LMFDB - Embedded newform 448.4.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 14 $ x^3 - x^2 - 11*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32447$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.64895 q^{3} +18.3341 q^{5} -7.00000 q^{7} -5.38731 q^{9} +39.5789 q^{11} -64.5130 q^{13} +85.2344 q^{15} +109.264 q^{17} +137.462 q^{19} -32.5426 q^{21} -45.2344 q^{23} +211.141 q^{25} -150.567 q^{27} +41.1964 q^{29} +262.984 q^{31} +184.000 q^{33} -128.339 q^{35} -125.630 q^{37} -299.917 q^{39} -299.598 q^{41} +36.9141 q^{43} -98.7717 q^{45} +122.770 q^{47} +49.0000 q^{49} +507.963 q^{51} +20.4689 q^{53} +725.645 q^{55} +639.054 q^{57} -60.8037 q^{59} -791.626 q^{61} +37.7112 q^{63} -1182.79 q^{65} +1046.45 q^{67} -210.292 q^{69} -407.738 q^{71} +562.215 q^{73} +981.582 q^{75} -277.052 q^{77} +601.491 q^{79} -554.520 q^{81} -652.836 q^{83} +2003.26 q^{85} +191.520 q^{87} -898.344 q^{89} +451.591 q^{91} +1222.60 q^{93} +2520.25 q^{95} -621.580 q^{97} -213.224 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{3} - 10 q^{5} - 21 q^{7} + 31 q^{9} + 24 q^{11} - 58 q^{13} + 8 q^{15} + 174 q^{17} + 64 q^{19} - 56 q^{21} + 112 q^{23} + 373 q^{25} + 104 q^{27} - 314 q^{29} + 280 q^{31} + 552 q^{33} + 70 q^{35}+ \cdots + 824 q^{99}+O(q^{100}) $ 3 * q + 8 * q^3 - 10 * q^5 - 21 * q^7 + 31 * q^9 + 24 * q^11 - 58 * q^13 + 8 * q^15 + 174 * q^17 + 64 * q^19 - 56 * q^21 + 112 * q^23 + 373 * q^25 + 104 * q^27 - 314 * q^29 + 280 * q^31 + 552 * q^33 + 70 * q^35 - 178 * q^37 + 648 * q^39 + 318 * q^41 + 632 * q^43 - 714 * q^45 + 664 * q^47 + 147 * q^49 + 896 * q^51 - 434 * q^53 + 808 * q^55 - 256 * q^57 + 816 * q^59 - 450 * q^61 - 217 * q^63 - 1604 * q^65 + 408 * q^67 + 1304 * q^69 - 184 * q^71 + 6 * q^73 + 2096 * q^75 - 168 * q^77 + 552 * q^79 - 1381 * q^81 - 1736 * q^83 + 988 * q^85 + 96 * q^87 - 218 * q^89 + 406 * q^91 + 2768 * q^93 + 3832 * q^95 - 1090 * q^97 + 824 * 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
$3$	4.64895	0.894690	0.447345	−	0.894361i	$-0.352370\pi$
$3$	4.64895	0.894690	0.447345	+	0.894361i	$0.352370\pi$
$4$	0	0
$5$	18.3341	1.63986	0.819928	−	0.572467i	$-0.194013\pi$
$5$	18.3341	1.63986	0.819928	+	0.572467i	$0.194013\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−5.38731	−0.199530
$10$	0	0
$11$	39.5789	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$11$	39.5789	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$12$	0	0
$13$	−64.5130	−1.37636	−0.688180	−	0.725540i	$-0.741590\pi$
$13$	−64.5130	−1.37636	−0.688180	+	0.725540i	$0.741590\pi$
$14$	0	0
$15$	85.2344	1.46716
$16$	0	0
$17$	109.264	1.55885	0.779424	−	0.626496i	$-0.215512\pi$
$17$	109.264	1.55885	0.779424	+	0.626496i	$0.215512\pi$
$18$	0	0
$19$	137.462	1.65979	0.829895	−	0.557920i	$-0.188400\pi$
$19$	137.462	1.65979	0.829895	+	0.557920i	$0.188400\pi$
$20$	0	0
$21$	−32.5426	−0.338161
$22$	0	0
$23$	−45.2344	−0.410088	−0.205044	−	0.978753i	$-0.565734\pi$
$23$	−45.2344	−0.410088	−0.205044	+	0.978753i	$0.565734\pi$
$24$	0	0
$25$	211.141	1.68913
$26$	0	0
$27$	−150.567	−1.07321
$28$	0	0
$29$	41.1964	0.263793	0.131896	−	0.991264i	$-0.457893\pi$
$29$	41.1964	0.263793	0.131896	+	0.991264i	$0.457893\pi$
$30$	0	0
$31$	262.984	1.52365	0.761827	−	0.647780i	$-0.224303\pi$
$31$	262.984	1.52365	0.761827	+	0.647780i	$0.224303\pi$
$32$	0	0
$33$	184.000	0.970615
$34$	0	0
$35$	−128.339	−0.619807
$36$	0	0
$37$	−125.630	−0.558202	−0.279101	−	0.960262i	$-0.590036\pi$
$37$	−125.630	−0.558202	−0.279101	+	0.960262i	$0.590036\pi$
$38$	0	0
$39$	−299.917	−1.23142
$40$	0	0
$41$	−299.598	−1.14120	−0.570601	−	0.821227i	$-0.693290\pi$
$41$	−299.598	−1.14120	−0.570601	+	0.821227i	$0.693290\pi$
$42$	0	0
$43$	36.9141	0.130915	0.0654575	−	0.997855i	$-0.479149\pi$
$43$	36.9141	0.130915	0.0654575	+	0.997855i	$0.479149\pi$
$44$	0	0
$45$	−98.7717	−0.327200
$46$	0	0
$47$	122.770	0.381017	0.190509	−	0.981686i	$-0.438986\pi$
$47$	122.770	0.381017	0.190509	+	0.981686i	$0.438986\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	507.963	1.39469
$52$	0	0
$53$	20.4689	0.0530493	0.0265247	−	0.999648i	$-0.491556\pi$
$53$	20.4689	0.0530493	0.0265247	+	0.999648i	$0.491556\pi$
$54$	0	0
$55$	725.645	1.77902
$56$	0	0
$57$	639.054	1.48500
$58$	0	0
$59$	−60.8037	−0.134169	−0.0670845	−	0.997747i	$-0.521370\pi$
$59$	−60.8037	−0.134169	−0.0670845	+	0.997747i	$0.521370\pi$
$60$	0	0
$61$	−791.626	−1.66160	−0.830798	−	0.556574i	$-0.812116\pi$
$61$	−791.626	−1.66160	−0.830798	+	0.556574i	$0.812116\pi$
$62$	0	0
$63$	37.7112	0.0754152
$64$	0	0
$65$	−1182.79	−2.25703
$66$	0	0
$67$	1046.45	1.90812	0.954058	−	0.299621i	$-0.0968601\pi$
$67$	1046.45	1.90812	0.954058	+	0.299621i	$0.0968601\pi$
$68$	0	0
