LMFDB - Embedded newform 448.4.a.u.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:	$1$
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)

$f(q)$	$=$	$q-4.64895 q^{3} +18.3341 q^{5} +7.00000 q^{7} -5.38731 q^{9} +O(q^{10})$	$q-4.64895 q^{3} +18.3341 q^{5} +7.00000 q^{7} -5.38731 q^{9} -39.5789 q^{11} -64.5130 q^{13} -85.2344 q^{15} +109.264 q^{17} -137.462 q^{19} -32.5426 q^{21} +45.2344 q^{23} +211.141 q^{25} +150.567 q^{27} +41.1964 q^{29} -262.984 q^{31} +184.000 q^{33} +128.339 q^{35} -125.630 q^{37} +299.917 q^{39} -299.598 q^{41} -36.9141 q^{43} -98.7717 q^{45} -122.770 q^{47} +49.0000 q^{49} -507.963 q^{51} +20.4689 q^{53} -725.645 q^{55} +639.054 q^{57} +60.8037 q^{59} -791.626 q^{61} -37.7112 q^{63} -1182.79 q^{65} -1046.45 q^{67} -210.292 q^{69} +407.738 q^{71} +562.215 q^{73} -981.582 q^{75} -277.052 q^{77} -601.491 q^{79} -554.520 q^{81} +652.836 q^{83} +2003.26 q^{85} -191.520 q^{87} -898.344 q^{89} -451.591 q^{91} +1222.60 q^{93} -2520.25 q^{95} -621.580 q^{97} +213.224 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{3} - 10 q^{5} + 21 q^{7} + 31 q^{9}+O(q^{10}) $ 3 * q - 8 * q^3 - 10 * q^5 + 21 * q^7 + 31 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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	$-0.647630\pi$
$3$	−4.64895	−0.894690	−0.447345	+	0.894361i	$0.647630\pi$
$4$	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	$-0.682496\pi$
$11$	−39.5789	−1.08486	−0.542431	+	0.840100i	$0.682496\pi$
$12$	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	$-0.811600\pi$
$19$	−137.462	−1.65979	−0.829895	+	0.557920i	$0.811600\pi$
$20$	0	0
$21$	−32.5426	−0.338161
$22$	0	0
$23$	45.2344	0.410088	0.205044	−	0.978753i	$-0.434266\pi$
$23$	45.2344	0.410088	0.205044	+	0.978753i	$0.434266\pi$
$24$	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	$-0.775697\pi$
$31$	−262.984	−1.52365	−0.761827	+	0.647780i	$0.775697\pi$
$32$	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	$-0.520851\pi$
$43$	−36.9141	−0.130915	−0.0654575	+	0.997855i	$0.520851\pi$
$44$	0	0
$45$	−98.7717	−0.327200
$46$	0	0
$47$	−122.770	−0.381017	−0.190509	−	0.981686i	$-0.561014\pi$
$47$	−122.770	−0.381017	−0.190509	+	0.981686i	$0.561014\pi$
$48$	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	$-0.478630\pi$
$59$	60.8037	0.134169	0.0670845	+	0.997747i	$0.478630\pi$
$60$	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	$-0.903140\pi$
$67$	−1046.45	−1.90812	−0.954058	+	0.299621i	$0.903140\pi$
$68$	0	0
$69$	−210.292	−0.366902
$70$	0	0
$71$	407.738	0.681543	0.340772	−	0.940146i	$-0.389312\pi$
$71$	407.738	0.681543	0.340772	+	0.940146i	$0.389312\pi$
$72$	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	$-0.640891\pi$
$79$	−601.491	−0.856621	−0.428311	+	0.903632i	$0.640891\pi$
$80$	0	0
$81$	−554.520	−0.760658
$82$	0	0
$83$	652.836	0.863351	0.431675	−	0.902029i	$-0.357923\pi$
$83$	652.836	0.863351	0.431675	+	0.902029i	$0.357923\pi$
$84$	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	$-0.478141\pi$
$103$	143.459	0.137238	0.0686188	+	0.997643i	$0.478141\pi$
$104$	0	0
$105$	−596.641	−0.554535
$106$	0	0
$107$	−951.778	−0.859924	−0.429962	−	0.902847i	$-0.641473\pi$
$107$	−951.778	−0.859924	−0.429962	+	0.902847i	$0.641473\pi$
$108$	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	$-0.427672\pi$
$127$	644.834	0.450549	0.225275	+	0.974295i	$0.427672\pi$
$128$	0	0
$129$	171.612	0.117128
$130$	0	0
$131$	1728.38	1.15274	0.576370	−	0.817189i	$-0.304469\pi$
$131$	1728.38	1.15274	0.576370	+	0.817189i	$0.304469\pi$
