LMFDB - Embedded newform 2112.4.a.bn.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 = [2,0,6,0,14,0,24] 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{97}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			$-4.42443$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +16.8489 q^{5} -7.69772 q^{7} +9.00000 q^{9} +11.0000 q^{11} -24.8489 q^{13} +50.5466 q^{15} -15.9420 q^{17} -15.1511 q^{19} -23.0931 q^{21} +17.7557 q^{23} +158.884 q^{25} +27.0000 q^{27} +128.547 q^{29} +219.395 q^{31} +33.0000 q^{33} -129.698 q^{35} -92.0703 q^{37} -74.5466 q^{39} -459.942 q^{41} -64.9648 q^{43} +151.640 q^{45} +497.408 q^{47} -283.745 q^{49} -47.8260 q^{51} +526.919 q^{53} +185.337 q^{55} -45.4534 q^{57} +578.443 q^{59} +221.569 q^{61} -69.2794 q^{63} -418.675 q^{65} +860.745 q^{67} +53.2671 q^{69} +580.919 q^{71} +510.116 q^{73} +476.652 q^{75} -84.6749 q^{77} +1035.12 q^{79} +81.0000 q^{81} -606.211 q^{83} -268.605 q^{85} +385.640 q^{87} -23.4411 q^{89} +191.279 q^{91} +658.186 q^{93} -255.279 q^{95} +719.490 q^{97} +99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 14 q^{5} + 24 q^{7} + 18 q^{9} + 22 q^{11} - 30 q^{13} + 42 q^{15} + 106 q^{17} - 50 q^{19} + 72 q^{21} + 134 q^{23} + 42 q^{25} + 54 q^{27} + 198 q^{29} + 360 q^{31} + 66 q^{33} - 220 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 14 * q^5 + 24 * q^7 + 18 * q^9 + 22 * q^11 - 30 * q^13 + 42 * q^15 + 106 * q^17 - 50 * q^19 + 72 * q^21 + 134 * q^23 + 42 * q^25 + 54 * q^27 + 198 * q^29 + 360 * q^31 + 66 * q^33 - 220 * q^35 + 328 * q^37 - 90 * q^39 - 782 * q^41 - 386 * q^43 + 126 * q^45 + 266 * q^47 + 378 * q^49 + 318 * q^51 + 522 * q^53 + 154 * q^55 - 150 * q^57 + 172 * q^59 + 778 * q^61 + 216 * q^63 - 404 * q^65 + 776 * q^67 + 402 * q^69 + 630 * q^71 + 1296 * q^73 + 126 * q^75 + 264 * q^77 + 652 * q^79 + 162 * q^81 + 324 * q^83 - 616 * q^85 + 594 * q^87 - 756 * q^89 + 28 * q^91 + 1080 * q^93 - 156 * q^95 - 452 * q^97 + 198 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	16.8489	1.50701	0.753504	−	0.657444i	$-0.228362\pi$
$5$	16.8489	1.50701	0.753504	+	0.657444i	$0.228362\pi$
$6$	0	0
$7$	−7.69772	−0.415638	−0.207819	−	0.978167i	$-0.566636\pi$
$7$	−7.69772	−0.415638	−0.207819	+	0.978167i	$0.566636\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−24.8489	−0.530141	−0.265071	−	0.964229i	$-0.585395\pi$
$13$	−24.8489	−0.530141	−0.265071	+	0.964229i	$0.585395\pi$
$14$	0	0
$15$	50.5466	0.870071
$16$	0	0
$17$	−15.9420	−0.227441	−0.113721	−	0.993513i	$-0.536277\pi$
$17$	−15.9420	−0.227441	−0.113721	+	0.993513i	$0.536277\pi$
$18$	0	0
$19$	−15.1511	−0.182943	−0.0914713	−	0.995808i	$-0.529157\pi$
$19$	−15.1511	−0.182943	−0.0914713	+	0.995808i	$0.529157\pi$
$20$	0	0
$21$	−23.0931	−0.239968
$22$	0	0
$23$	17.7557	0.160971	0.0804853	−	0.996756i	$-0.474353\pi$
$23$	17.7557	0.160971	0.0804853	+	0.996756i	$0.474353\pi$
$24$	0	0
$25$	158.884	1.27107
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	128.547	0.823121	0.411560	−	0.911383i	$-0.364984\pi$
$29$	128.547	0.823121	0.411560	+	0.911383i	$0.364984\pi$
$30$	0	0
$31$	219.395	1.27112	0.635558	−	0.772053i	$-0.280770\pi$
$31$	219.395	1.27112	0.635558	+	0.772053i	$0.280770\pi$
$32$	0	0
$33$	33.0000	0.174078
$34$	0	0
$35$	−129.698	−0.626369
$36$	0	0
$37$	−92.0703	−0.409088	−0.204544	−	0.978857i	$-0.565571\pi$
$37$	−92.0703	−0.409088	−0.204544	+	0.978857i	$0.565571\pi$
$38$	0	0
$39$	−74.5466	−0.306077
$40$	0	0
$41$	−459.942	−1.75197	−0.875986	−	0.482336i	$-0.839788\pi$
$41$	−459.942	−1.75197	−0.875986	+	0.482336i	$0.839788\pi$
$42$	0	0
$43$	−64.9648	−0.230396	−0.115198	−	0.993343i	$-0.536750\pi$
$43$	−64.9648	−0.230396	−0.115198	+	0.993343i	$0.536750\pi$
$44$	0	0
$45$	151.640	0.502336
$46$	0	0
$47$	497.408	1.54371	0.771855	−	0.635799i	$-0.219329\pi$
$47$	497.408	1.54371	0.771855	+	0.635799i	$0.219329\pi$
$48$	0	0
$49$	−283.745	−0.827245
$50$	0	0
$51$	−47.8260	−0.131313
$52$	0	0
$53$	526.919	1.36562	0.682811	−	0.730596i	$-0.260757\pi$
$53$	526.919	1.36562	0.682811	+	0.730596i	$0.260757\pi$
$54$	0	0
$55$	185.337	0.454380
$56$	0	0
$57$	−45.4534	−0.105622
$58$	0	0
$59$	578.443	1.27639	0.638194	−	0.769876i	$-0.279682\pi$
$59$	578.443	1.27639	0.638194	+	0.769876i	$0.279682\pi$
$60$	0	0
$61$	221.569	0.465067	0.232533	−	0.972588i	$-0.425299\pi$
$61$	221.569	0.465067	0.232533	+	0.972588i	$0.425299\pi$
$62$	0	0
$63$	−69.2794	−0.138546
$64$	0	0
$65$	−418.675	−0.798927
$66$	0	0
$67$	860.745	1.56950	0.784752	−	0.619810i	$-0.212790\pi$
$67$	860.745	1.56950	0.784752	+	0.619810i	$0.212790\pi$
$68$	0	0
$69$	53.2671	0.0929364
$70$	0	0
$71$	580.919	0.971020	0.485510	−	0.874231i	$-0.338634\pi$
$71$	580.919	0.971020	0.485510	+	0.874231i	$0.338634\pi$
$72$	0	0