$69$	−210.292	−0.366902
$70$	0	0
$71$	−407.738	−0.681543	−0.340772	−	0.940146i	$-0.610688\pi$
$71$	−407.738	−0.681543	−0.340772	+	0.940146i	$0.610688\pi$
$72$	0	0
$73$	562.215	0.901402	0.450701	−	0.892675i	$-0.351174\pi$
$73$	562.215	0.901402	0.450701	+	0.892675i	$0.351174\pi$
$74$	0	0
$75$	981.582	1.51124
$76$	0	0
$77$	−277.052	−0.410039
$78$	0	0
$79$	601.491	0.856621	0.428311	−	0.903632i	$-0.359109\pi$
$79$	601.491	0.856621	0.428311	+	0.903632i	$0.359109\pi$
$80$	0	0
$81$	−554.520	−0.760658
$82$	0	0
$83$	−652.836	−0.863351	−0.431675	−	0.902029i	$-0.642077\pi$
$83$	−652.836	−0.863351	−0.431675	+	0.902029i	$0.642077\pi$
$84$	0	0
$85$	2003.26	2.55629
$86$	0	0
$87$	191.520	0.236013
$88$	0	0
$89$	−898.344	−1.06994	−0.534968	−	0.844872i	$-0.679676\pi$
$89$	−898.344	−1.06994	−0.534968	+	0.844872i	$0.679676\pi$
$90$	0	0
$91$	451.591	0.520215
$92$	0	0
$93$	1222.60	1.36320
$94$	0	0
$95$	2520.25	2.72182
$96$	0	0
$97$	−621.580	−0.650638	−0.325319	−	0.945604i	$-0.605472\pi$
$97$	−621.580	−0.650638	−0.325319	+	0.945604i	$0.605472\pi$
$98$	0	0
$99$	−213.224	−0.216462
$100$	0	0
$101$	566.973	0.558574	0.279287	−	0.960208i	$-0.409902\pi$
$101$	566.973	0.558574	0.279287	+	0.960208i	$0.409902\pi$
$102$	0	0
$103$	−143.459	−0.137238	−0.0686188	−	0.997643i	$-0.521859\pi$
$103$	−143.459	−0.137238	−0.0686188	+	0.997643i	$0.521859\pi$
$104$	0	0
$105$	−596.641	−0.554535
$106$	0	0
$107$	951.778	0.859924	0.429962	−	0.902847i	$-0.358527\pi$
$107$	951.778	0.859924	0.429962	+	0.902847i	$0.358527\pi$
$108$	0	0
$109$	−1122.90	−0.986738	−0.493369	−	0.869820i	$-0.664235\pi$
$109$	−1122.90	−0.986738	−0.493369	+	0.869820i	$0.664235\pi$
$110$	0	0
$111$	−584.047	−0.499417
$112$	0	0
$113$	499.874	0.416143	0.208072	−	0.978114i	$-0.433281\pi$
$113$	499.874	0.416143	0.208072	+	0.978114i	$0.433281\pi$
$114$	0	0
$115$	−829.335	−0.672486
$116$	0	0
$117$	347.551	0.274625
$118$	0	0
$119$	−764.848	−0.589189
$120$	0	0
$121$	235.487	0.176925
$122$	0	0
$123$	−1392.81	−1.02102
$124$	0	0
$125$	1579.32	1.13007
$126$	0	0
$127$	−644.834	−0.450549	−0.225275	−	0.974295i	$-0.572328\pi$
$127$	−644.834	−0.450549	−0.225275	+	0.974295i	$0.572328\pi$
$128$	0	0
$129$	171.612	0.117128
$130$	0	0
$131$	−1728.38	−1.15274	−0.576370	−	0.817189i	$-0.695531\pi$
$131$	−1728.38	−1.15274	−0.576370	+	0.817189i	$0.695531\pi$
$132$	0	0
$133$	−962.236	−0.627341
$134$	0	0
$135$	−2760.51	−1.75991
$136$	0	0
$137$	1146.24	0.714815	0.357408	−	0.933949i	$-0.383661\pi$
$137$	1146.24	0.714815	0.357408	+	0.933949i	$0.383661\pi$
$138$	0	0
$139$	−1638.07	−0.999564	−0.499782	−	0.866151i	$-0.666587\pi$
$139$	−1638.07	−0.999564	−0.499782	+	0.866151i	$0.666587\pi$
$140$	0	0
$141$	570.750	0.340892
$142$	0	0
$143$	−2553.35	−1.49316
$144$	0	0
$145$	755.301	0.432582
$146$	0	0
$147$	227.798	0.127813
$148$	0	0
$149$	−1722.49	−0.947060	−0.473530	−	0.880778i	$-0.657020\pi$
$149$	−1722.49	−0.947060	−0.473530	+	0.880778i	$0.657020\pi$
$150$	0	0
$151$	3262.89	1.75848	0.879240	−	0.476380i	$-0.158051\pi$
$151$	3262.89	1.75848	0.879240	+	0.476380i	$0.158051\pi$
$152$	0	0
$153$	−588.639	−0.311037
$154$	0	0
$155$	4821.58	2.49857
$156$	0	0
$157$	−1027.98	−0.522559	−0.261280	−	0.965263i	$-0.584144\pi$
$157$	−1027.98	−0.522559	−0.261280	+	0.965263i	$0.584144\pi$
$158$	0	0
$159$	95.1586	0.0474627
$160$	0	0
$161$	316.641	0.154999
$162$	0	0
$163$	−623.566	−0.299641	−0.149820	−	0.988713i	$-0.547870\pi$
$163$	−623.566	−0.299641	−0.149820	+	0.988713i	$0.547870\pi$
$164$	0	0
$165$	3373.48	1.59167
$166$	0	0
$167$	1312.31	0.608083	0.304041	−	0.952659i	$-0.401664\pi$
$167$	1312.31	0.608083	0.304041	+	0.952659i	$0.401664\pi$
$168$	0	0
$169$	1964.92	0.894367
$170$	0	0
$171$	−740.551	−0.331178
$172$	0	0
$173$	1602.25	0.704145	0.352073	−	0.935973i	$-0.385477\pi$
$173$	1602.25	0.704145	0.352073	+	0.935973i	$0.385477\pi$
$174$	0	0
$175$	−1477.99	−0.638430
$176$	0	0
$177$	−282.673	−0.120040
$178$	0	0
$179$	−2117.12	−0.884030	−0.442015	−	0.897008i	$-0.645736\pi$
$179$	−2117.12	−0.884030	−0.442015	+	0.897008i	$0.645736\pi$
$180$	0	0
$181$	−4200.80	−1.72510	−0.862550	−	0.505972i	$-0.831134\pi$
$181$	−4200.80	−1.72510	−0.862550	+	0.505972i	$0.831134\pi$
$182$	0	0
$183$	−3680.23	−1.48661
$184$	0	0
$185$	−2303.32	−0.915370
$186$	0	0
$187$	4324.55	1.69114
$188$	0	0
$189$	1053.97	0.405634
$190$	0	0
$191$	1211.18	0.458838	0.229419	−	0.973328i	$-0.426317\pi$
$191$	1211.18	0.458838	0.229419	+	0.973328i	$0.426317\pi$
$192$	0	0
$193$	−1640.80	−0.611955	−0.305977	−	0.952039i	$-0.598983\pi$
$193$	−1640.80	−0.611955	−0.305977	+	0.952039i	$0.598983\pi$
$194$	0	0
$195$	−5498.73	−2.01934
$196$	0	0
$197$	−2945.77	−1.06537	−0.532683	−	0.846315i	$-0.678816\pi$
