i	$0.469562\pi$
$572$	0	0
$573$	−6409.24	−0.467277
$574$	0	0
$575$	−12636.5	−0.916485
$576$	0	0
$577$	6339.65	0.457406	0.228703	−	0.973496i	$-0.426552\pi$
$577$	6339.65	0.457406	0.228703	+	0.973496i	$0.426552\pi$
$578$	0	0
$579$	−11843.3	−0.850069
$580$	0	0
$581$	−3589.83	−0.256336
$582$	0	0
$583$	−4506.17	−0.320114
$584$	0	0
$585$	471.715	0.0333385
$586$	0	0
$587$	13370.6	0.940140	0.470070	−	0.882629i	$-0.344229\pi$
$587$	13370.6	0.940140	0.470070	+	0.882629i	$0.344229\pi$
$588$	0	0
$589$	21935.3	1.53452
$590$	0	0
$591$	2771.66	0.192912
$592$	0	0
$593$	14319.3	0.991608	0.495804	−	0.868434i	$-0.334873\pi$
$593$	14319.3	0.991608	0.495804	+	0.868434i	$0.334873\pi$
$594$	0	0
$595$	−1211.26	−0.0834567
$596$	0	0
$597$	1428.46	0.0979276
$598$	0	0
$599$	−5788.63	−0.394853	−0.197427	−	0.980318i	$-0.563258\pi$
$599$	−5788.63	−0.394853	−0.197427	+	0.980318i	$0.563258\pi$
$600$	0	0
$601$	23968.1	1.62675	0.813375	−	0.581739i	$-0.197628\pi$
$601$	23968.1	1.62675	0.813375	+	0.581739i	$0.197628\pi$
$602$	0	0
$603$	−8105.28	−0.547384
$604$	0	0
$605$	422.184	0.0283706
$606$	0	0
$607$	−23526.6	−1.57317	−0.786585	−	0.617482i	$-0.788153\pi$
$607$	−23526.6	−1.57317	−0.786585	+	0.617482i	$0.788153\pi$
$608$	0	0
$609$	−3460.56	−0.230261
$610$	0	0
$611$	−2551.74	−0.168956
$612$	0	0
$613$	−1228.07	−0.0809159	−0.0404579	−	0.999181i	$-0.512882\pi$
$613$	−1228.07	−0.0809159	−0.0404579	+	0.999181i	$0.512882\pi$
$614$	0	0
$615$	2525.31	0.165578
$616$	0	0
$617$	−9844.90	−0.642368	−0.321184	−	0.947017i	$-0.604081\pi$
$617$	−9844.90	−0.642368	−0.321184	+	0.947017i	$0.604081\pi$
$618$	0	0
$619$	6551.68	0.425419	0.212709	−	0.977115i	$-0.431771\pi$
$619$	6551.68	0.425419	0.212709	+	0.977115i	$0.431771\pi$
$620$	0	0
$621$	−3024.00	−0.195409
$622$	0	0
$623$	−2414.88	−0.155297
$624$	0	0
$625$	11208.0	0.717309
$626$	0	0
$627$	2597.14	0.165422
$628$	0	0
$629$	−7491.01	−0.474859
$630$	0	0
$631$	−26440.5	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$631$	−26440.5	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$632$	0	0
$633$	−14754.7	−0.926458
$634$	0	0
$635$	7994.48	0.499608
$636$	0	0
$637$	−4814.31	−0.299450
$638$	0	0
$639$	6808.50	0.421502
$640$	0	0
$641$	−27927.2	−1.72084	−0.860421	−	0.509584i	$-0.829799\pi$
$641$	−27927.2	−1.72084	−0.860421	+	0.509584i	$0.829799\pi$
$642$	0	0
$643$	16737.7	1.02655	0.513274	−	0.858225i	$-0.328432\pi$
$643$	16737.7	1.02655	0.513274	+	0.858225i	$0.328432\pi$
$644$	0	0
$645$	−2931.03	−0.178929
$646$	0	0
$647$	7818.70	0.475092	0.237546	−	0.971376i	$-0.423657\pi$
$647$	7818.70	0.475092	0.237546	+	0.971376i	$0.423657\pi$
$648$	0	0
$649$	2156.00	0.130401
$650$	0	0
$651$	3967.17	0.238842
$652$	0	0