$132$	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	$-0.333413\pi$
$139$	1638.07	0.999564	0.499782	+	0.866151i	$0.333413\pi$
$140$	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	$-0.841949\pi$
$151$	−3262.89	−1.75848	−0.879240	+	0.476380i	$0.841949\pi$
$152$	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	$-0.452130\pi$
$163$	623.566	0.299641	0.149820	+	0.988713i	$0.452130\pi$
$164$	0	0
$165$	3373.48	1.59167
$166$	0	0
$167$	−1312.31	−0.608083	−0.304041	−	0.952659i	$-0.598336\pi$
$167$	−1312.31	−0.608083	−0.304041	+	0.952659i	$0.598336\pi$
$168$	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	$-0.354264\pi$
$179$	2117.12	0.884030	0.442015	+	0.897008i	$0.354264\pi$
$180$	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	$-0.573683\pi$
$191$	−1211.18	−0.458838	−0.229419	+	0.973328i	$0.573683\pi$
$192$	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	$-0.402334\pi$
$199$	1695.77	0.604071	0.302036	+	0.953297i	$0.402334\pi$
$200$	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	$-0.204489\pi$
$211$	4907.89	1.60129	0.800647	+	0.599136i	$0.204489\pi$
$212$	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	$-0.767985\pi$
$223$	−4967.91	−1.49182	−0.745910	+	0.666046i	$0.767985\pi$
$224$	0	0
$225$	−1137.48	−0.337031
$226$	0	0
$227$	4980.74	1.45632	0.728158	−	0.685410i	$-0.240377\pi$
$227$	4980.74	1.45632	0.728158	+	0.685410i	$0.240377\pi$
$228$	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	$-0.306207\pi$
$239$	4226.15	1.14380	0.571898	+	0.820325i	$0.306207\pi$
$240$	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	$-0.134949\pi$
$251$	7249.07	1.82294	0.911469	+	0.411370i	$0.134949\pi$
$252$	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	$-0.324940\pi$
$263$	4458.42	1.04532	0.522658	+	0.852542i	$0.324940\pi$
$264$	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	$-0.175227\pi$
$271$	7604.31	1.70453	0.852267	+	0.523107i	$0.175227\pi$
$272$	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	$-0.500467\pi$
$283$	−13.9668	−0.00293371	−0.00146686	+	0.999999i	$0.500467\pi$
$284$	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	$-0.370174\pi$
$307$	4267.18	0.793293	0.396647	+	0.917971i	$0.370174\pi$
$308$	0	0
$309$	−666.934	−0.122785
$310$	0	0
$311$	−831.172	−0.151548	−0.0757740	−	0.997125i	$-0.524143\pi$
$311$	−831.172	−0.151548	−0.0757740	+	0.997125i	$0.524143\pi$
$312$	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	$-0.468151\pi$
$331$	1203.06	0.199777	0.0998884	+	0.994999i	$0.468151\pi$
$332$	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	$-0.791429\pi$
$347$	−10250.4	−1.58580	−0.792898	+	0.609354i	$0.791429\pi$
$348$	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	$-0.643907\pi$
$359$	−5943.02	−0.873707	−0.436853	+	0.899533i	$0.643907\pi$
$360$	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	$-0.477823\pi$
$367$	978.903	0.139232	0.0696162	+	0.997574i	$0.477823\pi$
$368$	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	$-0.554802\pi$
$379$	−2528.04	−0.342630	−0.171315	+	0.985216i	$0.554802\pi$
$380$	0	0
$381$	−2997.80	−0.403102
$382$	0	0
$383$	−9220.11	−1.23009	−0.615047	−	0.788491i	$-0.710863\pi$
$383$	−9220.11	−1.23009	−0.615047	+	0.788491i	$0.710863\pi$
$384$	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	$-0.709828\pi$
$419$	−10506.1	−1.22496	−0.612479	+	0.790487i	$0.709828\pi$
$420$	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	$-0.554839\pi$