$197$	−2945.77	−1.06537	−0.532683	+	0.846315i	$0.678816\pi$
$198$	0	0
$199$	−1695.77	−0.604071	−0.302036	−	0.953297i	$-0.597666\pi$
$199$	−1695.77	−0.604071	−0.302036	+	0.953297i	$0.597666\pi$
$200$	0	0
$201$	4864.87	1.70717
$202$	0	0
$203$	−288.375	−0.0997042
$204$	0	0
$205$	−5492.87	−1.87141
$206$	0	0
$207$	243.692	0.0818249
$208$	0	0
$209$	5440.60	1.80064
$210$	0	0
$211$	−4907.89	−1.60129	−0.800647	−	0.599136i	$-0.795511\pi$
$211$	−4907.89	−1.60129	−0.800647	+	0.599136i	$0.795511\pi$
$212$	0	0
$213$	−1895.55	−0.609770
$214$	0	0
$215$	676.788	0.214682
$216$	0	0
$217$	−1840.89	−0.575887
$218$	0	0
$219$	2613.71	0.806475
$220$	0	0
$221$	−7048.95	−2.14554
$222$	0	0
$223$	4967.91	1.49182	0.745910	−	0.666046i	$-0.232015\pi$
$223$	4967.91	1.49182	0.745910	+	0.666046i	$0.232015\pi$
$224$	0	0
$225$	−1137.48	−0.337031
$226$	0	0
$227$	−4980.74	−1.45632	−0.728158	−	0.685410i	$-0.759623\pi$
$227$	−4980.74	−1.45632	−0.728158	+	0.685410i	$0.759623\pi$
$228$	0	0
$229$	3630.32	1.04759	0.523795	−	0.851844i	$-0.324516\pi$
$229$	3630.32	1.04759	0.523795	+	0.851844i	$0.324516\pi$
$230$	0	0
$231$	−1288.00	−0.366858
$232$	0	0
$233$	−1138.20	−0.320026	−0.160013	−	0.987115i	$-0.551154\pi$
$233$	−1138.20	−0.320026	−0.160013	+	0.987115i	$0.551154\pi$
$234$	0	0
$235$	2250.88	0.624813
$236$	0	0
$237$	2796.30	0.766410
$238$	0	0
$239$	−4226.15	−1.14380	−0.571898	−	0.820325i	$-0.693793\pi$
$239$	−4226.15	−1.14380	−0.571898	+	0.820325i	$0.693793\pi$
$240$	0	0
$241$	−602.692	−0.161090	−0.0805452	−	0.996751i	$-0.525666\pi$
$241$	−602.692	−0.161090	−0.0805452	+	0.996751i	$0.525666\pi$
$242$	0	0
$243$	1487.37	0.392654
$244$	0	0
$245$	898.373	0.234265
$246$	0	0
$247$	−8868.10	−2.28447
$248$	0	0
$249$	−3035.00	−0.772431
$250$	0	0
$251$	−7249.07	−1.82294	−0.911469	−	0.411370i	$-0.865051\pi$
$251$	−7249.07	−1.82294	−0.911469	+	0.411370i	$0.865051\pi$
$252$	0	0
$253$	−1790.33	−0.444889
$254$	0	0
$255$	9313.06	2.28708
$256$	0	0
$257$	1373.20	0.333299	0.166650	−	0.986016i	$-0.446705\pi$
$257$	1373.20	0.333299	0.166650	+	0.986016i	$0.446705\pi$
$258$	0	0
$259$	879.411	0.210980
$260$	0	0
$261$	−221.938	−0.0526345
$262$	0	0
$263$	−4458.42	−1.04532	−0.522658	−	0.852542i	$-0.675060\pi$
$263$	−4458.42	−1.04532	−0.522658	+	0.852542i	$0.675060\pi$
$264$	0	0
$265$	375.279	0.0869932
$266$	0	0
$267$	−4176.35	−0.957261
$268$	0	0
$269$	6660.26	1.50960	0.754802	−	0.655953i	$-0.227733\pi$
$269$	6660.26	1.50960	0.754802	+	0.655953i	$0.227733\pi$
$270$	0	0
$271$	−7604.31	−1.70453	−0.852267	−	0.523107i	$-0.824773\pi$
$271$	−7604.31	−1.70453	−0.852267	+	0.523107i	$0.824773\pi$
$272$	0	0
$273$	2099.42	0.465431
$274$	0	0
$275$	8356.72	1.83247
$276$	0	0
$277$	−1853.26	−0.401990	−0.200995	−	0.979592i	$-0.564418\pi$
$277$	−1853.26	−0.401990	−0.200995	+	0.979592i	$0.564418\pi$
$278$	0	0
$279$	−1416.77	−0.304015
$280$	0	0
$281$	−6591.70	−1.39939	−0.699694	−	0.714443i	$-0.746680\pi$
$281$	−6591.70	−1.39939	−0.699694	+	0.714443i	$0.746680\pi$
$282$	0	0
$283$	13.9668	0.00293371	0.00146686	−	0.999999i	$-0.499533\pi$
$283$	13.9668	0.00293371	0.00146686	+	0.999999i	$0.499533\pi$
$284$	0	0
$285$	11716.5	2.43518
$286$	0	0
$287$	2097.18	0.431334
$288$	0	0
$289$	7025.64	1.43001
$290$	0	0
$291$	−2889.69	−0.582119
$292$	0	0
$293$	6043.17	1.20493	0.602467	−	0.798144i	$-0.294184\pi$
$293$	6043.17	1.20493	0.602467	+	0.798144i	$0.294184\pi$
$294$	0	0
$295$	−1114.78	−0.220018
$296$	0	0
$297$	−5959.26	−1.16428
$298$	0	0
$299$	2918.21	0.564429
$300$	0	0
$301$	−258.399	−0.0494812
$302$	0	0
$303$	2635.83	0.499750
$304$	0	0
$305$	−14513.8	−2.72478
$306$	0	0
$307$	−4267.18	−0.793293	−0.396647	−	0.917971i	$-0.629826\pi$
$307$	−4267.18	−0.793293	−0.396647	+	0.917971i	$0.629826\pi$
$308$	0	0
$309$	−666.934	−0.122785
$310$	0	0
$311$	831.172	0.151548	0.0757740	−	0.997125i	$-0.475857\pi$
$311$	831.172	0.151548	0.0757740	+	0.997125i	$0.475857\pi$
$312$	0	0
$313$	6078.60	1.09771	0.548854	−	0.835918i	$-0.315064\pi$
$313$	6078.60	1.09771	0.548854	+	0.835918i	$0.315064\pi$
$314$	0	0
$315$	691.402	0.123670
$316$	0	0
$317$	−942.333	−0.166961	−0.0834806	−	0.996509i	$-0.526604\pi$
$317$	−942.333	−0.166961	−0.0834806	+	0.996509i	$0.526604\pi$
$318$	0	0
$319$	1630.51	0.286178
$320$	0	0
$321$	4424.76	0.769365
$322$	0	0
$323$	15019.7	2.58736
$324$	0	0
$325$	−13621.3	−2.32485
$326$	0	0
$327$	−5220.31	−0.882825
$328$	0	0
$329$	−859.388	−0.144011
$330$	0	0
$331$	−1203.06	−0.199777	−0.0998884	−	0.994999i	$-0.531849\pi$
$331$	−1203.06	−0.199777	−0.0998884	+	0.994999i	$0.531849\pi$
$332$	0	0
$333$	676.808	0.111378
$334$	0	0
$335$	19185.7	3.12904
$336$	0	0
$337$	9139.85	1.47739	0.738693	−	0.674042i	$-0.235443\pi$
$337$	9139.85	1.47739	0.738693	+	0.674042i	$0.235443\pi$
$338$	0	0
