LMFDB - Embedded newform 448.4.a.q.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54138$ 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)

$f(q)$	$=$	$q-9.08276 q^{3} -3.08276 q^{5} -7.00000 q^{7} +55.4966 q^{9} +O(q^{10})$	$q-9.08276 q^{3} -3.08276 q^{5} -7.00000 q^{7} +55.4966 q^{9} -9.83447 q^{11} -0.917237 q^{13} +28.0000 q^{15} +108.828 q^{17} -30.2551 q^{19} +63.5793 q^{21} +173.324 q^{23} -115.497 q^{25} -258.828 q^{27} +270.814 q^{29} -124.152 q^{31} +89.3242 q^{33} +21.5793 q^{35} +140.166 q^{37} +8.33105 q^{39} -308.800 q^{41} -471.490 q^{43} -171.083 q^{45} -255.821 q^{47} +49.0000 q^{49} -988.455 q^{51} -483.297 q^{53} +30.3174 q^{55} +274.800 q^{57} +131.731 q^{59} +124.255 q^{61} -388.476 q^{63} +2.82763 q^{65} +775.918 q^{67} -1574.26 q^{69} -529.683 q^{71} -1019.60 q^{73} +1049.03 q^{75} +68.8413 q^{77} +1097.57 q^{79} +852.462 q^{81} -1276.16 q^{83} -335.490 q^{85} -2459.74 q^{87} -940.621 q^{89} +6.42066 q^{91} +1127.64 q^{93} +93.2694 q^{95} +528.855 q^{97} -545.780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{5} - 14 q^{7} + 38 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 6 * q^5 - 14 * q^7 + 38 * q^9	$ 2 q - 6 q^{3} + 6 q^{5} - 14 q^{7} + 38 q^{9} - 44 q^{11} - 14 q^{13} + 56 q^{15} + 96 q^{17} - 170 q^{19} + 42 q^{21} + 152 q^{23} - 158 q^{25} - 396 q^{27} + 128 q^{29} + 68 q^{31} - 16 q^{33} - 42 q^{35} + 256 q^{37} - 32 q^{39} + 88 q^{41} - 724 q^{43} - 330 q^{45} - 244 q^{47} + 98 q^{49} - 1028 q^{51} - 188 q^{53} - 280 q^{55} - 156 q^{57} - 138 q^{59} + 358 q^{61} - 266 q^{63} - 116 q^{65} - 200 q^{67} - 1640 q^{69} - 1400 q^{71} - 628 q^{73} + 918 q^{75} + 308 q^{77} + 200 q^{79} + 902 q^{81} - 618 q^{83} - 452 q^{85} - 2900 q^{87} - 908 q^{89} + 98 q^{91} + 1720 q^{93} - 1176 q^{95} + 1520 q^{97} + 52 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^5 - 14 * q^7 + 38 * q^9 - 44 * q^11 - 14 * q^13 + 56 * q^15 + 96 * q^17 - 170 * q^19 + 42 * q^21 + 152 * q^23 - 158 * q^25 - 396 * q^27 + 128 * q^29 + 68 * q^31 - 16 * q^33 - 42 * q^35 + 256 * q^37 - 32 * q^39 + 88 * q^41 - 724 * q^43 - 330 * q^45 - 244 * q^47 + 98 * q^49 - 1028 * q^51 - 188 * q^53 - 280 * q^55 - 156 * q^57 - 138 * q^59 + 358 * q^61 - 266 * q^63 - 116 * q^65 - 200 * q^67 - 1640 * q^69 - 1400 * q^71 - 628 * q^73 + 918 * q^75 + 308 * q^77 + 200 * q^79 + 902 * q^81 - 618 * q^83 - 452 * q^85 - 2900 * q^87 - 908 * q^89 + 98 * q^91 + 1720 * q^93 - 1176 * q^95 + 1520 * q^97 + 52 * 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
$3$	−9.08276	−1.74798	−0.873989	−	0.485945i	$-0.838475\pi$
$3$	−9.08276	−1.74798	−0.873989	+	0.485945i	$0.838475\pi$
$4$	0	0
$5$	−3.08276	−0.275731	−0.137865	−	0.990451i	$-0.544024\pi$
$5$	−3.08276	−0.275731	−0.137865	+	0.990451i	$0.544024\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	55.4966	2.05543
$10$	0	0
$11$	−9.83447	−0.269564	−0.134782	−	0.990875i	$-0.543033\pi$
$11$	−9.83447	−0.269564	−0.134782	+	0.990875i	$0.543033\pi$
$12$	0	0
$13$	−0.917237	−0.0195689	−0.00978446	−	0.999952i	$-0.503115\pi$
$13$	−0.917237	−0.0195689	−0.00978446	+	0.999952i	$0.503115\pi$
$14$	0	0
$15$	28.0000	0.481971
$16$	0	0
$17$	108.828	1.55262	0.776311	−	0.630350i	$-0.217089\pi$
$17$	108.828	1.55262	0.776311	+	0.630350i	$0.217089\pi$
$18$	0	0
$19$	−30.2551	−0.365316	−0.182658	−	0.983177i	$-0.558470\pi$
$19$	−30.2551	−0.365316	−0.182658	+	0.983177i	$0.558470\pi$
$20$	0	0
$21$	63.5793	0.660674
$22$	0	0
$23$	173.324	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$23$	173.324	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$24$	0	0
$25$	−115.497	−0.923973
$26$	0	0
$27$	−258.828	−1.84487
$28$	0	0
$29$	270.814	1.73410	0.867050	−	0.498222i	$-0.166013\pi$
$29$	270.814	1.73410	0.867050	+	0.498222i	$0.166013\pi$
$30$	0	0
$31$	−124.152	−0.719301	−0.359650	−	0.933087i	$-0.617104\pi$
$31$	−124.152	−0.719301	−0.359650	+	0.933087i	$0.617104\pi$
$32$	0	0
$33$	89.3242	0.471192
$34$	0	0
$35$	21.5793	0.104216
$36$	0	0
$37$	140.166	0.622786	0.311393	−	0.950281i	$-0.399205\pi$
$37$	140.166	0.622786	0.311393	+	0.950281i	$0.399205\pi$
$38$	0	0
$39$	8.33105	0.0342060
$40$	0	0
$41$	−308.800	−1.17626	−0.588128	−	0.808768i	$-0.700135\pi$
$41$	−308.800	−1.17626	−0.588128	+	0.808768i	$0.700135\pi$
$42$	0	0
$43$	−471.490	−1.67213	−0.836064	−	0.548632i	$-0.815149\pi$
$43$	−471.490	−1.67213	−0.836064	+	0.548632i	$0.815149\pi$
$44$	0	0
$45$	−171.083	−0.566745
$46$	0	0
$47$	−255.821	−0.793942	−0.396971	−	0.917831i	$-0.629939\pi$
$47$	−255.821	−0.793942	−0.396971	+	0.917831i	$0.629939\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−988.455	−2.71395
$52$	0	0
$53$	−483.297	−1.25256	−0.626282	−	0.779596i	$-0.715424\pi$
$53$	−483.297	−1.25256	−0.626282	+	0.779596i	$0.715424\pi$
$54$	0	0
$55$	30.3174	0.0743271
$56$	0	0
$57$	274.800	0.638565
$58$	0	0