$431$	−3067.87	−0.342864	−0.171432	+	0.985196i	$0.554839\pi$
$432$	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	$-0.624273\pi$
$439$	−7001.06	−0.761145	−0.380572	+	0.924751i	$0.624273\pi$
$440$	0	0
$441$	−263.978	−0.0285043
$442$	0	0
$443$	9583.07	1.02778	0.513889	−	0.857857i	$-0.328204\pi$
$443$	9583.07	1.02778	0.513889	+	0.857857i	$0.328204\pi$
$444$	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	$-0.581723\pi$
$463$	−5059.57	−0.507858	−0.253929	+	0.967223i	$0.581723\pi$
$464$	0	0
$465$	22415.3	2.23545
$466$	0	0
$467$	−5316.15	−0.526771	−0.263386	−	0.964691i	$-0.584839\pi$
$467$	−5316.15	−0.526771	−0.263386	+	0.964691i	$0.584839\pi$
$468$	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	$-0.343785\pi$
$479$	9881.68	0.942600	0.471300	+	0.881973i	$0.343785\pi$
$480$	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	$-0.409515\pi$
$487$	6028.14	0.560905	0.280453	+	0.959868i	$0.409515\pi$
$488$	0	0
$489$	−2898.92	−0.268086
$490$	0	0
$491$	−2441.52	−0.224408	−0.112204	−	0.993685i	$-0.535791\pi$
$491$	−2441.52	−0.224408	−0.112204	+	0.993685i	$0.535791\pi$
$492$	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	$-0.557136\pi$
$499$	−3980.18	−0.357069	−0.178534	+	0.983934i	$0.557136\pi$
$500$	0	0
$501$	6100.87	0.544046
$502$	0	0
$503$	−4203.45	−0.372610	−0.186305	−	0.982492i	$-0.559651\pi$
$503$	−4203.45	−0.372610	−0.186305	+	0.982492i	$0.559651\pi$
$504$	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	$-0.211616\pi$
$523$	18826.8	1.57407	0.787033	+	0.616911i	$0.211616\pi$
$524$	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	$-0.254977\pi$
$547$	17807.3	1.39193	0.695966	+	0.718075i	$0.254977\pi$
$548$	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	$-0.764670\pi$
$563$	−19742.3	−1.47787	−0.738934	+	0.673778i	$0.764670\pi$
$564$	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	$-0.639552\pi$
$571$	−11584.3	−0.849012	−0.424506	+	0.905425i	$0.639552\pi$
$572$	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	$-0.206279\pi$
$587$	22677.2	1.59453	0.797265	+	0.603629i	$0.206279\pi$
$588$	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	$-0.410585\pi$
$599$	8128.35	0.554450	0.277225	+	0.960805i	$0.410585\pi$
$600$	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	$-0.366817\pi$
$607$	12152.5	0.812608	0.406304	+	0.913738i	$0.366817\pi$
$608$	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	$-0.515250\pi$
$619$	−1475.11	−0.0957832	−0.0478916	+	0.998853i	$0.515250\pi$
$620$	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	$-0.543371\pi$
$631$	−4306.01	−0.271664	−0.135832	+	0.990732i	$0.543371\pi$
$632$	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	$-0.619725\pi$
$643$	−11978.2	−0.734643	−0.367322	+	0.930094i	$0.619725\pi$
$644$	0	0
$645$	3146.35	0.192074
$646$	0	0
$647$	−24522.4	−1.49007	−0.745035	−	0.667026i	$-0.767567\pi$
$647$	−24522.4	−1.49007	−0.745035	+	0.667026i	$0.767567\pi$
$648$	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	$-0.456398\pi$
$659$	4620.17	0.273105	0.136553	+	0.990633i	$0.456398\pi$
$660$	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	$-0.500640\pi$
$683$	−71.8070	−0.00402287	−0.00201143	+	0.999998i	$0.500640\pi$
$684$	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	$-0.510064\pi$
$691$	−1148.37	−0.0632212	−0.0316106	+	0.999500i	$0.510064\pi$
$692$	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	$-0.658984\pi$
$719$	−18467.9	−0.957910	−0.478955	+	0.877839i	$0.658984\pi$
$720$	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	$-0.600555\pi$