$339$	2323.89	0.372319
$340$	0	0
$341$	10408.6	1.65295
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3855.53	−0.601666
$346$	0	0
$347$	10250.4	1.58580	0.792898	−	0.609354i	$-0.208571\pi$
$347$	10250.4	1.58580	0.792898	+	0.609354i	$0.208571\pi$
$348$	0	0
$349$	8186.99	1.25570	0.627850	−	0.778334i	$-0.283935\pi$
$349$	8186.99	1.25570	0.627850	+	0.778334i	$0.283935\pi$
$350$	0	0
$351$	9713.51	1.47712
$352$	0	0
$353$	−5928.04	−0.893818	−0.446909	−	0.894580i	$-0.647475\pi$
$353$	−5928.04	−0.893818	−0.446909	+	0.894580i	$0.647475\pi$
$354$	0	0
$355$	−7475.52	−1.11763
$356$	0	0
$357$	−3555.74	−0.527142
$358$	0	0
$359$	5943.02	0.873707	0.436853	−	0.899533i	$-0.356093\pi$
$359$	5943.02	0.873707	0.436853	+	0.899533i	$0.356093\pi$
$360$	0	0
$361$	12036.9	1.75490
$362$	0	0
$363$	1094.77	0.158293
$364$	0	0
$365$	10307.7	1.47817
$366$	0	0
$367$	−978.903	−0.139232	−0.0696162	−	0.997574i	$-0.522177\pi$
$367$	−978.903	−0.139232	−0.0696162	+	0.997574i	$0.522177\pi$
$368$	0	0
$369$	1614.02	0.227704
$370$	0	0
$371$	−143.282	−0.0200508
$372$	0	0
$373$	12025.2	1.66928	0.834639	−	0.550798i	$-0.185676\pi$
$373$	12025.2	1.66928	0.834639	+	0.550798i	$0.185676\pi$
$374$	0	0
$375$	7342.16	1.01106
$376$	0	0
$377$	−2657.70	−0.363074
$378$	0	0
$379$	2528.04	0.342630	0.171315	−	0.985216i	$-0.445198\pi$
$379$	2528.04	0.342630	0.171315	+	0.985216i	$0.445198\pi$
$380$	0	0
$381$	−2997.80	−0.403102
$382$	0	0
$383$	9220.11	1.23009	0.615047	−	0.788491i	$-0.289137\pi$
$383$	9220.11	1.23009	0.615047	+	0.788491i	$0.289137\pi$
$384$	0	0
$385$	−5079.51	−0.672405
$386$	0	0
$387$	−198.868	−0.0261215
$388$	0	0
$389$	−3269.81	−0.426186	−0.213093	−	0.977032i	$-0.568354\pi$
$389$	−3269.81	−0.426186	−0.213093	+	0.977032i	$0.568354\pi$
$390$	0	0
$391$	−4942.50	−0.639266
$392$	0	0
$393$	−8035.13	−1.03135
$394$	0	0
$395$	11027.8	1.40473
$396$	0	0
$397$	−12910.1	−1.63208	−0.816042	−	0.577992i	$-0.803836\pi$
$397$	−12910.1	−1.63208	−0.816042	+	0.577992i	$0.803836\pi$
$398$	0	0
$399$	−4473.38	−0.561276
$400$	0	0
$401$	−15097.2	−1.88010	−0.940050	−	0.341036i	$-0.889222\pi$
$401$	−15097.2	−1.88010	−0.940050	+	0.341036i	$0.889222\pi$
$402$	0	0
$403$	−16965.9	−2.09710
$404$	0	0
$405$	−10166.6	−1.24737
$406$	0	0
$407$	−4972.30	−0.605572
$408$	0	0
$409$	−8329.99	−1.00707	−0.503535	−	0.863975i	$-0.667967\pi$
$409$	−8329.99	−1.00707	−0.503535	+	0.863975i	$0.667967\pi$
$410$	0	0
$411$	5328.80	0.639538
$412$	0	0
$413$	425.626	0.0507111
$414$	0	0
$415$	−11969.2	−1.41577
$416$	0	0
$417$	−7615.30	−0.894300
$418$	0	0
$419$	10506.1	1.22496	0.612479	−	0.790487i	$-0.290172\pi$
$419$	10506.1	1.22496	0.612479	+	0.790487i	$0.290172\pi$
$420$	0	0
$421$	−1795.85	−0.207897	−0.103948	−	0.994583i	$-0.533148\pi$
$421$	−1795.85	−0.207897	−0.103948	+	0.994583i	$0.533148\pi$
$422$	0	0
$423$	−661.398	−0.0760243
$424$	0	0
$425$	23070.1	2.63309
$426$	0	0
$427$	5541.38	0.628024
$428$	0	0
$429$	−11870.4	−1.33592
$430$	0	0
$431$	3067.87	0.342864	0.171432	−	0.985196i	$-0.445161\pi$
$431$	3067.87	0.342864	0.171432	+	0.985196i	$0.445161\pi$
$432$	0	0
$433$	2713.60	0.301172	0.150586	−	0.988597i	$-0.451884\pi$
$433$	2713.60	0.301172	0.150586	+	0.988597i	$0.451884\pi$
$434$	0	0
$435$	3511.35	0.387027
$436$	0	0
$437$	−6218.03	−0.680660
$438$	0	0
$439$	7001.06	0.761145	0.380572	−	0.924751i	$-0.375727\pi$
$439$	7001.06	0.761145	0.380572	+	0.924751i	$0.375727\pi$
$440$	0	0
$441$	−263.978	−0.0285043
$442$	0	0
$443$	−9583.07	−1.02778	−0.513889	−	0.857857i	$-0.671796\pi$
$443$	−9583.07	−1.02778	−0.513889	+	0.857857i	$0.671796\pi$
$444$	0	0
$445$	−16470.4	−1.75454
$446$	0	0
$447$	−8007.76	−0.847325
$448$	0	0
$449$	14091.2	1.48107	0.740537	−	0.672015i	$-0.234571\pi$
$449$	14091.2	1.48107	0.740537	+	0.672015i	$0.234571\pi$
$450$	0	0
$451$	−11857.7	−1.23805
$452$	0	0
$453$	15169.0	1.57329
$454$	0	0
$455$	8279.53	0.853078
$456$	0	0
$457$	−4134.59	−0.423212	−0.211606	−	0.977355i	$-0.567869\pi$
$457$	−4134.59	−0.423212	−0.211606	+	0.977355i	$0.567869\pi$
$458$	0	0
$459$	−16451.5	−1.67297
$460$	0	0
$461$	39.6890	0.00400977	0.00200488	−	0.999998i	$-0.499362\pi$
$461$	39.6890	0.00400977	0.00200488	+	0.999998i	$0.499362\pi$
$462$	0	0
$463$	5059.57	0.507858	0.253929	−	0.967223i	$-0.418277\pi$
$463$	5059.57	0.507858	0.253929	+	0.967223i	$0.418277\pi$
$464$	0	0
$465$	22415.3	2.23545
$466$	0	0
$467$	5316.15	0.526771	0.263386	−	0.964691i	$-0.415161\pi$
$467$	5316.15	0.526771	0.263386	+	0.964691i	$0.415161\pi$
$468$	0	0
$469$	−7325.13	−0.721200
$470$	0	0
$471$	−4779.03	−0.467529
$472$	0	0
$473$	1461.02	0.142025
$474$	0	0
$475$	29023.9	2.80359
$476$	0	0
$477$	−110.272	−0.0105849
$478$	0	0
$479$	−9881.68	−0.942600	−0.471300	−	0.881973i	$-0.656215\pi$