$59$	131.731	0.290677	0.145338	−	0.989382i	$-0.453573\pi$
$59$	131.731	0.290677	0.145338	+	0.989382i	$0.453573\pi$
$60$	0	0
$61$	124.255	0.260807	0.130404	−	0.991461i	$-0.458373\pi$
$61$	124.255	0.260807	0.130404	+	0.991461i	$0.458373\pi$
$62$	0	0
$63$	−388.476	−0.776879
$64$	0	0
$65$	2.82763	0.00539575
$66$	0	0
$67$	775.918	1.41483	0.707414	−	0.706800i	$-0.249862\pi$
$67$	775.918	1.41483	0.707414	+	0.706800i	$0.249862\pi$
$68$	0	0
$69$	−1574.26	−2.74665
$70$	0	0
$71$	−529.683	−0.885377	−0.442688	−	0.896675i	$-0.645975\pi$
$71$	−529.683	−0.885377	−0.442688	+	0.896675i	$0.645975\pi$
$72$	0	0
$73$	−1019.60	−1.63473	−0.817364	−	0.576121i	$-0.804566\pi$
$73$	−1019.60	−1.63473	−0.817364	+	0.576121i	$0.804566\pi$
$74$	0	0
$75$	1049.03	1.61508
$76$	0	0
$77$	68.8413	0.101886
$78$	0	0
$79$	1097.57	1.56312	0.781561	−	0.623829i	$-0.214424\pi$
$79$	1097.57	1.56312	0.781561	+	0.623829i	$0.214424\pi$
$80$	0	0
$81$	852.462	1.16936
$82$	0	0
$83$	−1276.16	−1.68767	−0.843835	−	0.536602i	$-0.819708\pi$
$83$	−1276.16	−1.68767	−0.843835	+	0.536602i	$0.819708\pi$
$84$	0	0
$85$	−335.490	−0.428106
$86$	0	0
$87$	−2459.74	−3.03117
$88$	0	0
$89$	−940.621	−1.12029	−0.560144	−	0.828395i	$-0.689254\pi$
$89$	−940.621	−1.12029	−0.560144	+	0.828395i	$0.689254\pi$
$90$	0	0
$91$	6.42066	0.00739635
$92$	0	0
$93$	1127.64	1.25732
$94$	0	0
$95$	93.2694	0.100729
$96$	0	0
$97$	528.855	0.553578	0.276789	−	0.960931i	$-0.410730\pi$
$97$	528.855	0.553578	0.276789	+	0.960931i	$0.410730\pi$
$98$	0	0
$99$	−545.780	−0.554070
$100$	0	0
$101$	995.180	0.980437	0.490218	−	0.871600i	$-0.336917\pi$
$101$	995.180	0.980437	0.490218	+	0.871600i	$0.336917\pi$
$102$	0	0
$103$	−782.497	−0.748560	−0.374280	−	0.927316i	$-0.622110\pi$
$103$	−782.497	−0.748560	−0.374280	+	0.927316i	$0.622110\pi$
$104$	0	0
$105$	−196.000	−0.182168
$106$	0	0
$107$	227.696	0.205722	0.102861	−	0.994696i	$-0.467200\pi$
$107$	227.696	0.205722	0.102861	+	0.994696i	$0.467200\pi$
$108$	0	0
$109$	895.725	0.787109	0.393555	−	0.919301i	$-0.371245\pi$
$109$	895.725	0.787109	0.393555	+	0.919301i	$0.371245\pi$
$110$	0	0
$111$	−1273.09	−1.08862
$112$	0	0
$113$	−1354.18	−1.12735	−0.563675	−	0.825997i	$-0.690613\pi$
$113$	−1354.18	−1.12735	−0.563675	+	0.825997i	$0.690613\pi$
$114$	0	0
$115$	−534.317	−0.433264
$116$	0	0
$117$	−50.9035	−0.0402225
$118$	0	0
$119$	−761.793	−0.586836
$120$	0	0
$121$	−1234.28	−0.927335
$122$	0	0
$123$	2804.76	2.05607
$124$	0	0
$125$	741.394	0.530498
$126$	0	0
$127$	67.0890	0.0468755	0.0234378	−	0.999725i	$-0.492539\pi$
$127$	67.0890	0.0468755	0.0234378	+	0.999725i	$0.492539\pi$
$128$	0	0
$129$	4282.43	2.92284
$130$	0	0
$131$	1469.97	0.980394	0.490197	−	0.871612i	$-0.336925\pi$
$131$	1469.97	0.980394	0.490197	+	0.871612i	$0.336925\pi$
$132$	0	0
$133$	211.786	0.138076
$134$	0	0
$135$	797.904	0.508686
$136$	0	0
$137$	1221.06	0.761477	0.380738	−	0.924683i	$-0.375670\pi$
$137$	1221.06	0.761477	0.380738	+	0.924683i	$0.375670\pi$
$138$	0	0
$139$	−1059.68	−0.646623	−0.323311	−	0.946293i	$-0.604796\pi$
$139$	−1059.68	−0.646623	−0.323311	+	0.946293i	$0.604796\pi$
$140$	0	0
$141$	2323.56	1.38779
$142$	0	0
$143$	9.02055	0.00527508
$144$	0	0
$145$	−834.855	−0.478144
$146$	0	0
$147$	−445.055	−0.249711
$148$	0	0
$149$	−315.546	−0.173493	−0.0867467	−	0.996230i	$-0.527647\pi$
$149$	−315.546	−0.173493	−0.0867467	+	0.996230i	$0.527647\pi$
$150$	0	0
$151$	3471.56	1.87094	0.935469	−	0.353410i	$-0.114978\pi$
$151$	3471.56	1.87094	0.935469	+	0.353410i	$0.114978\pi$
$152$	0	0
$153$	6039.56	3.19130
$154$	0	0
$155$	382.731	0.198333
$156$	0	0
$157$	−3232.70	−1.64330	−0.821648	−	0.569996i	$-0.806945\pi$
$157$	−3232.70	−1.64330	−0.821648	+	0.569996i	$0.806945\pi$
$158$	0	0
$159$	4389.67	2.18946
$160$	0	0
$161$	−1213.27	−0.593907
$162$	0	0
$163$	325.035	0.156189	0.0780943	−	0.996946i	$-0.475116\pi$
$163$	325.035	0.156189	0.0780943	+	0.996946i	$0.475116\pi$
$164$	0	0
$165$	−275.365	−0.129922
$166$	0	0
$167$	−1438.08	−0.666361	−0.333180	−	0.942863i	$-0.608122\pi$
$167$	−1438.08	−0.666361	−0.333180	+	0.942863i	$0.608122\pi$
$168$	0	0
$169$	−2196.16	−0.999617
$170$	0	0
$171$	−1679.06	−0.750881
$172$	0	0
$173$	−2366.52	−1.04002	−0.520009	−	0.854161i	$-0.674071\pi$
$173$	−2366.52	−1.04002	−0.520009	+	0.854161i	$0.674071\pi$
$174$	0	0
$175$	808.476	0.349229
$176$	0	0
$177$	−1196.48	−0.508097
$178$	0	0
$179$	1990.13	0.831000	0.415500	−	0.909593i	$-0.363607\pi$
$179$	1990.13	0.831000	0.415500	+	0.909593i	$0.363607\pi$
$180$	0	0
$181$	−1342.07	−0.551136	−0.275568	−	0.961282i	$-0.588866\pi$
$181$	−1342.07	−0.551136	−0.275568	+	0.961282i	$0.588866\pi$
$182$	0	0
$183$	−1128.58	−0.455885
$184$	0	0
$185$	−432.097	−0.171721
$186$	0	0
$187$	−1070.26	−0.418531
$188$	0	0
$189$	1811.79	0.697294
$190$	0	0
$191$	−1980.66	−0.750345	−0.375172	−	0.926955i	$-0.622416\pi$
$191$	−1980.66	−0.750345	−0.375172	+	0.926955i	$0.622416\pi$
$192$	0	0
