LMFDB - Embedded newform 448.4.a.w.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$-0.523976$ 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)

$f(q)$	$=$	$q-7.45090 q^{3} -7.54680 q^{5} +7.00000 q^{7} +28.5159 q^{9} +O(q^{10})$	$q-7.45090 q^{3} -7.54680 q^{5} +7.00000 q^{7} +28.5159 q^{9} +15.1323 q^{11} -73.6722 q^{13} +56.2305 q^{15} -96.9699 q^{17} -108.617 q^{19} -52.1563 q^{21} -7.44482 q^{23} -68.0458 q^{25} -11.2946 q^{27} -10.9064 q^{29} +212.244 q^{31} -112.749 q^{33} -52.8276 q^{35} +286.440 q^{37} +548.924 q^{39} -195.627 q^{41} +105.004 q^{43} -215.204 q^{45} +480.016 q^{47} +49.0000 q^{49} +722.513 q^{51} +228.590 q^{53} -114.200 q^{55} +809.296 q^{57} -660.893 q^{59} -81.4833 q^{61} +199.611 q^{63} +555.989 q^{65} +454.984 q^{67} +55.4706 q^{69} -339.122 q^{71} +718.313 q^{73} +507.002 q^{75} +105.926 q^{77} -866.989 q^{79} -685.774 q^{81} +1016.83 q^{83} +731.813 q^{85} +81.2624 q^{87} +32.1592 q^{89} -515.705 q^{91} -1581.41 q^{93} +819.713 q^{95} -1403.37 q^{97} +431.509 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9}+O(q^{10}) $ 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9	$ 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9} + 6 q^{13} + 168 q^{15} - 66 q^{17} - 168 q^{19} + 336 q^{23} + 69 q^{25} - 168 q^{27} - 90 q^{29} + 504 q^{31} + 120 q^{33} + 42 q^{35} - 18 q^{37} + 840 q^{39} - 450 q^{41} + 150 q^{45} + 504 q^{47} + 147 q^{49} + 336 q^{51} + 78 q^{53} + 1176 q^{55} + 48 q^{57} + 504 q^{59} - 498 q^{61} + 105 q^{63} + 1068 q^{65} + 1008 q^{67} + 1224 q^{69} + 504 q^{71} - 234 q^{73} + 1848 q^{75} + 168 q^{79} - 981 q^{81} + 3024 q^{83} - 1476 q^{85} - 336 q^{87} + 246 q^{89} + 42 q^{91} - 1200 q^{93} - 2184 q^{95} - 2514 q^{97} + 2688 q^{99}+O(q^{100}) $ 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9 + 6 * q^13 + 168 * q^15 - 66 * q^17 - 168 * q^19 + 336 * q^23 + 69 * q^25 - 168 * q^27 - 90 * q^29 + 504 * q^31 + 120 * q^33 + 42 * q^35 - 18 * q^37 + 840 * q^39 - 450 * q^41 + 150 * q^45 + 504 * q^47 + 147 * q^49 + 336 * q^51 + 78 * q^53 + 1176 * q^55 + 48 * q^57 + 504 * q^59 - 498 * q^61 + 105 * q^63 + 1068 * q^65 + 1008 * q^67 + 1224 * q^69 + 504 * q^71 - 234 * q^73 + 1848 * q^75 + 168 * q^79 - 981 * q^81 + 3024 * q^83 - 1476 * q^85 - 336 * q^87 + 246 * q^89 + 42 * q^91 - 1200 * q^93 - 2184 * q^95 - 2514 * q^97 + 2688 * 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
$3$	−7.45090	−1.43393	−0.716963	−	0.697111i	$-0.754468\pi$
$3$	−7.45090	−1.43393	−0.716963	+	0.697111i	$0.754468\pi$
$4$	0	0
$5$	−7.54680	−0.675007	−0.337503	−	0.941324i	$-0.609582\pi$
$5$	−7.54680	−0.675007	−0.337503	+	0.941324i	$0.609582\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	28.5159	1.05614
$10$	0	0
$11$	15.1323	0.414777	0.207388	−	0.978259i	$-0.433504\pi$
$11$	15.1323	0.414777	0.207388	+	0.978259i	$0.433504\pi$
$12$	0	0
$13$	−73.6722	−1.57177	−0.785884	−	0.618374i	$-0.787792\pi$
$13$	−73.6722	−1.57177	−0.785884	+	0.618374i	$0.787792\pi$
$14$	0	0
$15$	56.2305	0.967909