$73$	510.116	0.817871	0.408935	−	0.912563i	$-0.365900\pi$
$73$	510.116	0.817871	0.408935	+	0.912563i	$0.365900\pi$
$74$	0	0
$75$	476.652	0.733854
$76$	0	0
$77$	−84.6749	−0.125319
$78$	0	0
$79$	1035.12	1.47418	0.737088	−	0.675797i	$-0.236200\pi$
$79$	1035.12	1.47418	0.737088	+	0.675797i	$0.236200\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−606.211	−0.801690	−0.400845	−	0.916146i	$-0.631283\pi$
$83$	−606.211	−0.801690	−0.400845	+	0.916146i	$0.631283\pi$
$84$	0	0
$85$	−268.605	−0.342756
$86$	0	0
$87$	385.640	0.475229
$88$	0	0
$89$	−23.4411	−0.0279186	−0.0139593	−	0.999903i	$-0.504444\pi$
$89$	−23.4411	−0.0279186	−0.0139593	+	0.999903i	$0.504444\pi$
$90$	0	0
$91$	191.279	0.220347
$92$	0	0
$93$	658.186	0.733879
$94$	0	0
$95$	−255.279	−0.275696
$96$	0	0
$97$	719.490	0.753126	0.376563	−	0.926391i	$-0.377106\pi$
$97$	719.490	0.753126	0.376563	+	0.926391i	$0.377106\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	0	0
$101$	−1871.27	−1.84355	−0.921774	−	0.387727i	$-0.873260\pi$
$101$	−1871.27	−1.84355	−0.921774	+	0.387727i	$0.873260\pi$
$102$	0	0
$103$	−428.745	−0.410151	−0.205075	−	0.978746i	$-0.565744\pi$
$103$	−428.745	−0.410151	−0.205075	+	0.978746i	$0.565744\pi$
$104$	0	0
$105$	−389.093	−0.361634
$106$	0	0
$107$	−1148.02	−1.03723	−0.518616	−	0.855008i	$-0.673552\pi$
$107$	−1148.02	−1.03723	−0.518616	+	0.855008i	$0.673552\pi$
$108$	0	0
$109$	1828.32	1.60662	0.803308	−	0.595564i	$-0.203071\pi$
$109$	1828.32	1.60662	0.803308	+	0.595564i	$0.203071\pi$
$110$	0	0
$111$	−276.211	−0.236187
$112$	0	0
$113$	1126.40	0.937722	0.468861	−	0.883272i	$-0.344665\pi$
$113$	1126.40	0.937722	0.468861	+	0.883272i	$0.344665\pi$
$114$	0	0
$115$	299.163	0.242584
$116$	0	0
$117$	−223.640	−0.176714
$118$	0	0
$119$	122.717	0.0945332
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1379.83	−1.01150
$124$	0	0
$125$	570.907	0.408508
$126$	0	0
$127$	661.304	0.462057	0.231029	−	0.972947i	$-0.425791\pi$
$127$	661.304	0.462057	0.231029	+	0.972947i	$0.425791\pi$
$128$	0	0
$129$	−194.895	−0.133019
$130$	0	0
$131$	−622.186	−0.414967	−0.207483	−	0.978239i	$-0.566527\pi$
$131$	−622.186	−0.414967	−0.207483	+	0.978239i	$0.566527\pi$
$132$	0	0
$133$	116.629	0.0760378
$134$	0	0
$135$	454.919	0.290024
$136$	0	0
$137$	−1872.84	−1.16794	−0.583969	−	0.811776i	$-0.698501\pi$
$137$	−1872.84	−1.16794	−0.583969	+	0.811776i	$0.698501\pi$
$138$	0	0
$139$	954.058	0.582174	0.291087	−	0.956697i	$-0.405983\pi$
$139$	954.058	0.582174	0.291087	+	0.956697i	$0.405983\pi$
$140$	0	0
$141$	1492.22	0.891261
$142$	0	0
$143$	−273.337	−0.159844
$144$	0	0
$145$	2165.86	1.24045
$146$	0	0
$147$	−851.236	−0.477610
$148$	0	0
$149$	2047.01	1.12549	0.562745	−	0.826631i	$-0.309745\pi$
$149$	2047.01	1.12549	0.562745	+	0.826631i	$0.309745\pi$
$150$	0	0
$151$	475.863	0.256458	0.128229	−	0.991745i	$-0.459071\pi$
$151$	475.863	0.256458	0.128229	+	0.991745i	$0.459071\pi$
$152$	0	0
$153$	−143.478	−0.0758138
$154$	0	0
$155$	3696.56	1.91558
$156$	0	0
$157$	647.466	0.329130	0.164565	−	0.986366i	$-0.447378\pi$
$157$	647.466	0.329130	0.164565	+	0.986366i	$0.447378\pi$
$158$	0	0
$159$	1580.76	0.788442
$160$	0	0
$161$	−136.678	−0.0669054
$162$	0	0
$163$	−1093.23	−0.525329	−0.262665	−	0.964887i	$-0.584601\pi$
$163$	−1093.23	−0.525329	−0.262665	+	0.964887i	$0.584601\pi$
$164$	0	0
$165$	556.012	0.262336
$166$	0	0
$167$	1123.25	0.520479	0.260240	−	0.965544i	$-0.416198\pi$
$167$	1123.25	0.520479	0.260240	+	0.965544i	$0.416198\pi$
$168$	0	0
$169$	−1579.53	−0.718951
$170$	0	0
$171$	−136.360	−0.0609809
$172$	0	0
$173$	−46.0123	−0.0202211	−0.0101106	−	0.999949i	$-0.503218\pi$
$173$	−46.0123	−0.0202211	−0.0101106	+	0.999949i	$0.503218\pi$
$174$	0	0
$175$	−1223.04	−0.528305
$176$	0	0
$177$	1735.33	0.736923
$178$	0	0
$179$	831.975	0.347401	0.173700	−	0.984799i	$-0.444428\pi$
$179$	831.975	0.347401	0.173700	+	0.984799i	$0.444428\pi$
$180$	0	0
$181$	1810.63	0.743553	0.371776	−	0.928322i	$-0.378749\pi$
$181$	1810.63	0.743553	0.371776	+	0.928322i	$0.378749\pi$
$182$	0	0
$183$	664.708	0.268506
$184$	0	0
$185$	−1551.28	−0.616499
$186$	0	0
$187$	−175.362	−0.0685762
$188$	0	0
$189$	−207.838	−0.0799895
$190$	0	0
$191$	458.898	0.173847	0.0869233	−	0.996215i	$-0.472296\pi$
$191$	458.898	0.173847	0.0869233	+	0.996215i	$0.472296\pi$
$192$	0	0
$193$	1778.91	0.663465	0.331733	−	0.943373i	$-0.392367\pi$
$193$	1778.91	0.663465	0.331733	+	0.943373i	$0.392367\pi$
$194$	0	0
$195$	−1256.02	−0.461260
$196$	0	0
$197$	−5304.53	−1.91844	−0.959218	−	0.282666i	$-0.908781\pi$
$197$	−5304.53	−1.91844	−0.959218	+	0.282666i	$0.908781\pi$
$198$	0	0
$199$	−5138.40	−1.83041	−0.915205	−	0.402989i	$-0.867971\pi$
$199$	−5138.40	−1.83041	−0.915205	+	0.402989i	$0.867971\pi$
$200$	0	0
$201$	2582.24	0.906153