$653$	−19747.6	−1.18344	−0.591719	−	0.806144i	$-0.701550\pi$
$653$	−19747.6	−1.18344	−0.591719	+	0.806144i	$0.701550\pi$
$654$	0	0
$655$	4003.47	0.238822
$656$	0	0
$657$	−9178.33	−0.545024
$658$	0	0
$659$	−7867.72	−0.465072	−0.232536	−	0.972588i	$-0.574702\pi$
$659$	−7867.72	−0.465072	−0.232536	+	0.972588i	$0.574702\pi$
$660$	0	0
$661$	−4227.41	−0.248755	−0.124378	−	0.992235i	$-0.539693\pi$
$661$	−4227.41	−0.248755	−0.124378	+	0.992235i	$0.539693\pi$
$662$	0	0
$663$	−3297.35	−0.193150
$664$	0	0
$665$	−1302.85	−0.0759733
$666$	0	0
$667$	−27230.0	−1.58073
$668$	0	0
$669$	−6300.88	−0.364135
$670$	0	0
$671$	−7714.94	−0.443863
$672$	0	0
$673$	29397.6	1.68379	0.841897	−	0.539638i	$-0.181439\pi$
$673$	29397.6	1.68379	0.841897	+	0.539638i	$0.181439\pi$
$674$	0	0
$675$	3046.30	0.173707
$676$	0	0
$677$	−5737.14	−0.325696	−0.162848	−	0.986651i	$-0.552068\pi$
$677$	−5737.14	−0.325696	−0.162848	+	0.986651i	$0.552068\pi$
$678$	0	0
$679$	−2914.86	−0.164745
$680$	0	0
$681$	−6771.49	−0.381034
$682$	0	0
$683$	−32097.6	−1.79821	−0.899107	−	0.437729i	$-0.855783\pi$
$683$	−32097.6	−1.79821	−0.899107	+	0.437729i	$0.855783\pi$
$684$	0	0
$685$	4426.30	0.246891
$686$	0	0
$687$	−15933.2	−0.884848
$688$	0	0
$689$	6153.69	0.340257
$690$	0	0
$691$	16456.2	0.905965	0.452983	−	0.891519i	$-0.350360\pi$
$691$	16456.2	0.905965	0.452983	+	0.891519i	$0.350360\pi$
$692$	0	0
$693$	469.712	0.0257473
$694$	0	0
$695$	1696.72	0.0926047
$696$	0	0
$697$	−17652.3	−0.959294
$698$	0	0
$699$	−7398.80	−0.400355
$700$	0	0
$701$	−27238.1	−1.46758	−0.733788	−	0.679379i	$-0.762249\pi$
$701$	−27238.1	−1.46758	−0.733788	+	0.679379i	$0.762249\pi$
$702$	0	0
$703$	−8057.44	−0.432279
$704$	0	0
$705$	1778.09	0.0949882
$706$	0	0
$707$	−4820.09	−0.256405
$708$	0	0
$709$	−28761.4	−1.52349	−0.761747	−	0.647875i	$-0.775658\pi$
$709$	−28761.4	−1.52349	−0.761747	+	0.647875i	$0.775658\pi$
$710$	0	0
$711$	−2947.94	−0.155494
$712$	0	0
$713$	31216.3	1.63964
$714$	0	0
$715$	−576.540	−0.0301558
$716$	0	0
$717$	−4288.21	−0.223356
$718$	0	0
$719$	−27272.0	−1.41456	−0.707282	−	0.706931i	$-0.750079\pi$
$719$	−27272.0	−1.41456	−0.707282	+	0.706931i	$0.750079\pi$
$720$	0	0
$721$	−5229.28	−0.270109
$722$	0	0
$723$	2936.97	0.151075
$724$	0	0
$725$	27430.8	1.40518
$726$	0	0
$727$	3979.75	0.203027	0.101514	−	0.994834i	$-0.467631\pi$
$727$	3979.75	0.203027	0.101514	+	0.994834i	$0.467631\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	20488.3	1.03665
$732$	0	0
$733$	−9342.48	−0.470767	−0.235384	−	0.971903i	$-0.575635\pi$
$733$	−9342.48	−0.470767	−0.235384	+	0.971903i	$0.575635\pi$
$734$	0	0
$735$	3354.68	0.168353
$736$	0	0
$737$	9906.45	0.495127
$738$	0	0
$739$	28928.0	1.43997	0.719983	−	0.693992i	$-0.244150\pi$