$193$	−15.0628	−0.00561784	−0.00280892	−	0.999996i	$-0.500894\pi$
$193$	−15.0628	−0.00561784	−0.00280892	+	0.999996i	$0.500894\pi$
$194$	0	0
$195$	−25.6826	−0.00943165
$196$	0	0
$197$	−2120.73	−0.766985	−0.383492	−	0.923544i	$-0.625279\pi$
$197$	−2120.73	−0.766985	−0.383492	+	0.923544i	$0.625279\pi$
$198$	0	0
$199$	3173.83	1.13059	0.565294	−	0.824889i	$-0.308763\pi$
$199$	3173.83	1.13059	0.565294	+	0.824889i	$0.308763\pi$
$200$	0	0
$201$	−7047.48	−2.47309
$202$	0	0
$203$	−1895.70	−0.655428
$204$	0	0
$205$	951.958	0.324330
$206$	0	0
$207$	9618.90	3.22976
$208$	0	0
$209$	297.543	0.0984761
$210$	0	0
$211$	−3465.30	−1.13062	−0.565310	−	0.824878i	$-0.691243\pi$
$211$	−3465.30	−1.13062	−0.565310	+	0.824878i	$0.691243\pi$
$212$	0	0
$213$	4810.98	1.54762
$214$	0	0
$215$	1453.49	0.461057
$216$	0	0
$217$	869.063	0.271870
$218$	0	0
$219$	9260.79	2.85747
$220$	0	0
$221$	−99.8208	−0.0303831
$222$	0	0
$223$	951.253	0.285653	0.142827	−	0.989748i	$-0.454381\pi$
$223$	951.253	0.285653	0.142827	+	0.989748i	$0.454381\pi$
$224$	0	0
$225$	−6409.66	−1.89916
$226$	0	0
$227$	−4404.77	−1.28791	−0.643953	−	0.765065i	$-0.722707\pi$
$227$	−4404.77	−1.28791	−0.643953	+	0.765065i	$0.722707\pi$
$228$	0	0
$229$	653.345	0.188534	0.0942670	−	0.995547i	$-0.469949\pi$
$229$	653.345	0.188534	0.0942670	+	0.995547i	$0.469949\pi$
$230$	0	0
$231$	−625.269	−0.178094
$232$	0	0
$233$	−3443.85	−0.968301	−0.484150	−	0.874985i	$-0.660871\pi$
$233$	−3443.85	−0.968301	−0.484150	+	0.874985i	$0.660871\pi$
$234$	0	0
$235$	788.635	0.218914
$236$	0	0
$237$	−9969.00	−2.73230
$238$	0	0
$239$	−939.283	−0.254214	−0.127107	−	0.991889i	$-0.540569\pi$
$239$	−939.283	−0.254214	−0.127107	+	0.991889i	$0.540569\pi$
$240$	0	0
$241$	−2194.79	−0.586634	−0.293317	−	0.956015i	$-0.594759\pi$
$241$	−2194.79	−0.586634	−0.293317	+	0.956015i	$0.594759\pi$
$242$	0	0
$243$	−754.367	−0.199147
$244$	0	0
$245$	−151.055	−0.0393901
$246$	0	0
$247$	27.7511	0.00714884
$248$	0	0
$249$	11591.1	2.95001
$250$	0	0
$251$	4654.45	1.17046	0.585232	−	0.810866i	$-0.301004\pi$
$251$	4654.45	1.17046	0.585232	+	0.810866i	$0.301004\pi$
$252$	0	0
$253$	−1704.55	−0.423574
$254$	0	0
$255$	3047.17	0.748319
$256$	0	0
$257$	−1330.30	−0.322887	−0.161443	−	0.986882i	$-0.551615\pi$
$257$	−1330.30	−0.322887	−0.161443	+	0.986882i	$0.551615\pi$
$258$	0	0
$259$	−981.159	−0.235391
$260$	0	0
$261$	15029.2	3.56432
$262$	0	0
$263$	−6374.98	−1.49467	−0.747335	−	0.664448i	$-0.768667\pi$
$263$	−6374.98	−1.49467	−0.747335	+	0.664448i	$0.768667\pi$
$264$	0	0
$265$	1489.89	0.345370
$266$	0	0
$267$	8543.44	1.95824
$268$	0	0
$269$	−2687.02	−0.609034	−0.304517	−	0.952507i	$-0.598495\pi$
$269$	−2687.02	−0.609034	−0.304517	+	0.952507i	$0.598495\pi$
$270$	0	0
$271$	6963.92	1.56099	0.780494	−	0.625163i	$-0.214967\pi$
$271$	6963.92	1.56099	0.780494	+	0.625163i	$0.214967\pi$
$272$	0	0
$273$	−58.3174	−0.0129287
$274$	0	0
$275$	1135.85	0.249070
$276$	0	0
$277$	−5764.74	−1.25043	−0.625215	−	0.780452i	$-0.714989\pi$
$277$	−5764.74	−1.25043	−0.625215	+	0.780452i	$0.714989\pi$
$278$	0	0
$279$	−6890.00	−1.47847
$280$	0	0
$281$	3665.51	0.778170	0.389085	−	0.921202i	$-0.372791\pi$
$281$	3665.51	0.778170	0.389085	+	0.921202i	$0.372791\pi$
$282$	0	0
$283$	−2432.11	−0.510861	−0.255431	−	0.966827i	$-0.582217\pi$
$283$	−2432.11	−0.510861	−0.255431	+	0.966827i	$0.582217\pi$
$284$	0	0
$285$	−847.144	−0.176072
$286$	0	0
$287$	2161.60	0.444583
$288$	0	0
$289$	6930.45	1.41064
$290$	0	0
$291$	−4803.46	−0.967643
$292$	0	0
$293$	−8031.83	−1.60145	−0.800724	−	0.599033i	$-0.795552\pi$
$293$	−8031.83	−1.60145	−0.800724	+	0.599033i	$0.795552\pi$
$294$	0	0
$295$	−406.096	−0.0801485
$296$	0	0
$297$	2545.43	0.497310
$298$	0	0
$299$	−158.979	−0.0307492
$300$	0	0
$301$	3300.43	0.632005
$302$	0	0
$303$	−9038.98	−1.71378
$304$	0	0
$305$	−383.049	−0.0719125
$306$	0	0
$307$	−4990.45	−0.927752	−0.463876	−	0.885900i	$-0.653542\pi$
$307$	−4990.45	−0.927752	−0.463876	+	0.885900i	$0.653542\pi$
$308$	0	0
$309$	7107.23	1.30847
$310$	0	0
$311$	−6697.32	−1.22113	−0.610563	−	0.791967i	$-0.709057\pi$
$311$	−6697.32	−1.22113	−0.610563	+	0.791967i	$0.709057\pi$
$312$	0	0
$313$	4039.89	0.729547	0.364774	−	0.931096i	$-0.381146\pi$
$313$	4039.89	0.729547	0.364774	+	0.931096i	$0.381146\pi$
$314$	0	0
$315$	1197.58	0.214209
$316$	0	0
$317$	−246.856	−0.0437376	−0.0218688	−	0.999761i	$-0.506962\pi$
$317$	−246.856	−0.0437376	−0.0218688	+	0.999761i	$0.506962\pi$
$318$	0	0
$319$	−2663.31	−0.467451
$320$	0	0
$321$	−2068.11	−0.359597
$322$	0	0
$323$	−3292.59	−0.567198
$324$	0	0
$325$	105.938	0.0180811
$326$	0	0
$327$	−8135.66	−1.37585
$328$	0	0
$329$	1790.75	0.300082
$330$	0	0
$331$	2384.13	0.395902	0.197951	−	0.980212i	$-0.436571\pi$
$331$	2384.13	0.395902	0.197951	+	0.980212i	$0.436571\pi$
$332$	0	0
$333$	7778.71	1.28009
$334$	0	0
$335$	−2391.97	−0.390111
$336$	0	0