$16$	0	0
$17$	−96.9699	−1.38345	−0.691725	−	0.722161i	$-0.743149\pi$
$17$	−96.9699	−1.38345	−0.691725	+	0.722161i	$0.743149\pi$
$18$	0	0
$19$	−108.617	−1.31150	−0.655750	−	0.754978i	$-0.727647\pi$
$19$	−108.617	−1.31150	−0.655750	+	0.754978i	$0.727647\pi$
$20$	0	0
$21$	−52.1563	−0.541973
$22$	0	0
$23$	−7.44482	−0.0674935	−0.0337468	−	0.999430i	$-0.510744\pi$
$23$	−7.44482	−0.0674935	−0.0337468	+	0.999430i	$0.510744\pi$
$24$	0	0
$25$	−68.0458	−0.544366
$26$	0	0
$27$	−11.2946	−0.0805055
$28$	0	0
$29$	−10.9064	−0.0698368	−0.0349184	−	0.999390i	$-0.511117\pi$
$29$	−10.9064	−0.0698368	−0.0349184	+	0.999390i	$0.511117\pi$
$30$	0	0
$31$	212.244	1.22968	0.614842	−	0.788650i	$-0.289220\pi$
$31$	212.244	1.22968	0.614842	+	0.788650i	$0.289220\pi$
$32$	0	0
$33$	−112.749	−0.594759
$34$	0	0
$35$	−52.8276	−0.255129
$36$	0	0
$37$	286.440	1.27271	0.636356	−	0.771395i	$-0.280441\pi$
$37$	286.440	1.27271	0.636356	+	0.771395i	$0.280441\pi$
$38$	0	0
$39$	548.924	2.25380
$40$	0	0
$41$	−195.627	−0.745167	−0.372583	−	0.927999i	$-0.621528\pi$
$41$	−195.627	−0.745167	−0.372583	+	0.927999i	$0.621528\pi$
$42$	0	0
$43$	105.004	0.372394	0.186197	−	0.982512i	$-0.440384\pi$
$43$	105.004	0.372394	0.186197	+	0.982512i	$0.440384\pi$
$44$	0	0
$45$	−215.204	−0.712904
$46$	0	0
$47$	480.016	1.48973	0.744867	−	0.667213i	$-0.232513\pi$
$47$	480.016	1.48973	0.744867	+	0.667213i	$0.232513\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	722.513	1.98376
$52$	0	0
$53$	228.590	0.592438	0.296219	−	0.955120i	$-0.404274\pi$
$53$	228.590	0.592438	0.296219	+	0.955120i	$0.404274\pi$
$54$	0	0
$55$	−114.200	−0.279977
$56$	0	0
$57$	809.296	1.88059
$58$	0	0
$59$	−660.893	−1.45832	−0.729161	−	0.684342i	$-0.760089\pi$
$59$	−660.893	−1.45832	−0.729161	+	0.684342i	$0.760089\pi$
$60$	0	0
$61$	−81.4833	−0.171031	−0.0855153	−	0.996337i	$-0.527254\pi$
$61$	−81.4833	−0.171031	−0.0855153	+	0.996337i	$0.527254\pi$
$62$	0	0
$63$	199.611	0.399185
$64$	0	0
$65$	555.989	1.06095
$66$	0	0
$67$	454.984	0.829629	0.414814	−	0.909906i	$-0.363847\pi$
$67$	454.984	0.829629	0.414814	+	0.909906i	$0.363847\pi$
$68$	0	0
$69$	55.4706	0.0967807
$70$	0	0
$71$	−339.122	−0.566850	−0.283425	−	0.958994i	$-0.591471\pi$
$71$	−339.122	−0.566850	−0.283425	+	0.958994i	$0.591471\pi$
$72$	0	0
$73$	718.313	1.15167	0.575837	−	0.817564i	$-0.304676\pi$
$73$	718.313	1.15167	0.575837	+	0.817564i	$0.304676\pi$
$74$	0	0
$75$	507.002	0.780581
$76$	0	0
$77$	105.926	0.156771
$78$	0	0
$79$	−866.989	−1.23473	−0.617367	−	0.786676i	$-0.711800\pi$
$79$	−866.989	−1.23473	−0.617367	+	0.786676i	$0.711800\pi$
$80$	0	0
$81$	−685.774	−0.940704
$82$	0	0
$83$	1016.83	1.34472	0.672361	−	0.740223i	$-0.265280\pi$
$83$	1016.83	1.34472	0.672361	+	0.740223i	$0.265280\pi$
$84$	0	0
$85$	731.813	0.933838
$86$	0	0
$87$	81.2624	0.100141
$88$	0	0
$89$	32.1592	0.0383019	0.0191509	−	0.999817i	$-0.493904\pi$
$89$	32.1592	0.0383019	0.0191509	+	0.999817i	$0.493904\pi$
$90$	0	0
$91$	−515.705	−0.594072
$92$	0	0
$93$	−1581.41	−1.76328
$94$	0	0
$95$	819.713	0.885271