$739$	28928.0	1.43997	0.719983	+	0.693992i	$0.244150\pi$
$740$	0	0
$741$	−3546.68	−0.175831
$742$	0	0
$743$	4857.04	0.239822	0.119911	−	0.992785i	$-0.461739\pi$
$743$	4857.04	0.239822	0.119911	+	0.992785i	$0.461739\pi$
$744$	0	0
$745$	−8213.80	−0.403933
$746$	0	0
$747$	6809.57	0.333533
$748$	0	0
$749$	6536.00	0.318852
$750$	0	0
$751$	14355.4	0.697517	0.348759	−	0.937213i	$-0.386603\pi$
$751$	14355.4	0.697517	0.348759	+	0.937213i	$0.386603\pi$
$752$	0	0
$753$	−19591.9	−0.948165
$754$	0	0
$755$	−1989.05	−0.0958792
$756$	0	0
$757$	17714.9	0.850538	0.425269	−	0.905067i	$-0.360179\pi$
$757$	17714.9	0.850538	0.425269	+	0.905067i	$0.360179\pi$
$758$	0	0
$759$	3696.00	0.176754
$760$	0	0
$761$	−7945.82	−0.378497	−0.189248	−	0.981929i	$-0.560605\pi$
$761$	−7945.82	−0.378497	−0.189248	+	0.981929i	$0.560605\pi$
$762$	0	0
$763$	1519.29	0.0720866
$764$	0	0
$765$	2297.64	0.108590
$766$	0	0
$767$	−2944.26	−0.138606
$768$	0	0
$769$	27308.1	1.28057	0.640284	−	0.768139i	$-0.278817\pi$
$769$	27308.1	1.28057	0.640284	+	0.768139i	$0.278817\pi$
$770$	0	0
$771$	−24390.8	−1.13932
$772$	0	0
$773$	18872.6	0.878136	0.439068	−	0.898454i	$-0.355309\pi$
$773$	18872.6	0.878136	0.439068	+	0.898454i	$0.355309\pi$
$774$	0	0
$775$	−31446.6	−1.45754
$776$	0	0
$777$	−1457.25	−0.0672826
$778$	0	0
$779$	−18987.1	−0.873276
$780$	0	0
$781$	−8321.50	−0.381263
$782$	0	0
$783$	6564.37	0.299606
$784$	0	0
$785$	7231.82	0.328808
$786$	0	0
$787$	14512.1	0.657307	0.328654	−	0.944450i	$-0.393405\pi$
$787$	14512.1	0.657307	0.328654	+	0.944450i	$0.393405\pi$
$788$	0	0
$789$	13648.3	0.615832
$790$	0	0
$791$	7731.01	0.347513
$792$	0	0
$793$	10535.6	0.471792
$794$	0	0
$795$	−4287.98	−0.191294
$796$	0	0
$797$	−29108.9	−1.29371	−0.646856	−	0.762612i	$-0.723917\pi$
$797$	−29108.9	−1.29371	−0.646856	+	0.762612i	$0.723917\pi$
$798$	0	0
$799$	−12429.1	−0.550325
$800$	0	0
$801$	4580.80	0.202066
$802$	0	0
$803$	11218.0	0.492993
$804$	0	0
$805$	−1854.09	−0.0811777
$806$	0	0
$807$	−87.4567	−0.00381490
$808$	0	0
$809$	−3000.83	−0.130413	−0.0652063	−	0.997872i	$-0.520771\pi$
$809$	−3000.83	−0.130413	−0.0652063	+	0.997872i	$0.520771\pi$
$810$	0	0
$811$	−6239.39	−0.270154	−0.135077	−	0.990835i	$-0.543128\pi$
$811$	−6239.39	−0.270154	−0.135077	+	0.990835i	$0.543128\pi$
$812$	0	0
$813$	−23133.7	−0.997950
$814$	0	0
$815$	−9338.68	−0.401374
$816$	0	0
$817$	22037.6	0.943693
$818$	0	0
$819$	−641.445	−0.0273674
$820$	0	0
$821$	−14922.4	−0.634342	−0.317171	−	0.948368i	$-0.602733\pi$
$821$	−14922.4	−0.634342	−0.317171	+	0.948368i	$0.602733\pi$
$822$	0	0
$823$	−25737.8	−1.09011	−0.545057	−	0.838399i	$-0.683492\pi$
$823$	−25737.8	−1.09011	−0.545057	+	0.838399i	$0.683492\pi$
$824$	0	0
$825$	−3723.26	−0.157124