$337$	−4606.24	−0.744563	−0.372281	−	0.928120i	$-0.621424\pi$
$337$	−4606.24	−0.744563	−0.372281	+	0.928120i	$0.621424\pi$
$338$	0	0
$339$	12299.7	1.97058
$340$	0	0
$341$	1220.97	0.193898
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4853.08	0.757336
$346$	0	0
$347$	−1107.41	−0.171322	−0.0856610	−	0.996324i	$-0.527300\pi$
$347$	−1107.41	−0.171322	−0.0856610	+	0.996324i	$0.527300\pi$
$348$	0	0
$349$	742.918	0.113947	0.0569735	−	0.998376i	$-0.481855\pi$
$349$	742.918	0.113947	0.0569735	+	0.998376i	$0.481855\pi$
$350$	0	0
$351$	237.406	0.0361020
$352$	0	0
$353$	6754.33	1.01840	0.509202	−	0.860647i	$-0.329940\pi$
$353$	6754.33	1.01840	0.509202	+	0.860647i	$0.329940\pi$
$354$	0	0
$355$	1632.89	0.244126
$356$	0	0
$357$	6919.19	1.02578
$358$	0	0
$359$	644.363	0.0947303	0.0473651	−	0.998878i	$-0.484918\pi$
$359$	644.363	0.0947303	0.0473651	+	0.998878i	$0.484918\pi$
$360$	0	0
$361$	−5943.63	−0.866544
$362$	0	0
$363$	11210.7	1.62096
$364$	0	0
$365$	3143.19	0.450745
$366$	0	0
$367$	8414.65	1.19684	0.598421	−	0.801182i	$-0.295795\pi$
$367$	8414.65	1.19684	0.598421	+	0.801182i	$0.295795\pi$
$368$	0	0
$369$	−17137.4	−2.41771
$370$	0	0
$371$	3383.08	0.473425
$372$	0	0
$373$	2820.45	0.391522	0.195761	−	0.980652i	$-0.437282\pi$
$373$	2820.45	0.391522	0.195761	+	0.980652i	$0.437282\pi$
$374$	0	0
$375$	−6733.90	−0.927300
$376$	0	0
$377$	−248.401	−0.0339344
$378$	0	0
$379$	−4004.87	−0.542787	−0.271394	−	0.962468i	$-0.587485\pi$
$379$	−4004.87	−0.542787	−0.271394	+	0.962468i	$0.587485\pi$
$380$	0	0
$381$	−609.354	−0.0819374
$382$	0	0
$383$	−11247.1	−1.50052	−0.750261	−	0.661141i	$-0.770072\pi$
$383$	−11247.1	−1.50052	−0.750261	+	0.661141i	$0.770072\pi$
$384$	0	0
$385$	−212.221	−0.0280930
$386$	0	0
$387$	−26166.1	−3.43694
$388$	0	0
$389$	1741.60	0.226999	0.113499	−	0.993538i	$-0.463794\pi$
$389$	1741.60	0.226999	0.113499	+	0.993538i	$0.463794\pi$
$390$	0	0
$391$	18862.5	2.43968
$392$	0	0
$393$	−13351.4	−1.71371
$394$	0	0
$395$	−3383.56	−0.431001
$396$	0	0
$397$	−12354.7	−1.56188	−0.780940	−	0.624606i	$-0.785260\pi$
$397$	−12354.7	−1.56188	−0.780940	+	0.624606i	$0.785260\pi$
$398$	0	0
$399$	−1923.60	−0.241355
$400$	0	0
$401$	1091.86	0.135972	0.0679861	−	0.997686i	$-0.478343\pi$
$401$	1091.86	0.135972	0.0679861	+	0.997686i	$0.478343\pi$
$402$	0	0
$403$	113.877	0.0140759
$404$	0	0
$405$	−2627.94	−0.322428
$406$	0	0
$407$	−1378.45	−0.167881
$408$	0	0
$409$	8795.67	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$409$	8795.67	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$410$	0	0
$411$	−11090.6	−1.33105
$412$	0	0
$413$	−922.118	−0.109866
$414$	0	0
$415$	3934.10	0.465343
$416$	0	0
$417$	9624.79	1.13028
$418$	0	0
$419$	−13119.8	−1.52970	−0.764852	−	0.644206i	$-0.777188\pi$
$419$	−13119.8	−1.52970	−0.764852	+	0.644206i	$0.777188\pi$
$420$	0	0
$421$	−8585.97	−0.993954	−0.496977	−	0.867764i	$-0.665557\pi$
$421$	−8585.97	−0.993954	−0.496977	+	0.867764i	$0.665557\pi$
$422$	0	0
$423$	−14197.2	−1.63189
$424$	0	0
$425$	−12569.2	−1.43458
$426$	0	0
$427$	−869.786	−0.0985759
$428$	0	0
$429$	−81.9315	−0.00922072
$430$	0	0
$431$	−14398.9	−1.60922	−0.804608	−	0.593806i	$-0.797625\pi$
$431$	−14398.9	−1.60922	−0.804608	+	0.593806i	$0.797625\pi$
$432$	0	0
$433$	4048.50	0.449327	0.224663	−	0.974436i	$-0.427872\pi$
$433$	4048.50	0.449327	0.224663	+	0.974436i	$0.427872\pi$
$434$	0	0
$435$	7582.79	0.835786
$436$	0	0
$437$	−5243.95	−0.574032
$438$	0	0
$439$	−12241.6	−1.33088	−0.665441	−	0.746450i	$-0.731757\pi$
$439$	−12241.6	−1.33088	−0.665441	+	0.746450i	$0.731757\pi$
$440$	0	0
$441$	2719.33	0.293633
$442$	0	0
$443$	−418.123	−0.0448434	−0.0224217	−	0.999749i	$-0.507138\pi$
$443$	−418.123	−0.0448434	−0.0224217	+	0.999749i	$0.507138\pi$
$444$	0	0
$445$	2899.71	0.308898
$446$	0	0
$447$	2866.03	0.303263
$448$	0	0
$449$	4307.38	0.452735	0.226367	−	0.974042i	$-0.427315\pi$
$449$	4307.38	0.452735	0.226367	+	0.974042i	$0.427315\pi$
$450$	0	0
$451$	3036.89	0.317076
$452$	0	0
$453$	−31531.3	−3.27036
$454$	0	0
$455$	−19.7934	−0.00203940
$456$	0	0
$457$	−9250.54	−0.946876	−0.473438	−	0.880827i	$-0.656987\pi$
$457$	−9250.54	−0.946876	−0.473438	+	0.880827i	$0.656987\pi$
$458$	0	0
$459$	−28167.6	−2.86438
$460$	0	0
$461$	6335.44	0.640067	0.320034	−	0.947406i	$-0.396306\pi$
$461$	6335.44	0.640067	0.320034	+	0.947406i	$0.396306\pi$
$462$	0	0
$463$	11011.3	1.10527	0.552634	−	0.833424i	$-0.313623\pi$
$463$	11011.3	1.10527	0.552634	+	0.833424i	$0.313623\pi$
$464$	0	0
$465$	−3476.25	−0.346682
$466$	0	0
$467$	743.459	0.0736685	0.0368343	−	0.999321i	$-0.488273\pi$
$467$	743.459	0.0736685	0.0368343	+	0.999321i	$0.488273\pi$
$468$	0	0
$469$	−5431.42	−0.534755
$470$	0	0
$471$	29361.8	2.87245
$472$	0	0
$473$	4636.85	0.450746
$474$	0	0
$475$	3494.36	0.337542
$476$	0	0
$477$	−26821.3	−2.57456
$478$	0	0
$479$	8314.42	0.793101	0.396551	−	0.918013i	$-0.370207\pi$