$202$	0	0
$203$	−989.515	−0.342120
$204$	0	0
$205$	−7749.50	−2.64024
$206$	0	0
$207$	159.801	0.0536568
$208$	0	0
$209$	−166.663	−0.0551593
$210$	0	0
$211$	4262.36	1.39068	0.695339	−	0.718682i	$-0.255254\pi$
$211$	4262.36	1.39068	0.695339	+	0.718682i	$0.255254\pi$
$212$	0	0
$213$	1742.76	0.560619
$214$	0	0
$215$	−1094.58	−0.347209
$216$	0	0
$217$	−1688.84	−0.528323
$218$	0	0
$219$	1530.35	0.472198
$220$	0	0
$221$	396.141	0.120576
$222$	0	0
$223$	−1377.80	−0.413740	−0.206870	−	0.978368i	$-0.566328\pi$
$223$	−1377.80	−0.413740	−0.206870	+	0.978368i	$0.566328\pi$
$224$	0	0
$225$	1429.96	0.423691
$226$	0	0
$227$	1227.28	0.358843	0.179422	−	0.983772i	$-0.442577\pi$
$227$	1227.28	0.358843	0.179422	+	0.983772i	$0.442577\pi$
$228$	0	0
$229$	−3890.28	−1.12261	−0.561304	−	0.827610i	$-0.689700\pi$
$229$	−3890.28	−1.12261	−0.561304	+	0.827610i	$0.689700\pi$
$230$	0	0
$231$	−254.025	−0.0723532
$232$	0	0
$233$	−3218.14	−0.904837	−0.452419	−	0.891806i	$-0.649439\pi$
$233$	−3218.14	−0.904837	−0.452419	+	0.891806i	$0.649439\pi$
$234$	0	0
$235$	8380.75	2.32638
$236$	0	0
$237$	3105.35	0.851115
$238$	0	0
$239$	−428.098	−0.115864	−0.0579318	−	0.998321i	$-0.518451\pi$
$239$	−428.098	−0.115864	−0.0579318	+	0.998321i	$0.518451\pi$
$240$	0	0
$241$	1231.16	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$241$	1231.16	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−4780.78	−1.24667
$246$	0	0
$247$	376.489	0.0969854
$248$	0	0
$249$	−1818.63	−0.462856
$250$	0	0
$251$	−2838.22	−0.713732	−0.356866	−	0.934156i	$-0.616155\pi$
$251$	−2838.22	−0.713732	−0.356866	+	0.934156i	$0.616155\pi$
$252$	0	0
$253$	195.313	0.0485344
$254$	0	0
$255$	−805.814	−0.197890
$256$	0	0
$257$	342.007	0.0830110	0.0415055	−	0.999138i	$-0.486785\pi$
$257$	342.007	0.0830110	0.0415055	+	0.999138i	$0.486785\pi$
$258$	0	0
$259$	708.731	0.170032
$260$	0	0
$261$	1156.92	0.274374
$262$	0	0
$263$	5895.00	1.38213	0.691067	−	0.722791i	$-0.257141\pi$
$263$	5895.00	1.38213	0.691067	+	0.722791i	$0.257141\pi$
$264$	0	0
$265$	8877.99	2.05800
$266$	0	0
$267$	−70.3234	−0.0161188
$268$	0	0
$269$	2496.18	0.565779	0.282890	−	0.959152i	$-0.408707\pi$
$269$	2496.18	0.565779	0.282890	+	0.959152i	$0.408707\pi$
$270$	0	0
$271$	−2249.68	−0.504274	−0.252137	−	0.967692i	$-0.581133\pi$
$271$	−2249.68	−0.504274	−0.252137	+	0.967692i	$0.581133\pi$
$272$	0	0
$273$	573.838	0.127217
$274$	0	0
$275$	1747.72	0.383243
$276$	0	0
$277$	−4082.59	−0.885556	−0.442778	−	0.896631i	$-0.646007\pi$
$277$	−4082.59	−0.885556	−0.442778	+	0.896631i	$0.646007\pi$
$278$	0	0
$279$	1974.56	0.423705
$280$	0	0
$281$	−1033.79	−0.219468	−0.109734	−	0.993961i	$-0.535000\pi$
$281$	−1033.79	−0.219468	−0.109734	+	0.993961i	$0.535000\pi$
$282$	0	0
$283$	7809.14	1.64030	0.820150	−	0.572148i	$-0.193890\pi$
$283$	7809.14	1.64030	0.820150	+	0.572148i	$0.193890\pi$
$284$	0	0
$285$	−765.838	−0.159173
$286$	0	0
$287$	3540.50	0.728186
$288$	0	0
$289$	−4658.85	−0.948270
$290$	0	0
$291$	2158.47	0.434817
$292$	0	0
$293$	1949.19	0.388645	0.194323	−	0.980938i	$-0.437749\pi$
$293$	1949.19	0.388645	0.194323	+	0.980938i	$0.437749\pi$
$294$	0	0
$295$	9746.10	1.92353
$296$	0	0
$297$	297.000	0.0580259
$298$	0	0
$299$	−441.209	−0.0853371
$300$	0	0
$301$	500.081	0.0957614
$302$	0	0
$303$	−5613.81	−1.06437
$304$	0	0
$305$	3733.19	0.700859
$306$	0	0
$307$	−2364.09	−0.439497	−0.219748	−	0.975557i	$-0.570524\pi$
$307$	−2364.09	−0.439497	−0.219748	+	0.975557i	$0.570524\pi$
$308$	0	0
$309$	−1286.24	−0.236801
$310$	0	0
$311$	−1989.17	−0.362686	−0.181343	−	0.983420i	$-0.558044\pi$
$311$	−1989.17	−0.362686	−0.181343	+	0.983420i	$0.558044\pi$
$312$	0	0
$313$	−3878.67	−0.700433	−0.350216	−	0.936669i	$-0.613892\pi$
$313$	−3878.67	−0.700433	−0.350216	+	0.936669i	$0.613892\pi$
$314$	0	0
$315$	−1167.28	−0.208790
$316$	0	0
$317$	−2913.73	−0.516251	−0.258126	−	0.966111i	$-0.583105\pi$
$317$	−2913.73	−0.516251	−0.258126	+	0.966111i	$0.583105\pi$
$318$	0	0
$319$	1414.01	0.248180
$320$	0	0
$321$	−3444.07	−0.598846
$322$	0	0
$323$	241.540	0.0416087
$324$	0	0
$325$	−3948.09	−0.673847
$326$	0	0
$327$	5484.95	0.927580
$328$	0	0
$329$	−3828.90	−0.641624
$330$	0	0
$331$	−8104.46	−1.34580	−0.672902	−	0.739731i	$-0.734953\pi$
$331$	−8104.46	−1.34580	−0.672902	+	0.739731i	$0.734953\pi$
$332$	0	0
$333$	−828.633	−0.136363
$334$	0	0
$335$	14502.6	2.36525
$336$	0	0
$337$	5919.19	0.956792	0.478396	−	0.878144i	$-0.341218\pi$
$337$	5919.19	0.956792	0.478396	+	0.878144i	$0.341218\pi$
$338$	0	0
$339$	3379.19	0.541394
$340$	0	0
$341$	2413.35	0.383256
$342$	0	0
$343$	4824.51	0.759472
$344$	0	0
$345$	897.490	0.140056
$346$	0	0
$347$	8540.59	1.32128	0.660638	−	0.750705i	$-0.270286\pi$
$347$	8540.59	1.32128	0.660638	+	0.750705i	$0.270286\pi$
$348$	0	0