$96$	0	0
$97$	−1403.37	−1.46897	−0.734487	−	0.678623i	$-0.762577\pi$
$97$	−1403.37	−1.46897	−0.734487	+	0.678623i	$0.762577\pi$
$98$	0	0
$99$	431.509	0.438064
$100$	0	0
$101$	1718.67	1.69321	0.846604	−	0.532223i	$-0.178643\pi$
$101$	1718.67	1.69321	0.846604	+	0.532223i	$0.178643\pi$
$102$	0	0
$103$	1726.94	1.65204	0.826020	−	0.563641i	$-0.190600\pi$
$103$	1726.94	1.65204	0.826020	+	0.563641i	$0.190600\pi$
$104$	0	0
$105$	393.613	0.365835
$106$	0	0
$107$	2080.67	1.87987	0.939936	−	0.341350i	$-0.110884\pi$
$107$	2080.67	1.87987	0.939936	+	0.341350i	$0.110884\pi$
$108$	0	0
$109$	830.755	0.730017	0.365009	−	0.931004i	$-0.381066\pi$
$109$	830.755	0.730017	0.365009	+	0.931004i	$0.381066\pi$
$110$	0	0
$111$	−2134.23	−1.82498
$112$	0	0
$113$	−975.055	−0.811729	−0.405865	−	0.913933i	$-0.633030\pi$
$113$	−975.055	−0.811729	−0.405865	+	0.913933i	$0.633030\pi$
$114$	0	0
$115$	56.1846	0.0455586
$116$	0	0
$117$	−2100.83	−1.66001
$118$	0	0
$119$	−678.789	−0.522895
$120$	0	0
$121$	−1102.01	−0.827960
$122$	0	0
$123$	1457.60	1.06851
$124$	0	0
$125$	1456.88	1.04246
$126$	0	0
$127$	−1148.14	−0.802213	−0.401107	−	0.916031i	$-0.631374\pi$
$127$	−1148.14	−0.802213	−0.401107	+	0.916031i	$0.631374\pi$
$128$	0	0
$129$	−782.374	−0.533986
$130$	0	0
$131$	664.377	0.443106	0.221553	−	0.975148i	$-0.428887\pi$
$131$	664.377	0.443106	0.221553	+	0.975148i	$0.428887\pi$
$132$	0	0
$133$	−760.320	−0.495700
$134$	0	0
$135$	85.2382	0.0543418
$136$	0	0
$137$	2314.76	1.44353	0.721765	−	0.692138i	$-0.243331\pi$
$137$	2314.76	1.44353	0.721765	+	0.692138i	$0.243331\pi$
$138$	0	0
$139$	−2934.37	−1.79058	−0.895288	−	0.445487i	$-0.853031\pi$
$139$	−2934.37	−1.79058	−0.895288	+	0.445487i	$0.853031\pi$
$140$	0	0
$141$	−3576.55	−2.13617
$142$	0	0
$143$	−1114.83	−0.651933
$144$	0	0
$145$	82.3084	0.0471403
$146$	0	0
$147$	−365.094	−0.204847
$148$	0	0
$149$	−966.737	−0.531532	−0.265766	−	0.964038i	$-0.585625\pi$
$149$	−966.737	−0.531532	−0.265766	+	0.964038i	$0.585625\pi$
$150$	0	0
$151$	2756.66	1.48566	0.742828	−	0.669482i	$-0.233484\pi$
$151$	2756.66	1.48566	0.742828	+	0.669482i	$0.233484\pi$
$152$	0	0
$153$	−2765.18	−1.46112
$154$	0	0
$155$	−1601.77	−0.830045
$156$	0	0
$157$	114.138	0.0580205	0.0290103	−	0.999579i	$-0.490764\pi$
$157$	114.138	0.0580205	0.0290103	+	0.999579i	$0.490764\pi$
$158$	0	0
$159$	−1703.20	−0.849512
$160$	0	0
$161$	−52.1137	−0.0255102
$162$	0	0
$163$	−2389.49	−1.14822	−0.574108	−	0.818780i	$-0.694651\pi$
$163$	−2389.49	−1.14822	−0.574108	+	0.818780i	$0.694651\pi$
$164$	0	0
$165$	850.893	0.401466
$166$	0	0
$167$	285.948	0.132499	0.0662495	−	0.997803i	$-0.478897\pi$
$167$	285.948	0.132499	0.0662495	+	0.997803i	$0.478897\pi$
$168$	0	0
$169$	3230.59	1.47045
$170$	0	0
$171$	−3097.31	−1.38513
$172$	0	0
$173$	−1246.35	−0.547735	−0.273867	−	0.961767i	$-0.588303\pi$
$173$	−1246.35	−0.547735	−0.273867	+	0.961767i	$0.588303\pi$
$174$	0	0
$175$	−476.320	−0.205751
$176$	0	0
$177$	4924.25	2.09113
$178$	0	0
$179$	3194.28	1.33381	0.666905	−	0.745143i	$-0.267619\pi$