$479$	−9881.68	−0.942600	−0.471300	+	0.881973i	$0.656215\pi$
$480$	0	0
$481$	8104.77	0.768287
$482$	0	0
$483$	1472.05	0.138676
$484$	0	0
$485$	−11396.1	−1.06695
$486$	0	0
$487$	−6028.14	−0.560905	−0.280453	−	0.959868i	$-0.590485\pi$
$487$	−6028.14	−0.560905	−0.280453	+	0.959868i	$0.590485\pi$
$488$	0	0
$489$	−2898.92	−0.268086
$490$	0	0
$491$	2441.52	0.224408	0.112204	−	0.993685i	$-0.464209\pi$
$491$	2441.52	0.224408	0.112204	+	0.993685i	$0.464209\pi$
$492$	0	0
$493$	4501.29	0.411213
$494$	0	0
$495$	−3909.27	−0.354967
$496$	0	0
$497$	2854.16	0.257599
$498$	0	0
$499$	3980.18	0.357069	0.178534	−	0.983934i	$-0.442864\pi$
$499$	3980.18	0.357069	0.178534	+	0.983934i	$0.442864\pi$
$500$	0	0
$501$	6100.87	0.544046
$502$	0	0
$503$	4203.45	0.372610	0.186305	−	0.982492i	$-0.440349\pi$
$503$	4203.45	0.372610	0.186305	+	0.982492i	$0.440349\pi$
$504$	0	0
$505$	10395.0	0.915981
$506$	0	0
$507$	9134.83	0.800181
$508$	0	0
$509$	−7000.16	−0.609581	−0.304790	−	0.952419i	$-0.598586\pi$
$509$	−7000.16	−0.609581	−0.304790	+	0.952419i	$0.598586\pi$
$510$	0	0
$511$	−3935.51	−0.340698
$512$	0	0
$513$	−20697.3	−1.78130
$514$	0	0
$515$	−2630.20	−0.225050
$516$	0	0
$517$	4859.09	0.413351
$518$	0	0
$519$	7448.79	0.629992
$520$	0	0
$521$	19922.9	1.67531	0.837657	−	0.546197i	$-0.183925\pi$
$521$	19922.9	1.67531	0.837657	+	0.546197i	$0.183925\pi$
$522$	0	0
$523$	−18826.8	−1.57407	−0.787033	−	0.616911i	$-0.788384\pi$
$523$	−18826.8	−1.57407	−0.787033	+	0.616911i	$0.788384\pi$
$524$	0	0
$525$	−6871.08	−0.571197
$526$	0	0
$527$	28734.7	2.37515
$528$	0	0
$529$	−10120.8	−0.831828
$530$	0	0
$531$	327.568	0.0267707
$532$	0	0
$533$	19327.9	1.57071
$534$	0	0
$535$	17450.0	1.41015
$536$	0	0
$537$	−9842.40	−0.790933
$538$	0	0
$539$	1939.36	0.154980
$540$	0	0
$541$	−7267.56	−0.577555	−0.288777	−	0.957396i	$-0.593249\pi$
$541$	−7267.56	−0.577555	−0.288777	+	0.957396i	$0.593249\pi$
$542$	0	0
$543$	−19529.3	−1.54343
$544$	0	0
$545$	−20587.4	−1.61811
$546$	0	0
$547$	−17807.3	−1.39193	−0.695966	−	0.718075i	$-0.745023\pi$
$547$	−17807.3	−1.39193	−0.695966	+	0.718075i	$0.745023\pi$
$548$	0	0
$549$	4264.74	0.331538
$550$	0	0
$551$	5662.95	0.437840
$552$	0	0
$553$	−4210.44	−0.323772
$554$	0	0
$555$	−10708.0	−0.818973
$556$	0	0
$557$	−14127.4	−1.07468	−0.537342	−	0.843364i	$-0.680572\pi$
$557$	−14127.4	−1.07468	−0.537342	+	0.843364i	$0.680572\pi$
$558$	0	0
$559$	−2381.44	−0.180186
$560$	0	0
$561$	20104.6	1.51304
$562$	0	0
$563$	19742.3	1.47787	0.738934	−	0.673778i	$-0.235330\pi$
$563$	19742.3	1.47787	0.738934	+	0.673778i	$0.235330\pi$
$564$	0	0
$565$	9164.76	0.682415
$566$	0	0
$567$	3881.64	0.287502
$568$	0	0
$569$	−1074.96	−0.0791999	−0.0396000	−	0.999216i	$-0.512608\pi$
$569$	−1074.96	−0.0791999	−0.0396000	+	0.999216i	$0.512608\pi$
$570$	0	0
$571$	11584.3	0.849012	0.424506	−	0.905425i	$-0.360448\pi$
$571$	11584.3	0.849012	0.424506	+	0.905425i	$0.360448\pi$
$572$	0	0
$573$	5630.72	0.410518
$574$	0	0
$575$	−9550.84	−0.692691
$576$	0	0
$577$	−16784.8	−1.21103	−0.605513	−	0.795835i	$-0.707032\pi$
$577$	−16784.8	−1.21103	−0.605513	+	0.795835i	$0.707032\pi$
$578$	0	0
$579$	−7627.98	−0.547510
$580$	0	0
$581$	4569.86	0.326316
$582$	0	0
$583$	810.134	0.0575512
$584$	0	0
$585$	6372.06	0.450345
$586$	0	0
$587$	−22677.2	−1.59453	−0.797265	−	0.603629i	$-0.793721\pi$
$587$	−22677.2	−1.59453	−0.797265	+	0.603629i	$0.793721\pi$
$588$	0	0
$589$	36150.3	2.52895
$590$	0	0
$591$	−13694.7	−0.953172
$592$	0	0
$593$	2238.64	0.155025	0.0775126	−	0.996991i	$-0.475302\pi$
$593$	2238.64	0.155025	0.0775126	+	0.996991i	$0.475302\pi$
$594$	0	0
$595$	−14022.8	−0.966186
$596$	0	0
$597$	−7883.56	−0.540457
$598$	0	0
$599$	−8128.35	−0.554450	−0.277225	−	0.960805i	$-0.589415\pi$
$599$	−8128.35	−0.554450	−0.277225	+	0.960805i	$0.589415\pi$
$600$	0	0
$601$	27024.7	1.83421	0.917104	−	0.398647i	$-0.130520\pi$
$601$	27024.7	1.83421	0.917104	+	0.398647i	$0.130520\pi$
$602$	0	0
$603$	−5637.53	−0.380726
$604$	0	0
$605$	4317.45	0.290131
$606$	0	0
$607$	−12152.5	−0.812608	−0.406304	−	0.913738i	$-0.633183\pi$
$607$	−12152.5	−0.812608	−0.406304	+	0.913738i	$0.633183\pi$
$608$	0	0
$609$	−1340.64	−0.0892044
$610$	0	0
$611$	−7920.24	−0.524417
$612$	0	0
$613$	−3652.32	−0.240646	−0.120323	−	0.992735i	$-0.538393\pi$
$613$	−3652.32	−0.240646	−0.120323	+	0.992735i	$0.538393\pi$
$614$	0	0
$615$	−25536.0	−1.67433
$616$	0	0
$617$	6601.28	0.430725	0.215363	−	0.976534i	$-0.430907\pi$
$617$	6601.28	0.430725	0.215363	+	0.976534i	$0.430907\pi$
$618$	0	0
$619$	1475.11	0.0957832	0.0478916	−	0.998853i	$-0.484750\pi$
$619$	1475.11	0.0957832	0.0478916	+	0.998853i	$0.484750\pi$
$620$	0	0
$621$	6810.80	0.440110
$622$	0	0
$623$	6288.41	0.404398
$624$	0	0