$349$	−937.337	−0.143767	−0.0718833	−	0.997413i	$-0.522901\pi$
$349$	−937.337	−0.143767	−0.0718833	+	0.997413i	$0.522901\pi$
$350$	0	0
$351$	−670.919	−0.102026
$352$	0	0
$353$	−211.118	−0.0318319	−0.0159160	−	0.999873i	$-0.505066\pi$
$353$	−211.118	−0.0318319	−0.0159160	+	0.999873i	$0.505066\pi$
$354$	0	0
$355$	9787.82	1.46333
$356$	0	0
$357$	368.151	0.0545788
$358$	0	0
$359$	−1376.31	−0.202337	−0.101169	−	0.994869i	$-0.532258\pi$
$359$	−1376.31	−0.202337	−0.101169	+	0.994869i	$0.532258\pi$
$360$	0	0
$361$	−6629.44	−0.966532
$362$	0	0
$363$	363.000	0.0524864
$364$	0	0
$365$	8594.87	1.23254
$366$	0	0
$367$	1030.45	0.146564	0.0732821	−	0.997311i	$-0.476653\pi$
$367$	1030.45	0.146564	0.0732821	+	0.997311i	$0.476653\pi$
$368$	0	0
$369$	−4139.48	−0.583991
$370$	0	0
$371$	−4056.07	−0.567603
$372$	0	0
$373$	−9365.39	−1.30006	−0.650029	−	0.759909i	$-0.725243\pi$
$373$	−9365.39	−1.30006	−0.650029	+	0.759909i	$0.725243\pi$
$374$	0	0
$375$	1712.72	0.235852
$376$	0	0
$377$	−3194.24	−0.436370
$378$	0	0
$379$	7120.23	0.965017	0.482509	−	0.875891i	$-0.339726\pi$
$379$	7120.23	0.965017	0.482509	+	0.875891i	$0.339726\pi$
$380$	0	0
$381$	1983.91	0.266769
$382$	0	0
$383$	−1163.56	−0.155235	−0.0776176	−	0.996983i	$-0.524731\pi$
$383$	−1163.56	−0.155235	−0.0776176	+	0.996983i	$0.524731\pi$
$384$	0	0
$385$	−1426.67	−0.188857
$386$	0	0
$387$	−584.684	−0.0767988
$388$	0	0
$389$	10958.9	1.42838	0.714188	−	0.699954i	$-0.246796\pi$
$389$	10958.9	1.42838	0.714188	+	0.699954i	$0.246796\pi$
$390$	0	0
$391$	−283.062	−0.0366114
$392$	0	0
$393$	−1866.56	−0.239581
$394$	0	0
$395$	17440.6	2.22159
$396$	0	0
$397$	2172.09	0.274595	0.137298	−	0.990530i	$-0.456158\pi$
$397$	2172.09	0.274595	0.137298	+	0.990530i	$0.456158\pi$
$398$	0	0
$399$	349.888	0.0439005
$400$	0	0
$401$	7830.71	0.975180	0.487590	−	0.873073i	$-0.337876\pi$
$401$	7830.71	0.975180	0.487590	+	0.873073i	$0.337876\pi$
$402$	0	0
$403$	−5451.73	−0.673870
$404$	0	0
$405$	1364.76	0.167445
$406$	0	0
$407$	−1012.77	−0.123345
$408$	0	0
$409$	−10731.2	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$409$	−10731.2	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$410$	0	0
$411$	−5618.52	−0.674309
$412$	0	0
$413$	−4452.69	−0.530515
$414$	0	0
$415$	−10214.0	−1.20815
$416$	0	0
$417$	2862.17	0.336118
$418$	0	0
$419$	7315.88	0.852994	0.426497	−	0.904489i	$-0.359748\pi$
$419$	7315.88	0.852994	0.426497	+	0.904489i	$0.359748\pi$
$420$	0	0
$421$	12495.7	1.44657	0.723284	−	0.690551i	$-0.242632\pi$
$421$	12495.7	1.44657	0.723284	+	0.690551i	$0.242632\pi$
$422$	0	0
$423$	4476.67	0.514570
$424$	0	0
$425$	−2532.93	−0.289094
$426$	0	0
$427$	−1705.58	−0.193299
$428$	0	0
$429$	−820.012	−0.0922857
$430$	0	0
$431$	6075.01	0.678939	0.339470	−	0.940617i	$-0.389752\pi$
$431$	6075.01	0.678939	0.339470	+	0.940617i	$0.389752\pi$
$432$	0	0
$433$	5641.79	0.626160	0.313080	−	0.949727i	$-0.398639\pi$
$433$	5641.79	0.626160	0.313080	+	0.949727i	$0.398639\pi$
$434$	0	0
$435$	6497.59	0.716174
$436$	0	0
$437$	−269.019	−0.0294484
$438$	0	0
$439$	10897.0	1.18470	0.592351	−	0.805680i	$-0.298200\pi$
$439$	10897.0	1.18470	0.592351	+	0.805680i	$0.298200\pi$
$440$	0	0
$441$	−2553.71	−0.275748
$442$	0	0
$443$	7720.83	0.828054	0.414027	−	0.910265i	$-0.364122\pi$
$443$	7720.83	0.828054	0.414027	+	0.910265i	$0.364122\pi$
$444$	0	0
$445$	−394.956	−0.0420735
$446$	0	0
$447$	6141.04	0.649802
$448$	0	0
$449$	−7473.86	−0.785553	−0.392776	−	0.919634i	$-0.628485\pi$
$449$	−7473.86	−0.785553	−0.392776	+	0.919634i	$0.628485\pi$
$450$	0	0
$451$	−5059.36	−0.528240
$452$	0	0
$453$	1427.59	0.148066
$454$	0	0
$455$	3222.84	0.332064
$456$	0	0
$457$	11140.5	1.14033	0.570167	−	0.821529i	$-0.306879\pi$
$457$	11140.5	1.14033	0.570167	+	0.821529i	$0.306879\pi$
$458$	0	0
$459$	−430.434	−0.0437711
$460$	0	0
$461$	−14328.8	−1.44763	−0.723817	−	0.689992i	$-0.757614\pi$
$461$	−14328.8	−1.44763	−0.723817	+	0.689992i	$0.757614\pi$
$462$	0	0
$463$	11760.7	1.18049	0.590246	−	0.807223i	$-0.299031\pi$
$463$	11760.7	1.18049	0.590246	+	0.807223i	$0.299031\pi$
$464$	0	0
$465$	11089.7	1.10596
$466$	0	0
$467$	−11854.9	−1.17469	−0.587343	−	0.809338i	$-0.699826\pi$
$467$	−11854.9	−1.17469	−0.587343	+	0.809338i	$0.699826\pi$
$468$	0	0
$469$	−6625.77	−0.652345
$470$	0	0
$471$	1942.40	0.190023
$472$	0	0
$473$	−714.613	−0.0694671
$474$	0	0
$475$	−2407.27	−0.232533
$476$	0	0
$477$	4742.27	0.455207
$478$	0	0
$479$	−1324.68	−0.126359	−0.0631796	−	0.998002i	$-0.520124\pi$
$479$	−1324.68	−0.126359	−0.0631796	+	0.998002i	$0.520124\pi$
$480$	0	0
$481$	2287.84	0.216874
$482$	0	0
$483$	−410.035	−0.0386278
$484$	0	0
$485$	12122.6	1.13497
$486$	0	0
$487$	18636.4	1.73408	0.867040	−	0.498239i	$-0.166020\pi$
$487$	18636.4	1.73408	0.867040	+	0.498239i	$0.166020\pi$