$179$	3194.28	1.33381	0.666905	+	0.745143i	$0.267619\pi$
$180$	0	0
$181$	3996.41	1.64116	0.820582	−	0.571529i	$-0.193649\pi$
$181$	3996.41	1.64116	0.820582	+	0.571529i	$0.193649\pi$
$182$	0	0
$183$	607.124	0.245245
$184$	0	0
$185$	−2161.70	−0.859089
$186$	0	0
$187$	−1467.37	−0.573823
$188$	0	0
$189$	−79.0623	−0.0304282
$190$	0	0
$191$	3091.19	1.17105	0.585526	−	0.810654i	$-0.300888\pi$
$191$	3091.19	1.17105	0.585526	+	0.810654i	$0.300888\pi$
$192$	0	0
$193$	−87.5877	−0.0326668	−0.0163334	−	0.999867i	$-0.505199\pi$
$193$	−87.5877	−0.0326668	−0.0163334	+	0.999867i	$0.505199\pi$
$194$	0	0
$195$	−4142.62	−1.52133
$196$	0	0
$197$	−1893.85	−0.684931	−0.342466	−	0.939530i	$-0.611262\pi$
$197$	−1893.85	−0.684931	−0.342466	+	0.939530i	$0.611262\pi$
$198$	0	0
$199$	−2787.90	−0.993109	−0.496555	−	0.868005i	$-0.665402\pi$
$199$	−2787.90	−0.993109	−0.496555	+	0.868005i	$0.665402\pi$
$200$	0	0
$201$	−3390.04	−1.18963
$202$	0	0
$203$	−76.3448	−0.0263958
$204$	0	0
$205$	1476.36	0.502992
$206$	0	0
$207$	−212.295	−0.0712829
$208$	0	0
$209$	−1643.62	−0.543980
$210$	0	0
$211$	−5031.08	−1.64149	−0.820743	−	0.571298i	$-0.806440\pi$
$211$	−5031.08	−1.64149	−0.820743	+	0.571298i	$0.806440\pi$
$212$	0	0
$213$	2526.76	0.812821
$214$	0	0
$215$	−792.444	−0.251369
$216$	0	0
$217$	1485.71	0.464777
$218$	0	0
$219$	−5352.08	−1.65142
$220$	0	0
$221$	7143.98	2.17446
$222$	0	0
$223$	−1924.40	−0.577879	−0.288940	−	0.957347i	$-0.593303\pi$
$223$	−1924.40	−0.577879	−0.288940	+	0.957347i	$0.593303\pi$
$224$	0	0
$225$	−1940.38	−0.574929
$226$	0	0
$227$	2450.69	0.716556	0.358278	−	0.933615i	$-0.383364\pi$
$227$	2450.69	0.716556	0.358278	+	0.933615i	$0.383364\pi$
$228$	0	0
$229$	−4825.26	−1.39241	−0.696205	−	0.717843i	$-0.745130\pi$
$229$	−4825.26	−1.39241	−0.696205	+	0.717843i	$0.745130\pi$
$230$	0	0
$231$	−789.242	−0.224798
$232$	0	0
$233$	5702.82	1.60345	0.801725	−	0.597693i	$-0.203916\pi$
$233$	5702.82	1.60345	0.801725	+	0.597693i	$0.203916\pi$
$234$	0	0
$235$	−3622.59	−1.00558
$236$	0	0
$237$	6459.85	1.77052
$238$	0	0
$239$	−3394.11	−0.918605	−0.459303	−	0.888280i	$-0.651901\pi$
$239$	−3394.11	−0.918605	−0.459303	+	0.888280i	$0.651901\pi$
$240$	0	0
$241$	3968.61	1.06075	0.530374	−	0.847764i	$-0.322051\pi$
$241$	3968.61	1.06075	0.530374	+	0.847764i	$0.322051\pi$
$242$	0	0
$243$	5414.58	1.42941
$244$	0	0
$245$	−369.793	−0.0964295
$246$	0	0
$247$	8002.06	2.06137
$248$	0	0
$249$	−7576.32	−1.92823
$250$	0	0
$251$	603.807	0.151840	0.0759202	−	0.997114i	$-0.475811\pi$
$251$	603.807	0.151840	0.0759202	+	0.997114i	$0.475811\pi$
$252$	0	0
$253$	−112.657	−0.0279948
$254$	0	0
$255$	−5452.66	−1.33905
$256$	0	0
$257$	4591.44	1.11442	0.557210	−	0.830371i	$-0.311872\pi$
$257$	4591.44	1.11442	0.557210	+	0.830371i	$0.311872\pi$
$258$	0	0
$259$	2005.08	0.481040
$260$	0	0
$261$	−311.005	−0.0737577
$262$	0	0
$263$	−2057.20	−0.482328	−0.241164	−	0.970484i	$-0.577529\pi$
$263$	−2057.20	−0.482328	−0.241164	+	0.970484i	$0.577529\pi$
$264$	0	0
$265$	−1725.12	−0.399899
$266$	0	0