$826$	0	0
$827$	−27043.4	−1.13711	−0.568555	−	0.822645i	$-0.692497\pi$
$827$	−27043.4	−1.13711	−0.568555	+	0.822645i	$0.692497\pi$
$828$	0	0
$829$	9795.41	0.410384	0.205192	−	0.978722i	$-0.434218\pi$
$829$	9795.41	0.410384	0.205192	+	0.978722i	$0.434218\pi$
$830$	0	0
$831$	3382.56	0.141203
$832$	0	0
$833$	−23449.7	−0.975370
$834$	0	0
$835$	−4145.50	−0.171809
$836$	0	0
$837$	−7525.37	−0.310770
$838$	0	0
$839$	28875.5	1.18819	0.594095	−	0.804395i	$-0.297510\pi$
$839$	28875.5	1.18819	0.594095	+	0.804395i	$0.297510\pi$
$840$	0	0
$841$	34720.7	1.42362
$842$	0	0
$843$	5617.41	0.229506
$844$	0	0
$845$	−6878.28	−0.280024
$846$	0	0
$847$	−574.092	−0.0232893
$848$	0	0
$849$	6373.45	0.257640
$850$	0	0
$851$	−11466.6	−0.461892
$852$	0	0
$853$	47157.1	1.89288	0.946441	−	0.322878i	$-0.104650\pi$
$853$	47157.1	1.89288	0.946441	+	0.322878i	$0.104650\pi$
$854$	0	0
$855$	2471.38	0.0988531
$856$	0	0
$857$	5021.31	0.200145	0.100073	−	0.994980i	$-0.468092\pi$
$857$	5021.31	0.200145	0.100073	+	0.994980i	$0.468092\pi$
$858$	0	0
$859$	22921.1	0.910428	0.455214	−	0.890382i	$-0.349563\pi$
$859$	22921.1	0.910428	0.455214	+	0.890382i	$0.349563\pi$
$860$	0	0
$861$	−3433.95	−0.135922
$862$	0	0
$863$	−19488.1	−0.768693	−0.384347	−	0.923189i	$-0.625573\pi$
$863$	−19488.1	−0.768693	−0.384347	+	0.923189i	$0.625573\pi$
$864$	0	0
$865$	−2816.24	−0.110699
$866$	0	0
$867$	−1321.86	−0.0517794
$868$	0	0
$869$	3603.04	0.140650
$870$	0	0
$871$	−13528.4	−0.526282
$872$	0	0
$873$	5529.22	0.214360
$874$	0	0
$875$	3937.06	0.152111
$876$	0	0
$877$	8455.67	0.325573	0.162787	−	0.986661i	$-0.447952\pi$
$877$	8455.67	0.325573	0.162787	+	0.986661i	$0.447952\pi$
$878$	0	0
$879$	−9972.56	−0.382669
$880$	0	0
$881$	−11291.2	−0.431794	−0.215897	−	0.976416i	$-0.569268\pi$
$881$	−11291.2	−0.431794	−0.215897	+	0.976416i	$0.569268\pi$
$882$	0	0
$883$	−31818.1	−1.21264	−0.606322	−	0.795219i	$-0.707356\pi$
$883$	−31818.1	−1.21264	−0.606322	+	0.795219i	$0.707356\pi$
$884$	0	0
$885$	2051.61	0.0779254
$886$	0	0
$887$	17481.1	0.661732	0.330866	−	0.943678i	$-0.392659\pi$
$887$	17481.1	0.661732	0.330866	+	0.943678i	$0.392659\pi$
$888$	0	0
$889$	−10871.0	−0.410126
$890$	0	0
$891$	−891.000	−0.0335013
$892$	0	0
$893$	−13368.9	−0.500978
$894$	0	0
$895$	6805.16	0.254158
$896$	0	0
$897$	−5047.31	−0.187876
$898$	0	0
$899$	−67763.1	−2.51393
$900$	0	0
$901$	29973.6	1.10829
$902$	0	0
$903$	3985.66	0.146882
$904$	0	0
$905$	−3704.08	−0.136053
$906$	0	0
$907$	−10607.4	−0.388326	−0.194163	−	0.980969i	$-0.562199\pi$
$907$	−10607.4	−0.388326	−0.194163	+	0.980969i	$0.562199\pi$
$908$	0	0
$909$	9143.26	0.333623
$910$	0	0
$911$	−41249.2	−1.50016	−0.750080	−	0.661347i	$-0.769985\pi$