$488$	0	0
$489$	−3279.70	−0.303299
$490$	0	0
$491$	−124.552	−0.0114480	−0.00572398	−	0.999984i	$-0.501822\pi$
$491$	−124.552	−0.0114480	−0.00572398	+	0.999984i	$0.501822\pi$
$492$	0	0
$493$	−2049.29	−0.187212
$494$	0	0
$495$	1668.04	0.151460
$496$	0	0
$497$	−4471.75	−0.403592
$498$	0	0
$499$	−10230.2	−0.917768	−0.458884	−	0.888496i	$-0.651751\pi$
$499$	−10230.2	−0.917768	−0.458884	+	0.888496i	$0.651751\pi$
$500$	0	0
$501$	3369.76	0.300499
$502$	0	0
$503$	5150.81	0.456587	0.228294	−	0.973592i	$-0.426685\pi$
$503$	5150.81	0.456587	0.228294	+	0.973592i	$0.426685\pi$
$504$	0	0
$505$	−31528.8	−2.77824
$506$	0	0
$507$	−4738.60	−0.415086
$508$	0	0
$509$	22.7715	0.00198296	0.000991481	−	1.00000i	$-0.499684\pi$
$509$	22.7715	0.00198296	0.000991481		1.00000i	$0.499684\pi$
$510$	0	0
$511$	−3926.73	−0.339938
$512$	0	0
$513$	−409.081	−0.0352073
$514$	0	0
$515$	−7223.87	−0.618100
$516$	0	0
$517$	5471.49	0.465446
$518$	0	0
$519$	−138.037	−0.0116747
$520$	0	0
$521$	21521.7	1.80976	0.904879	−	0.425669i	$-0.139961\pi$
$521$	21521.7	1.80976	0.904879	+	0.425669i	$0.139961\pi$
$522$	0	0
$523$	−2923.36	−0.244416	−0.122208	−	0.992504i	$-0.538998\pi$
$523$	−2923.36	−0.244416	−0.122208	+	0.992504i	$0.538998\pi$
$524$	0	0
$525$	−3669.13	−0.305017
$526$	0	0
$527$	−3497.60	−0.289104
$528$	0	0
$529$	−11851.7	−0.974088
$530$	0	0
$531$	5205.99	0.425462
$532$	0	0
$533$	11429.0	0.928792
$534$	0	0
$535$	−19342.9	−1.56312
$536$	0	0
$537$	2495.93	0.200572
$538$	0	0
$539$	−3121.20	−0.249424
$540$	0	0
$541$	−21272.8	−1.69056	−0.845278	−	0.534327i	$-0.820565\pi$
$541$	−21272.8	−1.69056	−0.845278	+	0.534327i	$0.820565\pi$
$542$	0	0
$543$	5431.89	0.429290
$544$	0	0
$545$	30805.1	2.42118
$546$	0	0
$547$	18730.5	1.46409	0.732046	−	0.681256i	$-0.238566\pi$
$547$	18730.5	1.46409	0.732046	+	0.681256i	$0.238566\pi$
$548$	0	0
$549$	1994.12	0.155022
$550$	0	0
$551$	−1947.63	−0.150584
$552$	0	0
$553$	−7968.04	−0.612723
$554$	0	0
$555$	−4653.84	−0.355936
$556$	0	0
$557$	18885.0	1.43659	0.718297	−	0.695736i	$-0.244922\pi$
$557$	18885.0	1.43659	0.718297	+	0.695736i	$0.244922\pi$
$558$	0	0
$559$	1614.30	0.122143
$560$	0	0
$561$	−526.086	−0.0395925
$562$	0	0
$563$	−10285.1	−0.769922	−0.384961	−	0.922933i	$-0.625785\pi$
$563$	−10285.1	−0.769922	−0.384961	+	0.922933i	$0.625785\pi$
$564$	0	0
$565$	18978.5	1.41315
$566$	0	0
$567$	−623.515	−0.0461820
$568$	0	0
$569$	18008.8	1.32683	0.663415	−	0.748251i	$-0.269106\pi$
$569$	18008.8	1.32683	0.663415	+	0.748251i	$0.269106\pi$
$570$	0	0
$571$	7010.79	0.513822	0.256911	−	0.966435i	$-0.417295\pi$
$571$	7010.79	0.513822	0.256911	+	0.966435i	$0.417295\pi$
$572$	0	0
$573$	1376.69	0.100370
$574$	0	0
$575$	2821.10	0.204605
$576$	0	0
$577$	−16398.9	−1.18318	−0.591589	−	0.806240i	$-0.701499\pi$
$577$	−16398.9	−1.18318	−0.591589	+	0.806240i	$0.701499\pi$
$578$	0	0
$579$	5336.73	0.383052
$580$	0	0
$581$	4666.44	0.333213
$582$	0	0
$583$	5796.11	0.411750
$584$	0	0
$585$	−3768.07	−0.266309
$586$	0	0
$587$	−12823.5	−0.901671	−0.450836	−	0.892607i	$-0.648874\pi$
$587$	−12823.5	−0.901671	−0.450836	+	0.892607i	$0.648874\pi$
$588$	0	0
$589$	−3324.09	−0.232541
$590$	0	0
$591$	−15913.6	−1.10761
$592$	0	0
$593$	16899.5	1.17029	0.585144	−	0.810929i	$-0.301038\pi$
$593$	16899.5	1.17029	0.585144	+	0.810929i	$0.301038\pi$
$594$	0	0
$595$	2067.64	0.142462
$596$	0	0
$597$	−15415.2	−1.05679
$598$	0	0
$599$	−15074.9	−1.02829	−0.514143	−	0.857704i	$-0.671890\pi$
$599$	−15074.9	−1.02829	−0.514143	+	0.857704i	$0.671890\pi$
$600$	0	0
$601$	−11418.8	−0.775014	−0.387507	−	0.921867i	$-0.626664\pi$
$601$	−11418.8	−0.775014	−0.387507	+	0.921867i	$0.626664\pi$
$602$	0	0
$603$	7746.71	0.523168
$604$	0	0
$605$	2038.71	0.137001
$606$	0	0
$607$	−17952.8	−1.20046	−0.600232	−	0.799826i	$-0.704925\pi$
$607$	−17952.8	−1.20046	−0.600232	+	0.799826i	$0.704925\pi$
$608$	0	0
$609$	−2968.54	−0.197523
$610$	0	0
$611$	−12360.0	−0.818384
$612$	0	0
$613$	−12528.9	−0.825507	−0.412753	−	0.910843i	$-0.635433\pi$
$613$	−12528.9	−0.825507	−0.412753	+	0.910843i	$0.635433\pi$
$614$	0	0
$615$	−23248.5	−1.52434
$616$	0	0
$617$	−8586.10	−0.560232	−0.280116	−	0.959966i	$-0.590373\pi$
$617$	−8586.10	−0.560232	−0.280116	+	0.959966i	$0.590373\pi$
$618$	0	0
$619$	−18415.4	−1.19576	−0.597882	−	0.801584i	$-0.703991\pi$
$619$	−18415.4	−1.19576	−0.597882	+	0.801584i	$0.703991\pi$
$620$	0	0
$621$	479.404	0.0309788
$622$	0	0
$623$	180.443	0.0116040
$624$	0	0
$625$	−10241.4	−0.655448
$626$	0	0
$627$	−499.988	−0.0318462
$628$	0	0
$629$	1467.79	0.0930436
$630$	0	0
$631$	2374.38	0.149798	0.0748989	−	0.997191i	$-0.476137\pi$
$631$	2374.38	0.149798	0.0748989	+	0.997191i	$0.476137\pi$
$632$	0	0
$633$	12787.1	0.802909
$634$	0	0
$635$	11142.2	0.696324