$479$	8314.42	0.793101	0.396551	+	0.918013i	$0.370207\pi$
$480$	0	0
$481$	−128.565	−0.0121872
$482$	0	0
$483$	11019.8	1.03814
$484$	0	0
$485$	−1630.33	−0.152639
$486$	0	0
$487$	−19304.9	−1.79628	−0.898139	−	0.439712i	$-0.855080\pi$
$487$	−19304.9	−1.79628	−0.898139	+	0.439712i	$0.855080\pi$
$488$	0	0
$489$	−2952.22	−0.273014
$490$	0	0
$491$	−14548.7	−1.33721	−0.668607	−	0.743616i	$-0.733109\pi$
$491$	−14548.7	−1.33721	−0.668607	+	0.743616i	$0.733109\pi$
$492$	0	0
$493$	29472.0	2.69240
$494$	0	0
$495$	1682.51	0.152774
$496$	0	0
$497$	3707.78	0.334641
$498$	0	0
$499$	−18832.7	−1.68951	−0.844757	−	0.535150i	$-0.820255\pi$
$499$	−18832.7	−1.68951	−0.844757	+	0.535150i	$0.820255\pi$
$500$	0	0
$501$	13061.8	1.16478
$502$	0	0
$503$	15723.6	1.39380	0.696898	−	0.717170i	$-0.254563\pi$
$503$	15723.6	1.39380	0.696898	+	0.717170i	$0.254563\pi$
$504$	0	0
$505$	−3067.90	−0.270336
$506$	0	0
$507$	19947.2	1.74731
$508$	0	0
$509$	16492.0	1.43614	0.718070	−	0.695971i	$-0.245026\pi$
$509$	16492.0	1.43614	0.718070	+	0.695971i	$0.245026\pi$
$510$	0	0
$511$	7137.20	0.617869
$512$	0	0
$513$	7830.87	0.673959
$514$	0	0
$515$	2412.25	0.206401
$516$	0	0
$517$	2515.86	0.214018
$518$	0	0
$519$	21494.5	1.81793
$520$	0	0
$521$	−5892.14	−0.495469	−0.247734	−	0.968828i	$-0.579686\pi$
$521$	−5892.14	−0.495469	−0.247734	+	0.968828i	$0.579686\pi$
$522$	0	0
$523$	3788.64	0.316761	0.158380	−	0.987378i	$-0.449373\pi$
$523$	3788.64	0.316761	0.158380	+	0.987378i	$0.449373\pi$
$524$	0	0
$525$	−7343.20	−0.610444
$526$	0	0
$527$	−13511.1	−1.11680
$528$	0	0
$529$	17874.3	1.46908
$530$	0	0
$531$	7310.63	0.597466
$532$	0	0
$533$	283.243	0.0230181
$534$	0	0
$535$	−701.934	−0.0567238
$536$	0	0
$537$	−18075.8	−1.45257
$538$	0	0
$539$	−481.889	−0.0385092
$540$	0	0
$541$	11506.2	0.914399	0.457200	−	0.889364i	$-0.348852\pi$
$541$	11506.2	0.914399	0.457200	+	0.889364i	$0.348852\pi$
$542$	0	0
$543$	12189.7	0.963374
$544$	0	0
$545$	−2761.31	−0.217030
$546$	0	0
$547$	285.449	0.0223124	0.0111562	−	0.999938i	$-0.496449\pi$
$547$	285.449	0.0223124	0.0111562	+	0.999938i	$0.496449\pi$
$548$	0	0
$549$	6895.73	0.536071
$550$	0	0
$551$	−8193.51	−0.633494
$552$	0	0
$553$	−7683.01	−0.590804
$554$	0	0
$555$	3924.63	0.300165
$556$	0	0
$557$	15556.8	1.18342	0.591710	−	0.806151i	$-0.298453\pi$
$557$	15556.8	1.18342	0.591710	+	0.806151i	$0.298453\pi$
$558$	0	0
$559$	432.468	0.0327217
$560$	0	0
$561$	9720.94	0.731584
$562$	0	0
$563$	12429.1	0.930413	0.465207	−	0.885202i	$-0.345980\pi$
$563$	12429.1	0.930413	0.465207	+	0.885202i	$0.345980\pi$
$564$	0	0
$565$	4174.61	0.310845
$566$	0	0
$567$	−5967.24	−0.441976
$568$	0	0
$569$	14149.7	1.04251	0.521255	−	0.853401i	$-0.325464\pi$
$569$	14149.7	1.04251	0.521255	+	0.853401i	$0.325464\pi$
$570$	0	0
$571$	7606.55	0.557485	0.278743	−	0.960366i	$-0.410082\pi$
$571$	7606.55	0.557485	0.278743	+	0.960366i	$0.410082\pi$
$572$	0	0
$573$	17989.9	1.31159
$574$	0	0
$575$	−20018.4	−1.45187
$576$	0	0
$577$	−10153.3	−0.732562	−0.366281	−	0.930504i	$-0.619369\pi$
$577$	−10153.3	−0.732562	−0.366281	+	0.930504i	$0.619369\pi$
$578$	0	0
$579$	136.812	0.00981986
$580$	0	0
$581$	8933.11	0.637880
$582$	0	0
$583$	4752.97	0.337647
$584$	0	0
$585$	156.924	0.0110906
$586$	0	0
$587$	−8158.79	−0.573678	−0.286839	−	0.957979i	$-0.592605\pi$
$587$	−8158.79	−0.573678	−0.286839	+	0.957979i	$0.592605\pi$
$588$	0	0
$589$	3756.23	0.262772
$590$	0	0
$591$	19262.1	1.34067
$592$	0	0
$593$	12307.3	0.852277	0.426139	−	0.904658i	$-0.359874\pi$
$593$	12307.3	0.852277	0.426139	+	0.904658i	$0.359874\pi$
$594$	0	0
$595$	2348.43	0.161809
$596$	0	0
$597$	−28827.2	−1.97624
$598$	0	0
$599$	−17294.5	−1.17969	−0.589844	−	0.807518i	$-0.700811\pi$
$599$	−17294.5	−1.17969	−0.589844	+	0.807518i	$0.700811\pi$
$600$	0	0
$601$	−3523.35	−0.239136	−0.119568	−	0.992826i	$-0.538151\pi$
$601$	−3523.35	−0.239136	−0.119568	+	0.992826i	$0.538151\pi$
$602$	0	0
$603$	43060.8	2.90808
$604$	0	0
$605$	3805.00	0.255695
$606$	0	0
$607$	19402.8	1.29742	0.648710	−	0.761036i	$-0.275309\pi$
$607$	19402.8	1.29742	0.648710	+	0.761036i	$0.275309\pi$
$608$	0	0
$609$	17218.2	1.14567
$610$	0	0
$611$	234.648	0.0155366
$612$	0	0
$613$	27474.9	1.81028	0.905139	−	0.425115i	$-0.139766\pi$
$613$	27474.9	1.81028	0.905139	+	0.425115i	$0.139766\pi$
$614$	0	0
$615$	−8646.41	−0.566922
$616$	0	0
$617$	−9220.91	−0.601653	−0.300827	−	0.953679i	$-0.597263\pi$
$617$	−9220.91	−0.601653	−0.300827	+	0.953679i	$0.597263\pi$
$618$	0	0
$619$	−20137.1	−1.30756	−0.653779	−	0.756685i	$-0.726817\pi$
$619$	−20137.1	−1.30756	−0.653779	+	0.756685i	$0.726817\pi$
$620$	0	0
$621$	−44861.1	−2.89889
$622$	0	0
$623$	6584.35	0.423429
$624$	0	0
$625$	12151.5	0.777698
$626$	0	0
$627$	−2702.52	−0.172134
$628$	0	0
$629$	15253.9	0.966951
$630$	0	0
$631$	6810.84	0.429691	0.214846	−	0.976648i	$-0.431075\pi$