$911$	−41249.2	−1.50016	−0.750080	+	0.661347i	$0.769985\pi$
$912$	0	0
$913$	−8322.81	−0.301692
$914$	0	0
$915$	−7341.38	−0.265244
$916$	0	0
$917$	−5443.97	−0.196048
$918$	0	0
$919$	−13858.1	−0.497429	−0.248714	−	0.968577i	$-0.580008\pi$
$919$	−13858.1	−0.497429	−0.248714	+	0.968577i	$0.580008\pi$
$920$	0	0
$921$	−5096.83	−0.182352
$922$	0	0
$923$	11363.9	0.405253
$924$	0	0
$925$	11551.2	0.410595
$926$	0	0
$927$	9919.47	0.351454
$928$	0	0
$929$	20893.7	0.737890	0.368945	−	0.929451i	$-0.379719\pi$
$929$	20893.7	0.737890	0.368945	+	0.929451i	$0.379719\pi$
$930$	0	0
$931$	−25222.8	−0.887911
$932$	0	0
$933$	−20786.5	−0.729388
$934$	0	0
$935$	−2808.23	−0.0982236
$936$	0	0
$937$	3203.52	0.111691	0.0558454	−	0.998439i	$-0.482215\pi$
$937$	3203.52	0.111691	0.0558454	+	0.998439i	$0.482215\pi$
$938$	0	0
$939$	10682.2	0.371248
$940$	0	0
$941$	−19951.6	−0.691182	−0.345591	−	0.938385i	$-0.612322\pi$
$941$	−19951.6	−0.691182	−0.345591	+	0.938385i	$0.612322\pi$
$942$	0	0
$943$	−27020.6	−0.933099
$944$	0	0
$945$	446.968	0.0153861
$946$	0	0
$947$	38216.7	1.31138	0.655689	−	0.755031i	$-0.272378\pi$
$947$	38216.7	1.31138	0.655689	+	0.755031i	$0.272378\pi$
$948$	0	0
$949$	−15319.4	−0.524014
$950$	0	0
$951$	998.249	0.0340383
$952$	0	0
$953$	−47661.4	−1.62004	−0.810022	−	0.586399i	$-0.800545\pi$
$953$	−47661.4	−1.62004	−0.810022	+	0.586399i	$0.800545\pi$
$954$	0	0
$955$	7454.21	0.252579
$956$	0	0
$957$	−8023.12	−0.271004
$958$	0	0
$959$	−6018.94	−0.202671
$960$	0	0
$961$	47892.3	1.60761
$962$	0	0
$963$	−12398.2	−0.414876
$964$	0	0
$965$	13774.2	0.459490
$966$	0	0
$967$	18933.2	0.629628	0.314814	−	0.949153i	$-0.398058\pi$
$967$	18933.2	0.629628	0.314814	+	0.949153i	$0.398058\pi$
$968$	0	0
$969$	−17275.3	−0.572717
$970$	0	0
$971$	40660.3	1.34382	0.671911	−	0.740632i	$-0.265474\pi$
$971$	40660.3	1.34382	0.671911	+	0.740632i	$0.265474\pi$
$972$	0	0
$973$	−2307.23	−0.0760188
$974$	0	0
$975$	5084.53	0.167011
$976$	0	0
$977$	−22502.8	−0.736876	−0.368438	−	0.929652i	$-0.620107\pi$
$977$	−22502.8	−0.736876	−0.368438	+	0.929652i	$0.620107\pi$
$978$	0	0
$979$	−5598.76	−0.182775
$980$	0	0
$981$	−2881.96	−0.0937959
$982$	0	0
$983$	−4435.20	−0.143907	−0.0719536	−	0.997408i	$-0.522923\pi$
$983$	−4435.20	−0.143907	−0.0719536	+	0.997408i	$0.522923\pi$
$984$	0	0
$985$	−3223.55	−0.104275
$986$	0	0
$987$	−2417.87	−0.0779753
$988$	0	0
$989$	31361.8	1.00834
$990$	0	0
$991$	7362.76	0.236010	0.118005	−	0.993013i	$-0.462350\pi$
$991$	7362.76	0.236010	0.118005	+	0.993013i	$0.462350\pi$
$992$	0	0
$993$	−1624.33	−0.0519101
$994$	0	0
$995$	−1661.35	−0.0529331
$996$	0	0
$997$	53480.1	1.69883	0.849413	−	0.527728i	$-0.176956\pi$
$997$	53480.1	1.69883	0.849413	+	0.527728i	$0.176956\pi$
$998$	0	0