$636$	0	0
$637$	7050.74	0.438557
$638$	0	0
$639$	5228.27	0.323673
$640$	0	0
$641$	11086.0	0.683104	0.341552	−	0.939863i	$-0.389047\pi$
$641$	11086.0	0.683104	0.341552	+	0.939863i	$0.389047\pi$
$642$	0	0
$643$	−19934.1	−1.22259	−0.611294	−	0.791403i	$-0.709351\pi$
$643$	−19934.1	−1.22259	−0.611294	+	0.791403i	$0.709351\pi$
$644$	0	0
$645$	−3283.75	−0.200461
$646$	0	0
$647$	−30634.8	−1.86148	−0.930739	−	0.365684i	$-0.880835\pi$
$647$	−30634.8	−1.86148	−0.930739	+	0.365684i	$0.880835\pi$
$648$	0	0
$649$	6362.87	0.384845
$650$	0	0
$651$	−5066.53	−0.305028
$652$	0	0
$653$	9818.07	0.588378	0.294189	−	0.955747i	$-0.404951\pi$
$653$	9818.07	0.588378	0.294189	+	0.955747i	$0.404951\pi$
$654$	0	0
$655$	−10483.1	−0.625358
$656$	0	0
$657$	4591.04	0.272624
$658$	0	0
$659$	−16478.5	−0.974070	−0.487035	−	0.873383i	$-0.661922\pi$
$659$	−16478.5	−0.974070	−0.487035	+	0.873383i	$0.661922\pi$
$660$	0	0
$661$	−2958.12	−0.174066	−0.0870328	−	0.996205i	$-0.527738\pi$
$661$	−2958.12	−0.174066	−0.0870328	+	0.996205i	$0.527738\pi$
$662$	0	0
$663$	1188.42	0.0696146
$664$	0	0
$665$	1965.07	0.114590
$666$	0	0
$667$	2282.44	0.132498
$668$	0	0
$669$	−4133.39	−0.238873
$670$	0	0
$671$	2437.26	0.140223
$672$	0	0
$673$	−29960.3	−1.71602	−0.858012	−	0.513630i	$-0.828300\pi$
$673$	−29960.3	−1.71602	−0.858012	+	0.513630i	$0.828300\pi$
$674$	0	0
$675$	4289.87	0.244618
$676$	0	0
$677$	4514.73	0.256300	0.128150	−	0.991755i	$-0.459096\pi$
$677$	4514.73	0.256300	0.128150	+	0.991755i	$0.459096\pi$
$678$	0	0
$679$	−5538.43	−0.313027
$680$	0	0
$681$	3681.84	0.207178
$682$	0	0
$683$	13555.7	0.759438	0.379719	−	0.925102i	$-0.376021\pi$
$683$	13555.7	0.759438	0.379719	+	0.925102i	$0.376021\pi$
$684$	0	0
$685$	−31555.2	−1.76009
$686$	0	0
$687$	−11670.8	−0.648137
$688$	0	0
$689$	−13093.3	−0.723972
$690$	0	0
$691$	11471.3	0.631535	0.315768	−	0.948837i	$-0.397738\pi$
$691$	11471.3	0.631535	0.315768	+	0.948837i	$0.397738\pi$
$692$	0	0
$693$	−762.074	−0.0417731
$694$	0	0
$695$	16074.8	0.877340
$696$	0	0
$697$	7332.40	0.398471
$698$	0	0
$699$	−9654.41	−0.522408
$700$	0	0
$701$	−22229.0	−1.19769	−0.598843	−	0.800866i	$-0.704373\pi$
$701$	−22229.0	−1.19769	−0.598843	+	0.800866i	$0.704373\pi$
$702$	0	0
$703$	1394.97	0.0748397
$704$	0	0
$705$	25142.3	1.34314
$706$	0	0
$707$	14404.5	0.766248
$708$	0	0
$709$	15081.2	0.798851	0.399426	−	0.916766i	$-0.369210\pi$
$709$	15081.2	0.798851	0.399426	+	0.916766i	$0.369210\pi$
$710$	0	0
$711$	9316.06	0.491392
$712$	0	0
$713$	3895.52	0.204612
$714$	0	0
$715$	−4605.42	−0.240885
$716$	0	0
$717$	−1284.30	−0.0668938
$718$	0	0
$719$	−7399.80	−0.383819	−0.191910	−	0.981413i	$-0.561468\pi$
$719$	−7399.80	−0.383819	−0.191910	+	0.981413i	$0.561468\pi$
$720$	0	0
$721$	3300.36	0.170474
$722$	0	0
$723$	3693.48	0.189989
$724$	0	0
$725$	20424.0	1.04625
$726$	0	0
$727$	1705.77	0.0870202	0.0435101	−	0.999053i	$-0.486146\pi$
$727$	1705.77	0.0870202	0.0435101	+	0.999053i	$0.486146\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1035.67	0.0524017
$732$	0	0
$733$	−37122.6	−1.87061	−0.935303	−	0.353847i	$-0.884873\pi$
$733$	−37122.6	−1.87061	−0.935303	+	0.353847i	$0.884873\pi$
$734$	0	0
$735$	−14342.3	−0.719762
$736$	0	0
$737$	9468.20	0.473223
$738$	0	0
$739$	−34256.3	−1.70520	−0.852598	−	0.522568i	$-0.824974\pi$
$739$	−34256.3	−1.70520	−0.852598	+	0.522568i	$0.824974\pi$
$740$	0	0
$741$	1129.47	0.0559945
$742$	0	0
$743$	1567.88	0.0774160	0.0387080	−	0.999251i	$-0.487676\pi$
$743$	1567.88	0.0774160	0.0387080	+	0.999251i	$0.487676\pi$
$744$	0	0
$745$	34489.8	1.69612
$746$	0	0
$747$	−5455.90	−0.267230
$748$	0	0
$749$	8837.17	0.431112
$750$	0	0
$751$	−955.613	−0.0464325	−0.0232163	−	0.999730i	$-0.507391\pi$
$751$	−955.613	−0.0464325	−0.0232163	+	0.999730i	$0.507391\pi$
$752$	0	0
$753$	−8514.65	−0.412073
$754$	0	0
$755$	8017.75	0.386484
$756$	0	0
$757$	14015.4	0.672918	0.336459	−	0.941698i	$-0.390771\pi$
$757$	14015.4	0.672918	0.336459	+	0.941698i	$0.390771\pi$
$758$	0	0
$759$	585.938	0.0280214
$760$	0	0
$761$	−36271.0	−1.72776	−0.863879	−	0.503699i	$-0.831972\pi$
$761$	−36271.0	−1.72776	−0.863879	+	0.503699i	$0.831972\pi$
$762$	0	0
$763$	−14073.9	−0.667770
$764$	0	0
$765$	−2417.44	−0.114252
$766$	0	0
$767$	−14373.6	−0.676665
$768$	0	0
$769$	−18163.6	−0.851749	−0.425874	−	0.904782i	$-0.640033\pi$
$769$	−18163.6	−0.851749	−0.425874	+	0.904782i	$0.640033\pi$
$770$	0	0
$771$	1026.02	0.0479264
$772$	0	0
$773$	8345.65	0.388321	0.194160	−	0.980970i	$-0.437802\pi$
$773$	8345.65	0.388321	0.194160	+	0.980970i	$0.437802\pi$
$774$	0	0
$775$	34858.4	1.61568
$776$	0	0
$777$	2126.19	0.0981683
$778$	0	0
$779$	6968.65	0.320510
$780$	0	0
$781$	6390.11	0.292774
$782$	0	0
$783$	3470.76	0.158410
$784$	0	0
$785$	10909.1	0.496001