$625$	2562.85	0.164022
$626$	0	0
$627$	25293.1	1.61102
$628$	0	0
$629$	−13726.9	−0.870152
$630$	0	0
$631$	4306.01	0.271664	0.135832	−	0.990732i	$-0.456629\pi$
$631$	4306.01	0.271664	0.135832	+	0.990732i	$0.456629\pi$
$632$	0	0
$633$	−22816.5	−1.43266
$634$	0	0
$635$	−11822.5	−0.738835
$636$	0	0
$637$	−3161.14	−0.196623
$638$	0	0
$639$	2196.61	0.135988
$640$	0	0
$641$	5992.29	0.369238	0.184619	−	0.982810i	$-0.440895\pi$
$641$	5992.29	0.369238	0.184619	+	0.982810i	$0.440895\pi$
$642$	0	0
$643$	11978.2	0.734643	0.367322	−	0.930094i	$-0.380275\pi$
$643$	11978.2	0.734643	0.367322	+	0.930094i	$0.380275\pi$
$644$	0	0
$645$	3146.35	0.192074
$646$	0	0
$647$	24522.4	1.49007	0.745035	−	0.667026i	$-0.232433\pi$
$647$	24522.4	1.49007	0.745035	+	0.667026i	$0.232433\pi$
$648$	0	0
$649$	−2406.54	−0.145555
$650$	0	0
$651$	−8558.18	−0.515240
$652$	0	0
$653$	−2758.50	−0.165311	−0.0826557	−	0.996578i	$-0.526340\pi$
$653$	−2758.50	−0.165311	−0.0826557	+	0.996578i	$0.526340\pi$
$654$	0	0
$655$	−31688.3	−1.89033
$656$	0	0
$657$	−3028.83	−0.179857
$658$	0	0
$659$	−4620.17	−0.273105	−0.136553	−	0.990633i	$-0.543602\pi$
$659$	−4620.17	−0.273105	−0.136553	+	0.990633i	$0.543602\pi$
$660$	0	0
$661$	−464.202	−0.0273152	−0.0136576	−	0.999907i	$-0.504347\pi$
$661$	−464.202	−0.0273152	−0.0136576	+	0.999907i	$0.504347\pi$
$662$	0	0
$663$	−32770.2	−1.91959
$664$	0	0
$665$	−17641.8	−1.02875
$666$	0	0
$667$	−1863.50	−0.108178
$668$	0	0
$669$	23095.6	1.33472
$670$	0	0
$671$	−31331.7	−1.80260
$672$	0	0
$673$	−16725.1	−0.957954	−0.478977	−	0.877827i	$-0.658992\pi$
$673$	−16725.1	−0.957954	−0.478977	+	0.877827i	$0.658992\pi$
$674$	0	0
$675$	−31790.8	−1.81278
$676$	0	0
$677$	13671.5	0.776125	0.388063	−	0.921633i	$-0.373144\pi$
$677$	13671.5	0.776125	0.388063	+	0.921633i	$0.373144\pi$
$678$	0	0
$679$	4351.06	0.245918
$680$	0	0
$681$	−23155.2	−1.30295
$682$	0	0
$683$	71.8070	0.00402287	0.00201143	−	0.999998i	$-0.499360\pi$
$683$	71.8070	0.00402287	0.00201143	+	0.999998i	$0.499360\pi$
$684$	0	0
$685$	21015.3	1.17219
$686$	0	0
$687$	16877.2	0.937269
$688$	0	0
$689$	−1320.51	−0.0730150
$690$	0	0
$691$	1148.37	0.0632212	0.0316106	−	0.999500i	$-0.489936\pi$
$691$	1148.37	0.0632212	0.0316106	+	0.999500i	$0.489936\pi$
$692$	0	0
$693$	1492.56	0.0818151
$694$	0	0
$695$	−30032.6	−1.63914
$696$	0	0
$697$	−32735.3	−1.77896
$698$	0	0
$699$	−5291.44	−0.286324
$700$	0	0
$701$	−24434.2	−1.31650	−0.658249	−	0.752800i	$-0.728703\pi$
$701$	−24434.2	−1.31650	−0.658249	+	0.752800i	$0.728703\pi$
$702$	0	0
$703$	−17269.4	−0.926497
$704$	0	0
$705$	10464.2	0.559014
$706$	0	0
$707$	−3968.81	−0.211121
$708$	0	0
$709$	22082.3	1.16970	0.584850	−	0.811141i	$-0.301153\pi$
$709$	22082.3	1.16970	0.584850	+	0.811141i	$0.301153\pi$
$710$	0	0
$711$	−3240.42	−0.170922
$712$	0	0
$713$	−11895.9	−0.624833
$714$	0	0
$715$	−46813.5	−2.44857
$716$	0	0
$717$	−19647.2	−1.02334
$718$	0	0
$719$	18467.9	0.957910	0.478955	−	0.877839i	$-0.341016\pi$
$719$	18467.9	0.957910	0.478955	+	0.877839i	$0.341016\pi$
$720$	0	0
$721$	1004.22	0.0518709
$722$	0	0
$723$	−2801.88	−0.144126
$724$	0	0
$725$	8698.25	0.445579
$726$	0	0
$727$	12179.8	0.621352	0.310676	−	0.950516i	$-0.399445\pi$
$727$	12179.8	0.621352	0.310676	+	0.950516i	$0.399445\pi$
$728$	0	0
$729$	21886.7	1.11196
$730$	0	0
$731$	4033.38	0.204077
$732$	0	0
$733$	16641.6	0.838571	0.419286	−	0.907854i	$-0.362281\pi$
$733$	16641.6	0.838571	0.419286	+	0.907854i	$0.362281\pi$
$734$	0	0
$735$	4176.49	0.209595
$736$	0	0
$737$	41417.2	2.07004
$738$	0	0
$739$	18704.7	0.931076	0.465538	−	0.885028i	$-0.345861\pi$
$739$	18704.7	0.931076	0.465538	+	0.885028i	$0.345861\pi$
$740$	0	0
$741$	−41227.3	−2.04389
$742$	0	0
$743$	16296.5	0.804655	0.402328	−	0.915496i	$-0.368201\pi$
$743$	16296.5	0.804655	0.402328	+	0.915496i	$0.368201\pi$
$744$	0	0
$745$	−31580.4	−1.55304
$746$	0	0
$747$	3517.03	0.172264
$748$	0	0
$749$	−6662.45	−0.325021
$750$	0	0
$751$	17959.2	0.872625	0.436313	−	0.899795i	$-0.356284\pi$
$751$	17959.2	0.872625	0.436313	+	0.899795i	$0.356284\pi$
$752$	0	0
$753$	−33700.5	−1.63096
$754$	0	0
$755$	59822.3	2.88365
$756$	0	0
$757$	−12110.2	−0.581441	−0.290721	−	0.956808i	$-0.593895\pi$
$757$	−12110.2	−0.581441	−0.290721	+	0.956808i	$0.593895\pi$
$758$	0	0
$759$	−8323.14	−0.398038
$760$	0	0
$761$	1672.00	0.0796452	0.0398226	−	0.999207i	$-0.487321\pi$
$761$	1672.00	0.0796452	0.0398226	+	0.999207i	$0.487321\pi$
$762$	0	0
$763$	7860.31	0.372952
$764$	0	0
$765$	−10792.2	−0.510056
$766$	0	0
$767$	3922.63	0.184665
$768$	0	0
$769$	−1668.39	−0.0782365	−0.0391182	−	0.999235i	$-0.512455\pi$
$769$	−1668.39	−0.0782365	−0.0391182	+	0.999235i	$0.512455\pi$
$770$	0	0
$771$	6383.94	0.298200
$772$	0	0