$999$	2764.27	0.0875452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.ba.1.2		2
4.3	odd	2		2112.4.a.bh.1.2		2
8.3	odd	2		528.4.a.o.1.1		2
8.5	even	2		33.4.a.d.1.2	✓	2
24.5	odd	2		99.4.a.e.1.1		2
24.11	even	2		1584.4.a.x.1.2		2
40.13	odd	4		825.4.c.i.199.1		4
40.29	even	2		825.4.a.k.1.1		2
40.37	odd	4		825.4.c.i.199.4		4
56.13	odd	2		1617.4.a.j.1.2		2
88.21	odd	2		363.4.a.j.1.1		2
120.29	odd	2		2475.4.a.o.1.2		2
264.197	even	2		1089.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.d.1.2	✓	2	8.5	even	2
99.4.a.e.1.1		2	24.5	odd	2
363.4.a.j.1.1		2	88.21	odd	2
528.4.a.o.1.1		2	8.3	odd	2
825.4.a.k.1.1		2	40.29	even	2
825.4.c.i.199.1		4	40.13	odd	4
825.4.c.i.199.4		4	40.37	odd	4
1089.4.a.t.1.2		2	264.197	even	2
1584.4.a.x.1.2		2	24.11	even	2
1617.4.a.j.1.2		2	56.13	odd	2
2112.4.a.ba.1.2		2	1.1	even	1	trivial
2112.4.a.bh.1.2		2	4.3	odd	2
2475.4.a.o.1.2		2	120.29	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +3.48913 q^{5} -4.74456 q^{7} +9.00000 q^{9} -11.0000 q^{11} +15.0217 q^{13} -10.4674 q^{15} +73.1684 q^{17} +78.7011 q^{19} +14.2337 q^{21} +112.000 q^{23} -112.826 q^{25} -27.0000 q^{27} -243.125 q^{29} +278.717 q^{31} +33.0000 q^{33} -16.5544 q^{35} -102.380 q^{37} -45.0652 q^{39} -241.255 q^{41} +280.016 q^{43} +31.4021 q^{45} -169.870 q^{47} -320.489 q^{49} -219.505 q^{51} +409.652 q^{53} -38.3804 q^{55} -236.103 q^{57} -196.000 q^{59} +701.359 q^{61} -42.7011 q^{63} +52.4128 q^{65} -900.587 q^{67} -336.000 q^{69} +756.500 q^{71} -1019.81 q^{73} +338.478 q^{75} +52.1902 q^{77} -327.549 q^{79} +81.0000 q^{81} +756.619 q^{83} +255.294 q^{85} +729.375 q^{87} +508.978 q^{89} -71.2716 q^{91} -836.152 q^{93} +274.598 q^{95} +614.358 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 16 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} + 76 q^{13} + 48 q^{15} - 26 q^{17} + 54 q^{19} - 6 q^{21} + 224 q^{23} + 142 q^{25} - 54 q^{27} - 222 q^{29} - 40 q^{31} + 66 q^{33} - 148 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 16 * q^5 + 2 * q^7 + 18 * q^9 - 22 * q^11 + 76 * q^13 + 48 * q^15 - 26 * q^17 + 54 * q^19 - 6 * q^21 + 224 * q^23 + 142 * q^25 - 54 * q^27 - 222 * q^29 - 40 * q^31 + 66 * q^33 - 148 * q^35 + 48 * q^37 - 228 * q^39 - 494 * q^41 + 66 * q^43 - 144 * q^45 - 64 * q^47 - 618 * q^49 + 78 * q^51 + 84 * q^53 + 176 * q^55 - 162 * q^57 - 392 * q^59 + 1104 * q^61 + 18 * q^63 - 1136 * q^65 - 928 * q^67 - 672 * q^69 + 456 * q^71 - 592 * q^73 - 426 * q^75 - 22 * q^77 - 230 * q^79 + 162 * q^81 - 348 * q^83 + 2188 * q^85 + 666 * q^87 + 972 * q^89 + 340 * q^91 + 120 * q^93 + 756 * q^95 - 1184 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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