$786$	0	0
$787$	22996.2	1.04158	0.520791	−	0.853684i	$-0.325637\pi$
$787$	22996.2	1.04158	0.520791	+	0.853684i	$0.325637\pi$
$788$	0	0
$789$	17685.0	0.797975
$790$	0	0
$791$	−8670.69	−0.389752
$792$	0	0
$793$	−5505.75	−0.246551
$794$	0	0
$795$	26634.0	1.18819
$796$	0	0
$797$	2743.82	0.121946	0.0609730	−	0.998139i	$-0.480580\pi$
$797$	2743.82	0.121946	0.0609730	+	0.998139i	$0.480580\pi$
$798$	0	0
$799$	−7929.68	−0.351104
$800$	0	0
$801$	−210.970	−0.00930619
$802$	0	0
$803$	5611.28	0.246597
$804$	0	0
$805$	−2302.88	−0.100827
$806$	0	0
$807$	7488.53	0.326653
$808$	0	0
$809$	41241.7	1.79231	0.896156	−	0.443738i	$-0.146348\pi$
$809$	41241.7	1.79231	0.896156	+	0.443738i	$0.146348\pi$
$810$	0	0
$811$	−12832.9	−0.555641	−0.277820	−	0.960633i	$-0.589612\pi$
$811$	−12832.9	−0.555641	−0.277820	+	0.960633i	$0.589612\pi$
$812$	0	0
$813$	−6749.03	−0.291142
$814$	0	0
$815$	−18419.7	−0.791675
$816$	0	0
$817$	984.292	0.0421493
$818$	0	0
$819$	1721.51	0.0734488
$820$	0	0
$821$	−16368.5	−0.695817	−0.347908	−	0.937529i	$-0.613108\pi$
$821$	−16368.5	−0.695817	−0.347908	+	0.937529i	$0.613108\pi$
$822$	0	0
$823$	3869.53	0.163892	0.0819461	−	0.996637i	$-0.473886\pi$
$823$	3869.53	0.163892	0.0819461	+	0.996637i	$0.473886\pi$
$824$	0	0
$825$	5243.17	0.221265
$826$	0	0
$827$	−7388.69	−0.310677	−0.155339	−	0.987861i	$-0.549647\pi$
$827$	−7388.69	−0.310677	−0.155339	+	0.987861i	$0.549647\pi$
$828$	0	0
$829$	−23990.1	−1.00508	−0.502539	−	0.864554i	$-0.667601\pi$
$829$	−23990.1	−1.00508	−0.502539	+	0.864554i	$0.667601\pi$
$830$	0	0
$831$	−12247.8	−0.511276
$832$	0	0
$833$	4523.47	0.188150
$834$	0	0
$835$	18925.6	0.784367
$836$	0	0
$837$	5923.68	0.244626
$838$	0	0
$839$	−18228.3	−0.750074	−0.375037	−	0.927010i	$-0.622370\pi$
$839$	−18228.3	−0.750074	−0.375037	+	0.927010i	$0.622370\pi$
$840$	0	0
$841$	−7864.78	−0.322472
$842$	0	0
$843$	−3101.36	−0.126710
$844$	0	0
$845$	−26613.3	−1.08346
$846$	0	0
$847$	−931.424	−0.0377852
$848$	0	0
$849$	23427.4	0.947028
$850$	0	0
$851$	−1634.77	−0.0658511
$852$	0	0
$853$	21737.3	0.872534	0.436267	−	0.899817i	$-0.356300\pi$
$853$	21737.3	0.872534	0.436267	+	0.899817i	$0.356300\pi$
$854$	0	0
$855$	−2297.51	−0.0918987
$856$	0	0
$857$	−18712.2	−0.745852	−0.372926	−	0.927861i	$-0.621645\pi$
$857$	−18712.2	−0.745852	−0.372926	+	0.927861i	$0.621645\pi$
$858$	0	0
$859$	−30527.6	−1.21256	−0.606279	−	0.795252i	$-0.707339\pi$
$859$	−30527.6	−1.21256	−0.606279	+	0.795252i	$0.707339\pi$
$860$	0	0
$861$	10621.5	0.420418
$862$	0	0
$863$	10906.4	0.430196	0.215098	−	0.976592i	$-0.430993\pi$
$863$	10906.4	0.430196	0.215098	+	0.976592i	$0.430993\pi$
$864$	0	0
$865$	−775.255	−0.0304734
$866$	0	0
$867$	−13976.6	−0.547484
$868$	0	0
$869$	11386.3	0.444481
$870$	0	0
$871$	−21388.5	−0.832058
$872$	0	0
$873$	6475.41	0.251042
$874$	0	0
$875$	−4394.68	−0.169791
$876$	0	0
$877$	−21770.9	−0.838256	−0.419128	−	0.907927i	$-0.637664\pi$
$877$	−21770.9	−0.838256	−0.419128	+	0.907927i	$0.637664\pi$
$878$	0	0
$879$	5847.58	0.224384
$880$	0	0
$881$	47206.9	1.80527	0.902634	−	0.430409i	$-0.141631\pi$
$881$	47206.9	1.80527	0.902634	+	0.430409i	$0.141631\pi$
$882$	0	0
$883$	6059.68	0.230945	0.115473	−	0.993311i	$-0.463162\pi$
$883$	6059.68	0.230945	0.115473	+	0.993311i	$0.463162\pi$
$884$	0	0
$885$	29238.3	1.11055
$886$	0	0
$887$	−37130.2	−1.40553	−0.702767	−	0.711420i	$-0.748052\pi$
$887$	−37130.2	−1.40553	−0.702767	+	0.711420i	$0.748052\pi$
$888$	0	0
$889$	−5090.53	−0.192048
$890$	0	0
$891$	891.000	0.0335013
$892$	0	0
$893$	−7536.30	−0.282410
$894$	0	0
$895$	14017.8	0.523536
$896$	0	0
$897$	−1323.63	−0.0492694
$898$	0	0
$899$	28202.5	1.04628
$900$	0	0
$901$	−8400.15	−0.310599
$902$	0	0
$903$	1500.24	0.0552879
$904$	0	0
$905$	30507.0	1.12054
$906$	0	0
$907$	−1182.94	−0.0433064	−0.0216532	−	0.999766i	$-0.506893\pi$
$907$	−1182.94	−0.0433064	−0.0216532	+	0.999766i	$0.506893\pi$
$908$	0	0
$909$	−16841.4	−0.614516
$910$	0	0
$911$	37676.4	1.37022	0.685112	−	0.728438i	$-0.259753\pi$
$911$	37676.4	1.37022	0.685112	+	0.728438i	$0.259753\pi$
$912$	0	0
$913$	−6668.32	−0.241719
$914$	0	0
$915$	11199.6	0.404641
$916$	0	0
$917$	4789.41	0.172476
$918$	0	0
$919$	8697.82	0.312203	0.156101	−	0.987741i	$-0.450107\pi$
$919$	8697.82	0.312203	0.156101	+	0.987741i	$0.450107\pi$
$920$	0	0
$921$	−7092.26	−0.253744
$922$	0	0
$923$	−14435.2	−0.514778
$924$	0	0
$925$	−14628.5	−0.519981
$926$	0	0
$927$	−3858.71	−0.136717
$928$	0	0
$929$	−17247.5	−0.609119	−0.304559	−	0.952493i	$-0.598509\pi$
$929$	−17247.5	−0.609119	−0.304559	+	0.952493i	$0.598509\pi$
$930$	0	0
$931$	4299.06	0.151338
$932$	0	0
$933$	−5967.51	−0.209397
$934$	0	0
$935$	−2954.65	−0.103345
$936$	0	0
$937$	−41812.4	−1.45779	−0.728896	−	0.684624i	$-0.759966\pi$