$773$	19569.3	0.910555	0.455278	−	0.890350i	$-0.349540\pi$
$773$	19569.3	0.910555	0.455278	+	0.890350i	$0.349540\pi$
$774$	0	0
$775$	55526.6	2.57364
$776$	0	0
$777$	4088.33	0.188762
$778$	0	0
$779$	−41183.4	−1.89416
$780$	0	0
$781$	−16137.8	−0.739380
$782$	0	0
$783$	−6202.81	−0.283104
$784$	0	0
$785$	−18847.2	−0.856922
$786$	0	0
$787$	13612.2	0.616548	0.308274	−	0.951298i	$-0.400249\pi$
$787$	13612.2	0.616548	0.308274	+	0.951298i	$0.400249\pi$
$788$	0	0
$789$	−20727.0	−0.935234
$790$	0	0
$791$	−3499.12	−0.157287
$792$	0	0
$793$	51070.2	2.28695
$794$	0	0
$795$	1744.65	0.0778320
$796$	0	0
$797$	13828.6	0.614599	0.307299	−	0.951613i	$-0.400575\pi$
$797$	13828.6	0.614599	0.307299	+	0.951613i	$0.400575\pi$
$798$	0	0
$799$	13414.3	0.593948
$800$	0	0
$801$	4839.66	0.213484
$802$	0	0
$803$	22251.8	0.977896
$804$	0	0
$805$	5805.34	0.254176
$806$	0	0
$807$	30963.2	1.35063
$808$	0	0
$809$	30455.6	1.32356	0.661781	−	0.749697i	$-0.269801\pi$
$809$	30455.6	1.32356	0.661781	+	0.749697i	$0.269801\pi$
$810$	0	0
$811$	−23267.1	−1.00742	−0.503711	−	0.863872i	$-0.668032\pi$
$811$	−23267.1	−1.00742	−0.503711	+	0.863872i	$0.668032\pi$
$812$	0	0
$813$	−35352.0	−1.52503
$814$	0	0
$815$	−11432.6	−0.491368
$816$	0	0
$817$	5074.29	0.217291
$818$	0	0
$819$	−2432.86	−0.103799
$820$	0	0
$821$	−46489.9	−1.97626	−0.988130	−	0.153618i	$-0.950908\pi$
$821$	−46489.9	−1.97626	−0.988130	+	0.153618i	$0.950908\pi$
$822$	0	0
$823$	9843.10	0.416900	0.208450	−	0.978033i	$-0.433158\pi$
$823$	9843.10	0.416900	0.208450	+	0.978033i	$0.433158\pi$
$824$	0	0
$825$	38849.9	1.63949
$826$	0	0
$827$	11930.7	0.501656	0.250828	−	0.968032i	$-0.419297\pi$
$827$	11930.7	0.501656	0.250828	+	0.968032i	$0.419297\pi$
$828$	0	0
$829$	−7023.96	−0.294273	−0.147136	−	0.989116i	$-0.547006\pi$
$829$	−7023.96	−0.294273	−0.147136	+	0.989116i	$0.547006\pi$
$830$	0	0
$831$	−8615.68	−0.359657
$832$	0	0
$833$	5353.94	0.222693
$834$	0	0
$835$	24060.1	0.997168
$836$	0	0
$837$	−39596.6	−1.63520
$838$	0	0
$839$	−18204.6	−0.749097	−0.374548	−	0.927207i	$-0.622202\pi$
$839$	−18204.6	−0.749097	−0.374548	+	0.927207i	$0.622202\pi$
$840$	0	0
$841$	−22691.9	−0.930413
$842$	0	0
$843$	−30644.4	−1.25202
$844$	0	0
$845$	36025.2	1.46663
$846$	0	0
$847$	−1648.41	−0.0668713
$848$	0	0
$849$	64.9309	0.00262476
$850$	0	0
$851$	5682.81	0.228912
$852$	0	0
$853$	36774.6	1.47613	0.738064	−	0.674730i	$-0.235740\pi$
$853$	36774.6	1.47613	0.738064	+	0.674730i	$0.235740\pi$
$854$	0	0
$855$	−13577.4	−0.543084
$856$	0	0
$857$	29302.0	1.16796	0.583978	−	0.811770i	$-0.301496\pi$
$857$	29302.0	1.16796	0.583978	+	0.811770i	$0.301496\pi$
$858$	0	0
$859$	25701.0	1.02085	0.510423	−	0.859923i	$-0.329489\pi$
$859$	25701.0	1.02085	0.510423	+	0.859923i	$0.329489\pi$
$860$	0	0
$861$	9749.69	0.385910
$862$	0	0
$863$	36648.1	1.44556	0.722779	−	0.691079i	$-0.242864\pi$
$863$	36648.1	1.44556	0.722779	+	0.691079i	$0.242864\pi$
$864$	0	0
$865$	29376.0	1.15470
$866$	0	0
$867$	32661.8	1.27942
$868$	0	0
$869$	23806.3	0.929315
$870$	0	0
$871$	−67509.4	−2.62626
$872$	0	0
$873$	3348.64	0.129822
$874$	0	0
$875$	−11055.2	−0.427126
$876$	0	0
$877$	14241.2	0.548338	0.274169	−	0.961681i	$-0.411597\pi$
$877$	14241.2	0.548338	0.274169	+	0.961681i	$0.411597\pi$
$878$	0	0
$879$	28094.4	1.07804
$880$	0	0
$881$	13882.1	0.530873	0.265436	−	0.964128i	$-0.414484\pi$
$881$	13882.1	0.530873	0.265436	+	0.964128i	$0.414484\pi$
$882$	0	0
$883$	19084.7	0.727350	0.363675	−	0.931526i	$-0.381522\pi$
$883$	19084.7	0.727350	0.363675	+	0.931526i	$0.381522\pi$
$884$	0	0
$885$	−5182.57	−0.196848
$886$	0	0
$887$	32467.3	1.22902	0.614512	−	0.788907i	$-0.289353\pi$
$887$	32467.3	1.22902	0.614512	+	0.788907i	$0.289353\pi$
$888$	0	0
$889$	4513.83	0.170292
$890$	0	0
$891$	−21947.3	−0.825209
$892$	0	0
$893$	16876.2	0.632408
$894$	0	0
$895$	−38815.7	−1.44968
$896$	0	0
$897$	13566.6	0.504989
$898$	0	0
$899$	10834.0	0.401929
$900$	0	0
$901$	2236.51	0.0826959
$902$	0	0
$903$	−1201.28	−0.0442704
$904$	0	0
$905$	−77018.1	−2.82891
$906$	0	0
$907$	11625.3	0.425593	0.212796	−	0.977097i	$-0.431743\pi$
$907$	11625.3	0.425593	0.212796	+	0.977097i	$0.431743\pi$
$908$	0	0
$909$	−3054.46	−0.111452
$910$	0	0
$911$	−34260.3	−1.24599	−0.622994	−	0.782226i	$-0.714084\pi$
$911$	−34260.3	−1.24599	−0.622994	+	0.782226i	$0.714084\pi$
$912$	0	0
$913$	−25838.5	−0.936616
$914$	0	0
$915$	−67473.8	−2.43783
$916$	0	0
$917$	12098.6	0.435695
$918$	0	0
$919$	15872.6	0.569738	0.284869	−	0.958566i	$-0.408050\pi$
$919$	15872.6	0.569738	0.284869	+	0.958566i	$0.408050\pi$
$920$	0	0
$921$	−19837.9	−0.709751
$922$	0	0
$923$	26304.4	0.938049
$924$	0	0
$925$	−26525.6	−0.942873
$926$	0	0
$927$	772.859	0.0273830
$928$	0	0