$267$	−239.615	−0.0549220
$268$	0	0
$269$	−3753.45	−0.850750	−0.425375	−	0.905017i	$-0.639858\pi$
$269$	−3753.45	−0.850750	−0.425375	+	0.905017i	$0.639858\pi$
$270$	0	0
$271$	1096.12	0.245698	0.122849	−	0.992425i	$-0.460797\pi$
$271$	1096.12	0.245698	0.122849	+	0.992425i	$0.460797\pi$
$272$	0	0
$273$	3842.47	0.851856
$274$	0	0
$275$	−1029.69	−0.225790
$276$	0	0
$277$	−2810.03	−0.609524	−0.304762	−	0.952429i	$-0.598577\pi$
$277$	−2810.03	−0.609524	−0.304762	+	0.952429i	$0.598577\pi$
$278$	0	0
$279$	6052.34	1.29872
$280$	0	0
$281$	−2706.71	−0.574623	−0.287311	−	0.957837i	$-0.592761\pi$
$281$	−2706.71	−0.574623	−0.287311	+	0.957837i	$0.592761\pi$
$282$	0	0
$283$	−1774.75	−0.372785	−0.186393	−	0.982475i	$-0.559680\pi$
$283$	−1774.75	−0.372785	−0.186393	+	0.982475i	$0.559680\pi$
$284$	0	0
$285$	−6107.59	−1.26941
$286$	0	0
$287$	−1369.39	−0.281647
$288$	0	0
$289$	4490.16	0.913934
$290$	0	0
$291$	10456.4	2.10640
$292$	0	0
$293$	−4903.80	−0.977759	−0.488879	−	0.872351i	$-0.662594\pi$
$293$	−4903.80	−0.977759	−0.488879	+	0.872351i	$0.662594\pi$
$294$	0	0
$295$	4987.63	0.984377
$296$	0	0
$297$	−170.913	−0.0333918
$298$	0	0
$299$	548.476	0.106084
$300$	0	0
$301$	735.028	0.140752
$302$	0	0
$303$	−12805.6	−2.42794
$304$	0	0
$305$	614.938	0.115447
$306$	0	0
$307$	−1129.74	−0.210025	−0.105012	−	0.994471i	$-0.533488\pi$
$307$	−1129.74	−0.210025	−0.105012	+	0.994471i	$0.533488\pi$
$308$	0	0
$309$	−12867.2	−2.36890
$310$	0	0
$311$	−1254.18	−0.228675	−0.114338	−	0.993442i	$-0.536475\pi$
$311$	−1254.18	−0.228675	−0.114338	+	0.993442i	$0.536475\pi$
$312$	0	0
$313$	−8335.43	−1.50526	−0.752630	−	0.658444i	$-0.771215\pi$
$313$	−8335.43	−1.50526	−0.752630	+	0.658444i	$0.771215\pi$
$314$	0	0
$315$	−1506.43	−0.269452
$316$	0	0
$317$	−133.701	−0.0236890	−0.0118445	−	0.999930i	$-0.503770\pi$
$317$	−133.701	−0.0236890	−0.0118445	+	0.999930i	$0.503770\pi$
$318$	0	0
$319$	−165.038	−0.0289667
$320$	0	0
$321$	−15502.9	−2.69560
$322$	0	0
$323$	10532.6	1.81439
$324$	0	0
$325$	5013.08	0.855617
$326$	0	0
$327$	−6189.87	−1.04679
$328$	0	0
$329$	3360.11	0.563067
$330$	0	0
$331$	3242.40	0.538423	0.269212	−	0.963081i	$-0.413237\pi$
$331$	3242.40	0.538423	0.269212	+	0.963081i	$0.413237\pi$
$332$	0	0
$333$	8168.07	1.34417
$334$	0	0
$335$	−3433.67	−0.560005
$336$	0	0
$337$	5500.02	0.889037	0.444518	−	0.895770i	$-0.353375\pi$
$337$	5500.02	0.889037	0.444518	+	0.895770i	$0.353375\pi$
$338$	0	0
$339$	7265.03	1.16396
$340$	0	0
$341$	3211.74	0.510045
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−418.625	−0.0653276
$346$	0	0
$347$	−7304.97	−1.13012	−0.565060	−	0.825050i	$-0.691147\pi$
$347$	−7304.97	−1.13012	−0.565060	+	0.825050i	$0.691147\pi$
$348$	0	0
$349$	−1856.57	−0.284756	−0.142378	−	0.989812i	$-0.545475\pi$
$349$	−1856.57	−0.284756	−0.142378	+	0.989812i	$0.545475\pi$
$350$	0	0
$351$	832.099	0.126536
$352$	0	0
$353$	−3768.83	−0.568257	−0.284128	−	0.958786i	$-0.591704\pi$
$353$	−3768.83	−0.568257	−0.284128	+	0.958786i	$0.591704\pi$
$354$	0	0
$355$	2559.29	0.382628
$356$	0	0