$631$	6810.84	0.429691	0.214846	+	0.976648i	$0.431075\pi$
$632$	0	0
$633$	31474.5	1.97630
$634$	0	0
$635$	−206.820	−0.0129250
$636$	0	0
$637$	−44.9446	−0.00279556
$638$	0	0
$639$	−29395.6	−1.81983
$640$	0	0
$641$	1260.97	0.0776994	0.0388497	−	0.999245i	$-0.487631\pi$
$641$	1260.97	0.0776994	0.0388497	+	0.999245i	$0.487631\pi$
$642$	0	0
$643$	25637.3	1.57237	0.786187	−	0.617989i	$-0.212052\pi$
$643$	25637.3	1.57237	0.786187	+	0.617989i	$0.212052\pi$
$644$	0	0
$645$	−13201.7	−0.805918
$646$	0	0
$647$	1039.32	0.0631530	0.0315765	−	0.999501i	$-0.489947\pi$
$647$	1039.32	0.0631530	0.0315765	+	0.999501i	$0.489947\pi$
$648$	0	0
$649$	−1295.51	−0.0783561
$650$	0	0
$651$	−7893.49	−0.475223
$652$	0	0
$653$	12304.4	0.737379	0.368689	−	0.929553i	$-0.379807\pi$
$653$	12304.4	0.737379	0.368689	+	0.929553i	$0.379807\pi$
$654$	0	0
$655$	−4531.56	−0.270325
$656$	0	0
$657$	−56584.3	−3.36007
$658$	0	0
$659$	−17228.1	−1.01838	−0.509190	−	0.860654i	$-0.670055\pi$
$659$	−17228.1	−1.01838	−0.509190	+	0.860654i	$0.670055\pi$
$660$	0	0
$661$	−29787.3	−1.75278	−0.876392	−	0.481598i	$-0.840056\pi$
$661$	−29787.3	−1.75278	−0.876392	+	0.481598i	$0.840056\pi$
$662$	0	0
$663$	906.648	0.0531091
$664$	0	0
$665$	−652.886	−0.0380719
$666$	0	0
$667$	46938.6	2.72484
$668$	0	0
$669$	−8640.01	−0.499315
$670$	0	0
$671$	−1221.98	−0.0703043
$672$	0	0
$673$	23585.1	1.35087	0.675437	−	0.737417i	$-0.263955\pi$
$673$	23585.1	1.35087	0.675437	+	0.737417i	$0.263955\pi$
$674$	0	0
$675$	29893.7	1.70461
$676$	0	0
$677$	−23050.0	−1.30854	−0.654271	−	0.756260i	$-0.727025\pi$
$677$	−23050.0	−1.30854	−0.654271	+	0.756260i	$0.727025\pi$
$678$	0	0
$679$	−3701.99	−0.209233
$680$	0	0
$681$	40007.4	2.25123
$682$	0	0
$683$	−5419.17	−0.303600	−0.151800	−	0.988411i	$-0.548507\pi$
$683$	−5419.17	−0.303600	−0.151800	+	0.988411i	$0.548507\pi$
$684$	0	0
$685$	−3764.24	−0.209963
$686$	0	0
$687$	−5934.18	−0.329553
$688$	0	0
$689$	443.298	0.0245113
$690$	0	0
$691$	33495.2	1.84402	0.922009	−	0.387168i	$-0.126547\pi$
$691$	33495.2	1.84402	0.922009	+	0.387168i	$0.126547\pi$
$692$	0	0
$693$	3820.46	0.209419
$694$	0	0
$695$	3266.73	0.178294
$696$	0	0
$697$	−33606.0	−1.82628
$698$	0	0
$699$	31279.7	1.69257
$700$	0	0
$701$	−453.567	−0.0244380	−0.0122190	−	0.999925i	$-0.503890\pi$
$701$	−453.567	−0.0244380	−0.0122190	+	0.999925i	$0.503890\pi$
$702$	0	0
$703$	−4240.73	−0.227514
$704$	0	0
$705$	−7162.98	−0.382657
$706$	0	0
$707$	−6966.26	−0.370570
$708$	0	0
$709$	4561.63	0.241630	0.120815	−	0.992675i	$-0.461449\pi$
$709$	4561.63	0.241630	0.120815	+	0.992675i	$0.461449\pi$
$710$	0	0
$711$	60911.5	3.21289
$712$	0	0
$713$	−21518.5	−1.13026
$714$	0	0
$715$	−27.8082	−0.00145450
$716$	0	0
$717$	8531.29	0.444361
$718$	0	0
$719$	−3442.44	−0.178555	−0.0892777	−	0.996007i	$-0.528456\pi$
$719$	−3442.44	−0.178555	−0.0892777	+	0.996007i	$0.528456\pi$
$720$	0	0
$721$	5477.48	0.282929
$722$	0	0
$723$	19934.7	1.02542
$724$	0	0
$725$	−31278.1	−1.60226
$726$	0	0
$727$	18529.0	0.945256	0.472628	−	0.881262i	$-0.343305\pi$
$727$	18529.0	0.945256	0.472628	+	0.881262i	$0.343305\pi$
$728$	0	0
$729$	−16164.7	−0.821254
$730$	0	0
$731$	−51311.1	−2.59618
$732$	0	0
$733$	−13434.7	−0.676974	−0.338487	−	0.940971i	$-0.609915\pi$
$733$	−13434.7	−0.676974	−0.338487	+	0.940971i	$0.609915\pi$
$734$	0	0
$735$	1372.00	0.0688530
$736$	0	0
$737$	−7630.74	−0.381387
$738$	0	0
$739$	24980.0	1.24345	0.621723	−	0.783238i	$-0.286433\pi$
$739$	24980.0	1.24345	0.621723	+	0.783238i	$0.286433\pi$
$740$	0	0
$741$	−252.057	−0.0124960
$742$	0	0
$743$	−2704.11	−0.133518	−0.0667591	−	0.997769i	$-0.521266\pi$
$743$	−2704.11	−0.133518	−0.0667591	+	0.997769i	$0.521266\pi$
$744$	0	0
$745$	972.752	0.0478374
$746$	0	0
$747$	−70822.5	−3.46889
$748$	0	0
$749$	−1593.87	−0.0777555
$750$	0	0
$751$	−28896.3	−1.40405	−0.702024	−	0.712154i	$-0.747720\pi$
$751$	−28896.3	−1.40405	−0.702024	+	0.712154i	$0.747720\pi$
$752$	0	0
$753$	−42275.3	−2.04594
$754$	0	0
$755$	−10702.0	−0.515875
$756$	0	0
$757$	30491.2	1.46397	0.731984	−	0.681322i	$-0.238595\pi$
$757$	30491.2	1.46397	0.731984	+	0.681322i	$0.238595\pi$
$758$	0	0
$759$	15482.0	0.740399
$760$	0	0
$761$	2472.70	0.117786	0.0588931	−	0.998264i	$-0.481243\pi$
$761$	2472.70	0.117786	0.0588931	+	0.998264i	$0.481243\pi$
$762$	0	0
$763$	−6270.07	−0.297499
$764$	0	0
$765$	−18618.5	−0.879940
$766$	0	0
$767$	−120.829	−0.00568823
$768$	0	0
$769$	−23423.3	−1.09840	−0.549198	−	0.835692i	$-0.685067\pi$
$769$	−23423.3	−1.09840	−0.549198	+	0.835692i	$0.685067\pi$
$770$	0	0
$771$	12082.8	0.564399
$772$	0	0
$773$	22395.4	1.04205	0.521025	−	0.853541i	$-0.325550\pi$
$773$	22395.4	1.04205	0.521025	+	0.853541i	$0.325550\pi$
$774$	0	0
$775$	14339.1	0.664614
$776$	0	0
$777$	8911.63	0.411458
$778$	0	0
$779$	9342.79	0.429705
$780$	0	0
$781$	5209.15	0.238666
$782$	0	0
$783$	−70094.1	−3.19918
$784$	0	0
$785$	9965.64	0.453107
$786$	0	0