$929$	37178.4	1.31301	0.656503	−	0.754323i	$-0.272035\pi$
$929$	37178.4	1.31301	0.656503	+	0.754323i	$0.272035\pi$
$930$	0	0
$931$	6735.65	0.237113
$932$	0	0
$933$	3864.07	0.135589
$934$	0	0
$935$	79286.9	2.77322
$936$	0	0
$937$	14608.9	0.509340	0.254670	−	0.967028i	$-0.418033\pi$
$937$	14608.9	0.509340	0.254670	+	0.967028i	$0.418033\pi$
$938$	0	0
$939$	28259.1	0.982109
$940$	0	0
$941$	−6991.39	−0.242203	−0.121101	−	0.992640i	$-0.538643\pi$
$941$	−6991.39	−0.242203	−0.121101	+	0.992640i	$0.538643\pi$
$942$	0	0
$943$	13552.1	0.467994
$944$	0	0
$945$	19323.6	0.665182
$946$	0	0
$947$	41762.5	1.43305	0.716525	−	0.697562i	$-0.245732\pi$
$947$	41762.5	1.43305	0.716525	+	0.697562i	$0.245732\pi$
$948$	0	0
$949$	−36270.2	−1.24065
$950$	0	0
$951$	−4380.86	−0.149379
$952$	0	0
$953$	−33451.8	−1.13705	−0.568526	−	0.822665i	$-0.692486\pi$
$953$	−33451.8	−1.13705	−0.568526	+	0.822665i	$0.692486\pi$
$954$	0	0
$955$	22206.0	0.752429
$956$	0	0
$957$	7580.14	0.256041
$958$	0	0
$959$	−8023.66	−0.270175
$960$	0	0
$961$	39369.5	1.32152
$962$	0	0
$963$	−5127.52	−0.171581
$964$	0	0
$965$	−30082.6	−1.00352
$966$	0	0
$967$	−3559.92	−0.118386	−0.0591931	−	0.998247i	$-0.518853\pi$
$967$	−3559.92	−0.118386	−0.0591931	+	0.998247i	$0.518853\pi$
$968$	0	0
$969$	69825.7	2.31489
$970$	0	0
$971$	17768.8	0.587258	0.293629	−	0.955919i	$-0.405137\pi$
$971$	17768.8	0.587258	0.293629	+	0.955919i	$0.405137\pi$
$972$	0	0
$973$	11466.5	0.377800
$974$	0	0
$975$	−63324.8	−2.08002
$976$	0	0
$977$	27096.9	0.887313	0.443657	−	0.896197i	$-0.353681\pi$
$977$	27096.9	0.887313	0.443657	+	0.896197i	$0.353681\pi$
$978$	0	0
$979$	−35555.5	−1.16073
$980$	0	0
$981$	6049.42	0.196884
$982$	0	0
$983$	47107.4	1.52848	0.764239	−	0.644933i	$-0.223115\pi$
$983$	47107.4	1.52848	0.764239	+	0.644933i	$0.223115\pi$
$984$	0	0
$985$	−54008.1	−1.74705
$986$	0	0
$987$	−3995.25	−0.128845
$988$	0	0
$989$	−1669.79	−0.0536867
$990$	0	0
$991$	−6337.58	−0.203148	−0.101574	−	0.994828i	$-0.532388\pi$
$991$	−6337.58	−0.203148	−0.101574	+	0.994828i	$0.532388\pi$
$992$	0	0
$993$	−5592.96	−0.178738
$994$	0	0
$995$	−31090.6	−0.990590
$996$	0	0
$997$	−33027.7	−1.04915	−0.524573	−	0.851365i	$-0.675775\pi$
$997$	−33027.7	−1.04915	−0.524573	+	0.851365i	$0.675775\pi$
$998$	0	0
$999$	18915.7	0.599066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.x.1.2		3
4.3	odd	2		448.4.a.u.1.2		3
8.3	odd	2		224.4.a.h.1.2	yes	3
8.5	even	2		224.4.a.e.1.2	✓	3
24.5	odd	2		2016.4.a.s.1.3		3
24.11	even	2		2016.4.a.t.1.3		3
56.13	odd	2		1568.4.a.y.1.2		3
56.27	even	2		1568.4.a.v.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.e.1.2	✓	3	8.5	even	2
224.4.a.h.1.2	yes	3	8.3	odd	2
448.4.a.u.1.2		3	4.3	odd	2
448.4.a.x.1.2		3	1.1	even	1	trivial
1568.4.a.v.1.2		3	56.27	even	2
1568.4.a.y.1.2		3	56.13	odd	2
2016.4.a.s.1.3		3	24.5	odd	2
2016.4.a.t.1.3		3	24.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+4.64895 q^{3} +18.3341 q^{5} -7.00000 q^{7} -5.38731 q^{9} +39.5789 q^{11} -64.5130 q^{13} +85.2344 q^{15} +109.264 q^{17} +137.462 q^{19} -32.5426 q^{21} -45.2344 q^{23} +211.141 q^{25} -150.567 q^{27} +41.1964 q^{29} +262.984 q^{31} +184.000 q^{33} -128.339 q^{35} -125.630 q^{37} -299.917 q^{39} -299.598 q^{41} +36.9141 q^{43} -98.7717 q^{45} +122.770 q^{47} +49.0000 q^{49} +507.963 q^{51} +20.4689 q^{53} +725.645 q^{55} +639.054 q^{57} -60.8037 q^{59} -791.626 q^{61} +37.7112 q^{63} -1182.79 q^{65} +1046.45 q^{67} -210.292 q^{69} -407.738 q^{71} +562.215 q^{73} +981.582 q^{75} -277.052 q^{77} +601.491 q^{79} -554.520 q^{81} -652.836 q^{83} +2003.26 q^{85} +191.520 q^{87} -898.344 q^{89} +451.591 q^{91} +1222.60 q^{93} +2520.25 q^{95} -621.580 q^{97} -213.224 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{3} - 10 q^{5} - 21 q^{7} + 31 q^{9} + 24 q^{11} - 58 q^{13} + 8 q^{15} + 174 q^{17} + 64 q^{19} - 56 q^{21} + 112 q^{23} + 373 q^{25} + 104 q^{27} - 314 q^{29} + 280 q^{31} + 552 q^{33} + 70 q^{35}+ \cdots + 824 q^{99}+O(q^{100}) \) 3 * q + 8 * q^3 - 10 * q^5 - 21 * q^7 + 31 * q^9 + 24 * q^11 - 58 * q^13 + 8 * q^15 + 174 * q^17 + 64 * q^19 - 56 * q^21 + 112 * q^23 + 373 * q^25 + 104 * q^27 - 314 * q^29 + 280 * q^31 + 552 * q^33 + 70 * q^35 - 178 * q^37 + 648 * q^39 + 318 * q^41 + 632 * q^43 - 714 * q^45 + 664 * q^47 + 147 * q^49 + 896 * q^51 - 434 * q^53 + 808 * q^55 - 256 * q^57 + 816 * q^59 - 450 * q^61 - 217 * q^63 - 1604 * q^65 + 408 * q^67 + 1304 * q^69 - 184 * q^71 + 6 * q^73 + 2096 * q^75 - 168 * q^77 + 552 * q^79 - 1381 * q^81 - 1736 * q^83 + 988 * q^85 + 96 * q^87 - 218 * q^89 + 406 * q^91 + 2768 * q^93 + 3832 * q^95 - 1090 * q^97 + 824 * q^99

Introduction

L-functions

Modular forms

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