$357$	5057.59	0.749793
$358$	0	0
$359$	7413.61	1.08990	0.544952	−	0.838467i	$-0.316548\pi$
$359$	7413.61	1.08990	0.544952	+	0.838467i	$0.316548\pi$
$360$	0	0
$361$	4938.69	0.720031
$362$	0	0
$363$	8211.00	1.18723
$364$	0	0
$365$	−5420.97	−0.777388
$366$	0	0
$367$	12680.4	1.80357	0.901787	−	0.432180i	$-0.142256\pi$
$367$	12680.4	1.80357	0.901787	+	0.432180i	$0.142256\pi$
$368$	0	0
$369$	−5578.48	−0.787003
$370$	0	0
$371$	1600.13	0.223920
$372$	0	0
$373$	−656.291	−0.0911031	−0.0455515	−	0.998962i	$-0.514505\pi$
$373$	−656.291	−0.0911031	−0.0455515	+	0.998962i	$0.514505\pi$
$374$	0	0
$375$	−10855.1	−1.49481
$376$	0	0
$377$	803.498	0.109767
$378$	0	0
$379$	1506.08	0.204122	0.102061	−	0.994778i	$-0.467456\pi$
$379$	1506.08	0.204122	0.102061	+	0.994778i	$0.467456\pi$
$380$	0	0
$381$	8554.69	1.15031
$382$	0	0
$383$	5863.33	0.782251	0.391126	−	0.920337i	$-0.372086\pi$
$383$	5863.33	0.782251	0.391126	+	0.920337i	$0.372086\pi$
$384$	0	0
$385$	−799.401	−0.105821
$386$	0	0
$387$	2994.28	0.393302
$388$	0	0
$389$	7845.09	1.02252	0.511262	−	0.859425i	$-0.329178\pi$
$389$	7845.09	1.02252	0.511262	+	0.859425i	$0.329178\pi$
$390$	0	0
$391$	721.923	0.0933739
$392$	0	0
$393$	−4950.20	−0.635381
$394$	0	0
$395$	6543.00	0.833453
$396$	0	0
$397$	−5445.26	−0.688387	−0.344194	−	0.938899i	$-0.611848\pi$
$397$	−5445.26	−0.688387	−0.344194	+	0.938899i	$0.611848\pi$
$398$	0	0
$399$	5665.07	0.710797
$400$	0	0
$401$	214.249	0.0266810	0.0133405	−	0.999911i	$-0.495753\pi$
$401$	214.249	0.0266810	0.0133405	+	0.999911i	$0.495753\pi$
$402$	0	0
$403$	−15636.5	−1.93278
$404$	0	0
$405$	5175.40	0.634982
$406$	0	0
$407$	4334.48	0.527892
$408$	0	0
$409$	1057.44	0.127841	0.0639203	−	0.997955i	$-0.479640\pi$
$409$	1057.44	0.127841	0.0639203	+	0.997955i	$0.479640\pi$
$410$	0	0
$411$	−17247.1	−2.06991
$412$	0	0
$413$	−4626.25	−0.551194
$414$	0	0
$415$	−7673.84	−0.907697
$416$	0	0
$417$	21863.7	2.56755
$418$	0	0
$419$	12843.9	1.49753	0.748764	−	0.662836i	$-0.230647\pi$
$419$	12843.9	1.49753	0.748764	+	0.662836i	$0.230647\pi$
$420$	0	0
$421$	2078.04	0.240565	0.120282	−	0.992740i	$-0.461620\pi$
$421$	2078.04	0.240565	0.120282	+	0.992740i	$0.461620\pi$
$422$	0	0
$423$	13688.1	1.57337
$424$	0	0
$425$	6598.39	0.753103
$426$	0	0
$427$	−570.383	−0.0646435
$428$	0	0
$429$	8306.45	0.934823
$430$	0	0
$431$	11388.4	1.27276	0.636382	−	0.771374i	$-0.280430\pi$
$431$	11388.4	1.27276	0.636382	+	0.771374i	$0.280430\pi$
$432$	0	0
$433$	−730.090	−0.0810298	−0.0405149	−	0.999179i	$-0.512900\pi$
$433$	−730.090	−0.0810298	−0.0405149	+	0.999179i	$0.512900\pi$
$434$	0	0
$435$	−613.271	−0.0675957
$436$	0	0
$437$	808.635	0.0885177
$438$	0	0
$439$	4852.78	0.527587	0.263793	−	0.964579i	$-0.415026\pi$
$439$	4852.78	0.527587	0.263793	+	0.964579i	$0.415026\pi$
$440$	0	0
$441$	1397.28	0.150878
$442$	0	0
$443$	17688.6	1.89709	0.948546	−	0.316638i	$-0.102554\pi$
$443$	17688.6	1.89709	0.948546	+	0.316638i	$0.102554\pi$
$444$	0	0
$445$	−242.699	−0.0258540
$446$	0	0
$447$	7203.06	0.762177
$448$	0	0