$787$	−6245.12	−0.282865	−0.141432	−	0.989948i	$-0.545171\pi$
$787$	−6245.12	−0.282865	−0.141432	+	0.989948i	$0.545171\pi$
$788$	0	0
$789$	57902.4	2.61265
$790$	0	0
$791$	9479.25	0.426098
$792$	0	0
$793$	−113.971	−0.00510371
$794$	0	0
$795$	−13532.3	−0.603700
$796$	0	0
$797$	6727.27	0.298986	0.149493	−	0.988763i	$-0.452236\pi$
$797$	6727.27	0.298986	0.149493	+	0.988763i	$0.452236\pi$
$798$	0	0
$799$	−27840.4	−1.23269
$800$	0	0
$801$	−52201.2	−2.30267
$802$	0	0
$803$	10027.2	0.440664
$804$	0	0
$805$	3740.22	0.163758
$806$	0	0
$807$	24405.5	1.06458
$808$	0	0
$809$	27749.8	1.20597	0.602986	−	0.797752i	$-0.293978\pi$
$809$	27749.8	1.20597	0.602986	+	0.797752i	$0.293978\pi$
$810$	0	0
$811$	−25564.5	−1.10689	−0.553447	−	0.832885i	$-0.686688\pi$
$811$	−25564.5	−1.10689	−0.553447	+	0.832885i	$0.686688\pi$
$812$	0	0
$813$	−63251.6	−2.72857
$814$	0	0
$815$	−1002.01	−0.0430660
$816$	0	0
$817$	14265.0	0.610855
$818$	0	0
$819$	356.325	0.0152027
$820$	0	0
$821$	−643.073	−0.0273367	−0.0136683	−	0.999907i	$-0.504351\pi$
$821$	−643.073	−0.0273367	−0.0136683	+	0.999907i	$0.504351\pi$
$822$	0	0
$823$	−8931.20	−0.378277	−0.189138	−	0.981950i	$-0.560570\pi$
$823$	−8931.20	−0.378277	−0.189138	+	0.981950i	$0.560570\pi$
$824$	0	0
$825$	−10316.6	−0.435369
$826$	0	0
$827$	−19248.5	−0.809356	−0.404678	−	0.914459i	$-0.632616\pi$
$827$	−19248.5	−0.809356	−0.404678	+	0.914459i	$0.632616\pi$
$828$	0	0
$829$	18531.2	0.776376	0.388188	−	0.921580i	$-0.373101\pi$
$829$	18531.2	0.776376	0.388188	+	0.921580i	$0.373101\pi$
$830$	0	0
$831$	52359.7	2.18573
$832$	0	0
$833$	5332.55	0.221803
$834$	0	0
$835$	4433.27	0.183736
$836$	0	0
$837$	32133.9	1.32701
$838$	0	0
$839$	−7354.87	−0.302644	−0.151322	−	0.988485i	$-0.548353\pi$
$839$	−7354.87	−0.302644	−0.151322	+	0.988485i	$0.548353\pi$
$840$	0	0
$841$	48951.2	2.00710
$842$	0	0
$843$	−33292.9	−1.36022
$844$	0	0
$845$	6770.24	0.275625
$846$	0	0
$847$	8639.98	0.350500
$848$	0	0
$849$	22090.2	0.892974
$850$	0	0
$851$	24294.1	0.978602
$852$	0	0
$853$	−18103.0	−0.726654	−0.363327	−	0.931662i	$-0.618359\pi$
$853$	−18103.0	−0.726654	−0.363327	+	0.931662i	$0.618359\pi$
$854$	0	0
$855$	5176.13	0.207041
$856$	0	0
$857$	−17019.9	−0.678400	−0.339200	−	0.940714i	$-0.610156\pi$
$857$	−17019.9	−0.678400	−0.339200	+	0.940714i	$0.610156\pi$
$858$	0	0
$859$	33487.5	1.33013	0.665064	−	0.746787i	$-0.268404\pi$
$859$	33487.5	1.33013	0.665064	+	0.746787i	$0.268404\pi$
$860$	0	0
$861$	−19633.3	−0.777121
$862$	0	0
$863$	−22721.2	−0.896222	−0.448111	−	0.893978i	$-0.647903\pi$
$863$	−22721.2	−0.896222	−0.448111	+	0.893978i	$0.647903\pi$
$864$	0	0
$865$	7295.41	0.286765
$866$	0	0
$867$	−62947.7	−2.46576
$868$	0	0
$869$	−10794.1	−0.421362
$870$	0	0
$871$	−711.701	−0.0276866
$872$	0	0
$873$	29349.6	1.13784
$874$	0	0
$875$	−5189.76	−0.200509
$876$	0	0
$877$	−4277.59	−0.164702	−0.0823512	−	0.996603i	$-0.526243\pi$
$877$	−4277.59	−0.164702	−0.0823512	+	0.996603i	$0.526243\pi$
$878$	0	0
$879$	72951.2	2.79930
$880$	0	0
$881$	28019.8	1.07152	0.535761	−	0.844369i	$-0.320025\pi$
$881$	28019.8	1.07152	0.535761	+	0.844369i	$0.320025\pi$
$882$	0	0
$883$	−19342.3	−0.737169	−0.368585	−	0.929594i	$-0.620158\pi$
$883$	−19342.3	−0.737169	−0.368585	+	0.929594i	$0.620158\pi$
$884$	0	0
$885$	3688.47	0.140098
$886$	0	0
$887$	23806.5	0.901176	0.450588	−	0.892732i	$-0.351214\pi$
$887$	23806.5	0.901176	0.450588	+	0.892732i	$0.351214\pi$
$888$	0	0
$889$	−469.623	−0.0177173
$890$	0	0
$891$	−8383.52	−0.315217
$892$	0	0
$893$	7739.89	0.290040
$894$	0	0
$895$	−6135.08	−0.229132
$896$	0	0
$897$	1443.97	0.0537490
$898$	0	0
$899$	−33622.0	−1.24734
$900$	0	0
$901$	−52596.0	−1.94476
$902$	0	0
$903$	−29977.0	−1.10473
$904$	0	0
$905$	4137.30	0.151965
$906$	0	0
$907$	13453.0	0.492504	0.246252	−	0.969206i	$-0.420801\pi$
$907$	13453.0	0.492504	0.246252	+	0.969206i	$0.420801\pi$
$908$	0	0
$909$	55229.1	2.01522
$910$	0	0
$911$	23475.7	0.853770	0.426885	−	0.904306i	$-0.359611\pi$
$911$	23475.7	0.853770	0.426885	+	0.904306i	$0.359611\pi$
$912$	0	0
$913$	12550.4	0.454936
$914$	0	0
$915$	3479.14	0.125702
$916$	0	0
$917$	−10289.8	−0.370554
$918$	0	0
$919$	8465.88	0.303878	0.151939	−	0.988390i	$-0.451448\pi$
$919$	8465.88	0.303878	0.151939	+	0.988390i	$0.451448\pi$
$920$	0	0
$921$	45327.1	1.62169
$922$	0	0
$923$	485.845	0.0173259
$924$	0	0
$925$	−16188.6	−0.575437
$926$	0	0
$927$	−43425.9	−1.53861
$928$	0	0
$929$	−22244.4	−0.785592	−0.392796	−	0.919626i	$-0.628492\pi$
$929$	−22244.4	−0.785592	−0.392796	+	0.919626i	$0.628492\pi$
$930$	0	0
$931$	−1482.50	−0.0521880
$932$	0	0
$933$	60830.2	2.13450
$934$	0	0
$935$	3299.37	0.115402
$936$	0	0
$937$	46121.3	1.60802	0.804012	−	0.594613i	$-0.202695\pi$
$937$	46121.3	1.60802	0.804012	+	0.594613i	$0.202695\pi$
$938$	0	0
$939$	−36693.4	−1.27523
$940$	0	0
$941$	−33287.7	−1.15319	−0.576593	−	0.817032i	$-0.695618\pi$