$727$	−12179.8	−0.621352	−0.310676	+	0.950516i	$0.600555\pi$
$728$	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	$-0.654139\pi$
$739$	−18704.7	−0.931076	−0.465538	+	0.885028i	$0.654139\pi$
$740$	0	0
$741$	−41227.3	−2.04389
$742$	0	0
$743$	−16296.5	−0.804655	−0.402328	−	0.915496i	$-0.631799\pi$
$743$	−16296.5	−0.804655	−0.402328	+	0.915496i	$0.631799\pi$
$744$	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	$-0.643716\pi$
$751$	−17959.2	−0.872625	−0.436313	+	0.899795i	$0.643716\pi$
$752$	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	$-0.599751\pi$
$787$	−13612.2	−0.616548	−0.308274	+	0.951298i	$0.599751\pi$
$788$	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	$-0.331968\pi$
$811$	23267.1	1.00742	0.503711	+	0.863872i	$0.331968\pi$
$812$	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	$-0.566842\pi$
$823$	−9843.10	−0.416900	−0.208450	+	0.978033i	$0.566842\pi$
$824$	0	0
$825$	38849.9	1.63949
$826$	0	0
$827$	−11930.7	−0.501656	−0.250828	−	0.968032i	$-0.580703\pi$
$827$	−11930.7	−0.501656	−0.250828	+	0.968032i	$0.580703\pi$
$828$	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	$-0.377798\pi$
$839$	18204.6	0.749097	0.374548	+	0.927207i	$0.377798\pi$
$840$	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	$-0.670511\pi$
$859$	−25701.0	−1.02085	−0.510423	+	0.859923i	$0.670511\pi$
$860$	0	0
$861$	9749.69	0.385910
$862$	0	0
$863$	−36648.1	−1.44556	−0.722779	−	0.691079i	$-0.757136\pi$
$863$	−36648.1	−1.44556	−0.722779	+	0.691079i	$0.757136\pi$
$864$	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	$-0.618478\pi$
$883$	−19084.7	−0.727350	−0.363675	+	0.931526i	$0.618478\pi$
$884$	0	0
$885$	−5182.57	−0.196848
$886$	0	0
$887$	−32467.3	−1.22902	−0.614512	−	0.788907i	$-0.710647\pi$
$887$	−32467.3	−1.22902	−0.614512	+	0.788907i	$0.710647\pi$
$888$	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	$-0.568257\pi$
$907$	−11625.3	−0.425593	−0.212796	+	0.977097i	$0.568257\pi$
$908$	0	0
$909$	−3054.46	−0.111452
$910$	0	0
$911$	34260.3	1.24599	0.622994	−	0.782226i	$-0.285916\pi$
$911$	34260.3	1.24599	0.622994	+	0.782226i	$0.285916\pi$
$912$	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	$-0.591950\pi$
$919$	−15872.6	−0.569738	−0.284869	+	0.958566i	$0.591950\pi$
$920$	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	$-0.754268\pi$
$947$	−41762.5	−1.43305	−0.716525	+	0.697562i	$0.754268\pi$
$948$	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	$-0.481147\pi$
$967$	3559.92	0.118386	0.0591931	+	0.998247i	$0.481147\pi$
$968$	0	0
$969$	69825.7	2.31489
$970$	0	0
$971$	−17768.8	−0.587258	−0.293629	−	0.955919i	$-0.594863\pi$
$971$	−17768.8	−0.587258	−0.293629	+	0.955919i	$0.594863\pi$
$972$	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	$-0.776885\pi$
$983$	−47107.4	−1.52848	−0.764239	+	0.644933i	$0.776885\pi$
$984$	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	$-0.467612\pi$
$991$	6337.58	0.203148	0.101574	+	0.994828i	$0.467612\pi$
$992$	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.u.1.2		3
4.3	odd	2		448.4.a.x.1.2		3
8.3	odd	2		224.4.a.e.1.2	✓	3
8.5	even	2		224.4.a.h.1.2	yes	3
24.5	odd	2		2016.4.a.t.1.3		3
24.11	even	2		2016.4.a.s.1.3		3
56.13	odd	2		1568.4.a.v.1.2		3
56.27	even	2		1568.4.a.y.1.2		3

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

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

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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more