$449$	−9642.16	−1.01346	−0.506728	−	0.862106i	$-0.669145\pi$
$449$	−9642.16	−1.01346	−0.506728	+	0.862106i	$0.669145\pi$
$450$	0	0
$451$	−2960.28	−0.309078
$452$	0	0
$453$	−20539.6	−2.13032
$454$	0	0
$455$	3891.92	0.401003
$456$	0	0
$457$	16763.3	1.71588	0.857939	−	0.513751i	$-0.171745\pi$
$457$	16763.3	1.71588	0.857939	+	0.513751i	$0.171745\pi$
$458$	0	0
$459$	1095.24	0.111375
$460$	0	0
$461$	−4026.48	−0.406793	−0.203397	−	0.979096i	$-0.565198\pi$
$461$	−4026.48	−0.406793	−0.203397	+	0.979096i	$0.565198\pi$
$462$	0	0
$463$	−6303.00	−0.632668	−0.316334	−	0.948648i	$-0.602452\pi$
$463$	−6303.00	−0.632668	−0.316334	+	0.948648i	$0.602452\pi$
$464$	0	0
$465$	11934.6	1.19022
$466$	0	0
$467$	1033.80	0.102438	0.0512191	−	0.998687i	$-0.483689\pi$
$467$	1033.80	0.102438	0.0512191	+	0.998687i	$0.483689\pi$
$468$	0	0
$469$	3184.89	0.313570
$470$	0	0
$471$	−850.432	−0.0831972
$472$	0	0
$473$	1588.95	0.154460
$474$	0	0
$475$	7390.94	0.713936
$476$	0	0
$477$	6518.43	0.625699
$478$	0	0
$479$	14356.8	1.36948	0.684739	−	0.728788i	$-0.259916\pi$
$479$	14356.8	1.36948	0.684739	+	0.728788i	$0.259916\pi$
$480$	0	0
$481$	−21102.6	−2.00041
$482$	0	0
$483$	388.294	0.0365797
$484$	0	0
$485$	10590.9	0.991567
$486$	0	0
$487$	2205.48	0.205215	0.102608	−	0.994722i	$-0.467281\pi$
$487$	2205.48	0.205215	0.102608	+	0.994722i	$0.467281\pi$
$488$	0	0
$489$	17803.8	1.64646
$490$	0	0
$491$	2313.81	0.212670	0.106335	−	0.994330i	$-0.466088\pi$
$491$	2313.81	0.212670	0.106335	+	0.994330i	$0.466088\pi$
$492$	0	0
$493$	1057.59	0.0966157
$494$	0	0
$495$	−3256.52	−0.295696
$496$	0	0
$497$	−2373.85	−0.214249
$498$	0	0
$499$	−11276.4	−1.01163	−0.505813	−	0.862643i	$-0.668807\pi$
$499$	−11276.4	−1.01163	−0.505813	+	0.862643i	$0.668807\pi$
$500$	0	0
$501$	−2130.57	−0.189994
$502$	0	0
$503$	11781.3	1.04434	0.522171	−	0.852841i	$-0.325122\pi$
$503$	11781.3	1.04434	0.522171	+	0.852841i	$0.325122\pi$
$504$	0	0
$505$	−12970.5	−1.14293
$506$	0	0
$507$	−24070.8	−2.10852
$508$	0	0
$509$	−20887.6	−1.81891	−0.909456	−	0.415799i	$-0.863502\pi$
$509$	−20887.6	−1.81891	−0.909456	+	0.415799i	$0.863502\pi$
$510$	0	0
$511$	5028.19	0.435292
$512$	0	0
$513$	1226.79	0.105583
$514$	0	0
$515$	−13032.9	−1.11514
$516$	0	0
$517$	7263.72	0.617907
$518$	0	0
$519$	9286.42	0.785411
$520$	0	0
$521$	9372.49	0.788131	0.394065	−	0.919082i	$-0.371068\pi$
$521$	9372.49	0.788131	0.394065	+	0.919082i	$0.371068\pi$
$522$	0	0
$523$	−14959.8	−1.25076	−0.625378	−	0.780322i	$-0.715055\pi$
$523$	−14959.8	−1.25076	−0.625378	+	0.780322i	$0.715055\pi$
$524$	0	0
$525$	3549.01	0.295032
$526$	0	0
$527$	−20581.3	−1.70121
$528$	0	0
$529$	−12111.6	−0.995445
$530$	0	0
$531$	−18845.9	−1.54020
$532$	0	0
$533$	14412.3	1.17123
$534$	0	0
$535$	−15702.4	−1.26893
$536$	0	0
$537$	−23800.3	−1.91259
$538$	0	0
$539$	741.480	0.0592538
$540$	0	0
$541$	−7226.04	−0.574254	−0.287127	−	0.957892i	$-0.592700\pi$
$541$	−7226.04	−0.574254	−0.287127	+	0.957892i	$0.592700\pi$
$542$	0	0
$543$	−29776.8	−2.35331
$544$	0	0
$545$	−6269.54	−0.492766
$546$	0	0