$941$	−33287.7	−1.15319	−0.576593	+	0.817032i	$0.695618\pi$
$942$	0	0
$943$	−53522.6	−1.84829
$944$	0	0
$945$	−5585.33	−0.192265
$946$	0	0
$947$	−39766.4	−1.36456	−0.682278	−	0.731093i	$-0.739011\pi$
$947$	−39766.4	−1.36456	−0.682278	+	0.731093i	$0.739011\pi$
$948$	0	0
$949$	935.216	0.0319899
$950$	0	0
$951$	2242.14	0.0764524
$952$	0	0
$953$	−36676.9	−1.24667	−0.623337	−	0.781953i	$-0.714223\pi$
$953$	−36676.9	−1.24667	−0.623337	+	0.781953i	$0.714223\pi$
$954$	0	0
$955$	6105.92	0.206893
$956$	0	0
$957$	24190.2	0.817094
$958$	0	0
$959$	−8547.43	−0.287811
$960$	0	0
$961$	−14377.3	−0.482606
$962$	0	0
$963$	12636.4	0.422847
$964$	0	0
$965$	46.4350	0.00154901
$966$	0	0
$967$	−52869.3	−1.75818	−0.879091	−	0.476654i	$-0.841849\pi$
$967$	−52869.3	−1.75818	−0.879091	+	0.476654i	$0.841849\pi$
$968$	0	0
$969$	29905.9	0.991449
$970$	0	0
$971$	−12334.1	−0.407643	−0.203821	−	0.979008i	$-0.565336\pi$
$971$	−12334.1	−0.407643	−0.203821	+	0.979008i	$0.565336\pi$
$972$	0	0
$973$	7417.73	0.244400
$974$	0	0
$975$	−962.208	−0.0316054
$976$	0	0
$977$	−6952.90	−0.227680	−0.113840	−	0.993499i	$-0.536315\pi$
$977$	−6952.90	−0.227680	−0.113840	+	0.993499i	$0.536315\pi$
$978$	0	0
$979$	9250.51	0.301989
$980$	0	0
$981$	49709.7	1.61785
$982$	0	0
$983$	−48940.4	−1.58795	−0.793976	−	0.607949i	$-0.791993\pi$
$983$	−48940.4	−1.58795	−0.793976	+	0.607949i	$0.791993\pi$
$984$	0	0
$985$	6537.72	0.211481
$986$	0	0
$987$	−16264.9	−0.524537
$988$	0	0
$989$	−81720.6	−2.62747
$990$	0	0
$991$	−14919.9	−0.478250	−0.239125	−	0.970989i	$-0.576861\pi$
$991$	−14919.9	−0.478250	−0.239125	+	0.970989i	$0.576861\pi$
$992$	0	0
$993$	−21654.5	−0.692028
$994$	0	0
$995$	−9784.18	−0.311738
$996$	0	0
$997$	−2469.82	−0.0784554	−0.0392277	−	0.999230i	$-0.512490\pi$
$997$	−2469.82	−0.0784554	−0.0392277	+	0.999230i	$0.512490\pi$
$998$	0	0
$999$	−36278.7	−1.14896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.q.1.1		2
4.3	odd	2		448.4.a.t.1.2		2
8.3	odd	2		224.4.a.c.1.1	✓	2
8.5	even	2		224.4.a.d.1.2	yes	2
24.5	odd	2		2016.4.a.o.1.1		2
24.11	even	2		2016.4.a.p.1.1		2
56.13	odd	2		1568.4.a.p.1.1		2
56.27	even	2		1568.4.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.c.1.1	✓	2	8.3	odd	2
224.4.a.d.1.2	yes	2	8.5	even	2
448.4.a.q.1.1		2	1.1	even	1	trivial
448.4.a.t.1.2		2	4.3	odd	2
1568.4.a.p.1.1		2	56.13	odd	2
1568.4.a.u.1.2		2	56.27	even	2
2016.4.a.o.1.1		2	24.5	odd	2
2016.4.a.p.1.1		2	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)

\(f(q)\)	\(=\)	\(q-9.08276 q^{3} -3.08276 q^{5} -7.00000 q^{7} +55.4966 q^{9} +O(q^{10})\)	\(q-9.08276 q^{3} -3.08276 q^{5} -7.00000 q^{7} +55.4966 q^{9} -9.83447 q^{11} -0.917237 q^{13} +28.0000 q^{15} +108.828 q^{17} -30.2551 q^{19} +63.5793 q^{21} +173.324 q^{23} -115.497 q^{25} -258.828 q^{27} +270.814 q^{29} -124.152 q^{31} +89.3242 q^{33} +21.5793 q^{35} +140.166 q^{37} +8.33105 q^{39} -308.800 q^{41} -471.490 q^{43} -171.083 q^{45} -255.821 q^{47} +49.0000 q^{49} -988.455 q^{51} -483.297 q^{53} +30.3174 q^{55} +274.800 q^{57} +131.731 q^{59} +124.255 q^{61} -388.476 q^{63} +2.82763 q^{65} +775.918 q^{67} -1574.26 q^{69} -529.683 q^{71} -1019.60 q^{73} +1049.03 q^{75} +68.8413 q^{77} +1097.57 q^{79} +852.462 q^{81} -1276.16 q^{83} -335.490 q^{85} -2459.74 q^{87} -940.621 q^{89} +6.42066 q^{91} +1127.64 q^{93} +93.2694 q^{95} +528.855 q^{97} -545.780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} - 14 q^{7} + 38 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 6 * q^5 - 14 * q^7 + 38 * q^9	\( 2 q - 6 q^{3} + 6 q^{5} - 14 q^{7} + 38 q^{9} - 44 q^{11} - 14 q^{13} + 56 q^{15} + 96 q^{17} - 170 q^{19} + 42 q^{21} + 152 q^{23} - 158 q^{25} - 396 q^{27} + 128 q^{29} + 68 q^{31} - 16 q^{33} - 42 q^{35} + 256 q^{37} - 32 q^{39} + 88 q^{41} - 724 q^{43} - 330 q^{45} - 244 q^{47} + 98 q^{49} - 1028 q^{51} - 188 q^{53} - 280 q^{55} - 156 q^{57} - 138 q^{59} + 358 q^{61} - 266 q^{63} - 116 q^{65} - 200 q^{67} - 1640 q^{69} - 1400 q^{71} - 628 q^{73} + 918 q^{75} + 308 q^{77} + 200 q^{79} + 902 q^{81} - 618 q^{83} - 452 q^{85} - 2900 q^{87} - 908 q^{89} + 98 q^{91} + 1720 q^{93} - 1176 q^{95} + 1520 q^{97} + 52 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^5 - 14 * q^7 + 38 * q^9 - 44 * q^11 - 14 * q^13 + 56 * q^15 + 96 * q^17 - 170 * q^19 + 42 * q^21 + 152 * q^23 - 158 * q^25 - 396 * q^27 + 128 * q^29 + 68 * q^31 - 16 * q^33 - 42 * q^35 + 256 * q^37 - 32 * q^39 + 88 * q^41 - 724 * q^43 - 330 * q^45 - 244 * q^47 + 98 * q^49 - 1028 * q^51 - 188 * q^53 - 280 * q^55 - 156 * q^57 - 138 * q^59 + 358 * q^61 - 266 * q^63 - 116 * q^65 - 200 * q^67 - 1640 * q^69 - 1400 * q^71 - 628 * q^73 + 918 * q^75 + 308 * q^77 + 200 * q^79 + 902 * q^81 - 618 * q^83 - 452 * q^85 - 2900 * q^87 - 908 * q^89 + 98 * q^91 + 1720 * q^93 - 1176 * q^95 + 1520 * q^97 + 52 * q^99

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