$937$	−41812.4	−1.45779	−0.728896	+	0.684624i	$0.759966\pi$
$938$	0	0
$939$	−11636.0	−0.404395
$940$	0	0
$941$	−37655.9	−1.30451	−0.652257	−	0.757998i	$-0.726178\pi$
$941$	−37655.9	−1.30451	−0.652257	+	0.757998i	$0.726178\pi$
$942$	0	0
$943$	−8166.60	−0.282016
$944$	0	0
$945$	−3501.84	−0.120545
$946$	0	0
$947$	21244.4	0.728986	0.364493	−	0.931206i	$-0.381242\pi$
$947$	21244.4	0.728986	0.364493	+	0.931206i	$0.381242\pi$
$948$	0	0
$949$	−12675.8	−0.433587
$950$	0	0
$951$	−8741.20	−0.298058
$952$	0	0
$953$	1324.27	0.0450130	0.0225065	−	0.999747i	$-0.492835\pi$
$953$	1324.27	0.0450130	0.0225065	+	0.999747i	$0.492835\pi$
$954$	0	0
$955$	7731.91	0.261988
$956$	0	0
$957$	4242.04	0.143287
$958$	0	0
$959$	14416.6	0.485439
$960$	0	0
$961$	18343.4	0.615735
$962$	0	0
$963$	−10332.2	−0.345744
$964$	0	0
$965$	29972.6	0.999847
$966$	0	0
$967$	52267.1	1.73815	0.869077	−	0.494676i	$-0.164713\pi$
$967$	52267.1	1.73815	0.869077	+	0.494676i	$0.164713\pi$
$968$	0	0
$969$	724.619	0.0240228
$970$	0	0
$971$	52489.8	1.73479	0.867394	−	0.497622i	$-0.165793\pi$
$971$	52489.8	1.73479	0.867394	+	0.497622i	$0.165793\pi$
$972$	0	0
$973$	−7344.07	−0.241973
$974$	0	0
$975$	−11844.3	−0.389046
$976$	0	0
$977$	8324.11	0.272581	0.136291	−	0.990669i	$-0.456482\pi$
$977$	8324.11	0.272581	0.136291	+	0.990669i	$0.456482\pi$
$978$	0	0
$979$	−257.852	−0.00841777
$980$	0	0
$981$	16454.9	0.535539
$982$	0	0
$983$	−44407.1	−1.44086	−0.720431	−	0.693527i	$-0.756056\pi$
$983$	−44407.1	−1.44086	−0.720431	+	0.693527i	$0.756056\pi$
$984$	0	0
$985$	−89375.3	−2.89110
$986$	0	0
$987$	−11486.7	−0.370442
$988$	0	0
$989$	−1153.50	−0.0370870
$990$	0	0
$991$	−45124.7	−1.44645	−0.723226	−	0.690612i	$-0.757341\pi$
$991$	−45124.7	−1.44645	−0.723226	+	0.690612i	$0.757341\pi$
$992$	0	0
$993$	−24313.4	−0.777001
$994$	0	0
$995$	−86576.2	−2.75844
$996$	0	0
$997$	−5480.61	−0.174095	−0.0870474	−	0.996204i	$-0.527743\pi$
$997$	−5480.61	−0.174095	−0.0870474	+	0.996204i	$0.527743\pi$
$998$	0	0
$999$	−2485.90	−0.0787291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.bn.1.2		2
4.3	odd	2		2112.4.a.bg.1.2		2
8.3	odd	2		528.4.a.p.1.1		2
8.5	even	2		33.4.a.c.1.2	✓	2
24.5	odd	2		99.4.a.f.1.1		2
24.11	even	2		1584.4.a.bj.1.2		2
40.13	odd	4		825.4.c.h.199.1		4
40.29	even	2		825.4.a.l.1.1		2
40.37	odd	4		825.4.c.h.199.4		4
56.13	odd	2		1617.4.a.k.1.2		2
88.21	odd	2		363.4.a.i.1.1		2
120.29	odd	2		2475.4.a.p.1.2		2
264.197	even	2		1089.4.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.c.1.2	✓	2	8.5	even	2
99.4.a.f.1.1		2	24.5	odd	2
363.4.a.i.1.1		2	88.21	odd	2
528.4.a.p.1.1		2	8.3	odd	2
825.4.a.l.1.1		2	40.29	even	2
825.4.c.h.199.1		4	40.13	odd	4
825.4.c.h.199.4		4	40.37	odd	4
1089.4.a.u.1.2		2	264.197	even	2
1584.4.a.bj.1.2		2	24.11	even	2
1617.4.a.k.1.2		2	56.13	odd	2
2112.4.a.bg.1.2		2	4.3	odd	2
2112.4.a.bn.1.2		2	1.1	even	1	trivial
2475.4.a.p.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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +16.8489 q^{5} -7.69772 q^{7} +9.00000 q^{9} +11.0000 q^{11} -24.8489 q^{13} +50.5466 q^{15} -15.9420 q^{17} -15.1511 q^{19} -23.0931 q^{21} +17.7557 q^{23} +158.884 q^{25} +27.0000 q^{27} +128.547 q^{29} +219.395 q^{31} +33.0000 q^{33} -129.698 q^{35} -92.0703 q^{37} -74.5466 q^{39} -459.942 q^{41} -64.9648 q^{43} +151.640 q^{45} +497.408 q^{47} -283.745 q^{49} -47.8260 q^{51} +526.919 q^{53} +185.337 q^{55} -45.4534 q^{57} +578.443 q^{59} +221.569 q^{61} -69.2794 q^{63} -418.675 q^{65} +860.745 q^{67} +53.2671 q^{69} +580.919 q^{71} +510.116 q^{73} +476.652 q^{75} -84.6749 q^{77} +1035.12 q^{79} +81.0000 q^{81} -606.211 q^{83} -268.605 q^{85} +385.640 q^{87} -23.4411 q^{89} +191.279 q^{91} +658.186 q^{93} -255.279 q^{95} +719.490 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 14 q^{5} + 24 q^{7} + 18 q^{9} + 22 q^{11} - 30 q^{13} + 42 q^{15} + 106 q^{17} - 50 q^{19} + 72 q^{21} + 134 q^{23} + 42 q^{25} + 54 q^{27} + 198 q^{29} + 360 q^{31} + 66 q^{33} - 220 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 14 * q^5 + 24 * q^7 + 18 * q^9 + 22 * q^11 - 30 * q^13 + 42 * q^15 + 106 * q^17 - 50 * q^19 + 72 * q^21 + 134 * q^23 + 42 * q^25 + 54 * q^27 + 198 * q^29 + 360 * q^31 + 66 * q^33 - 220 * q^35 + 328 * q^37 - 90 * q^39 - 782 * q^41 - 386 * q^43 + 126 * q^45 + 266 * q^47 + 378 * q^49 + 318 * q^51 + 522 * q^53 + 154 * q^55 - 150 * q^57 + 172 * q^59 + 778 * q^61 + 216 * q^63 - 404 * q^65 + 776 * q^67 + 402 * q^69 + 630 * q^71 + 1296 * q^73 + 126 * q^75 + 264 * q^77 + 652 * q^79 + 162 * q^81 + 324 * q^83 - 616 * q^85 + 594 * q^87 - 756 * q^89 + 28 * q^91 + 1080 * q^93 - 156 * q^95 - 452 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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