$547$	−2760.25	−0.215759	−0.107879	−	0.994164i	$-0.534406\pi$
$547$	−2760.25	−0.215759	−0.107879	+	0.994164i	$0.534406\pi$
$548$	0	0
$549$	−2323.57	−0.180633
$550$	0	0
$551$	1184.62	0.0915909
$552$	0	0
$553$	−6068.93	−0.466685
$554$	0	0
$555$	16106.6	1.23187
$556$	0	0
$557$	3507.15	0.266791	0.133396	−	0.991063i	$-0.457412\pi$
$557$	3507.15	0.266791	0.133396	+	0.991063i	$0.457412\pi$
$558$	0	0
$559$	−7735.87	−0.585317
$560$	0	0
$561$	10933.2	0.822820
$562$	0	0
$563$	21375.4	1.60011	0.800057	−	0.599924i	$-0.204802\pi$
$563$	21375.4	1.60011	0.800057	+	0.599924i	$0.204802\pi$
$564$	0	0
$565$	7358.55	0.547923
$566$	0	0
$567$	−4800.41	−0.355553
$568$	0	0
$569$	−19816.3	−1.46001	−0.730003	−	0.683444i	$-0.760482\pi$
$569$	−19816.3	−1.46001	−0.730003	+	0.683444i	$0.760482\pi$
$570$	0	0
$571$	−21798.6	−1.59762	−0.798811	−	0.601582i	$-0.794537\pi$
$571$	−21798.6	−1.59762	−0.798811	+	0.601582i	$0.794537\pi$
$572$	0	0
$573$	−23032.2	−1.67920
$574$	0	0
$575$	506.588	0.0367412
$576$	0	0
$577$	−12782.0	−0.922223	−0.461112	−	0.887342i	$-0.652549\pi$
$577$	−12782.0	−0.922223	−0.461112	+	0.887342i	$0.652549\pi$
$578$	0	0
$579$	652.607	0.0468418
$580$	0	0
$581$	7117.84	0.508257
$582$	0	0
$583$	3459.08	0.245730
$584$	0	0
$585$	15854.5	1.12052
$586$	0	0
$587$	24178.8	1.70011	0.850057	−	0.526691i	$-0.176568\pi$
$587$	24178.8	1.70011	0.850057	+	0.526691i	$0.176568\pi$
$588$	0	0
$589$	−23053.4	−1.61273
$590$	0	0
$591$	14110.9	0.982141
$592$	0	0
$593$	−23640.2	−1.63708	−0.818540	−	0.574449i	$-0.805216\pi$
$593$	−23640.2	−1.63708	−0.818540	+	0.574449i	$0.805216\pi$
$594$	0	0
$595$	5122.69	0.352958
$596$	0	0
$597$	20772.3	1.42405
$598$	0	0
$599$	802.585	0.0547458	0.0273729	−	0.999625i	$-0.491286\pi$
$599$	802.585	0.0547458	0.0273729	+	0.999625i	$0.491286\pi$
$600$	0	0
$601$	9328.65	0.633151	0.316575	−	0.948567i	$-0.397467\pi$
$601$	9328.65	0.633151	0.316575	+	0.948567i	$0.397467\pi$
$602$	0	0
$603$	12974.3	0.876207
$604$	0	0
$605$	8316.69	0.558879
$606$	0	0
$607$	4490.94	0.300300	0.150150	−	0.988663i	$-0.452024\pi$
$607$	4490.94	0.300300	0.150150	+	0.988663i	$0.452024\pi$
$608$	0	0
$609$	568.837	0.0378496
$610$	0	0
$611$	−35363.8	−2.34152
$612$	0	0
$613$	−4957.18	−0.326621	−0.163311	−	0.986575i	$-0.552217\pi$
$613$	−4957.18	−0.326621	−0.163311	+	0.986575i	$0.552217\pi$
$614$	0	0
$615$	−11000.2	−0.721254
$616$	0	0
$617$	−3661.95	−0.238938	−0.119469	−	0.992838i	$-0.538119\pi$
$617$	−3661.95	−0.238938	−0.119469	+	0.992838i	$0.538119\pi$
$618$	0	0
$619$	7709.34	0.500589	0.250295	−	0.968170i	$-0.419473\pi$
$619$	7709.34	0.500589	0.250295	+	0.968170i	$0.419473\pi$
$620$	0	0
$621$	84.0863	0.00543360
$622$	0	0
$623$	225.114	0.0144767
$624$	0	0
$625$	−2489.05	−0.159299
$626$	0	0
$627$	12246.5	0.780027
$628$	0	0
$629$	−27776.0	−1.76073
$630$	0	0
$631$	−19351.0	−1.22084	−0.610422	−	0.792076i	$-0.709000\pi$
$631$	−19351.0	−1.22084	−0.610422	+	0.792076i	$0.709000\pi$
$632$	0	0
$633$	37486.0	2.35377
$634$	0	0
$635$	8664.80	0.541499
$636$	0	0
$637$	−3609.94	−0.224538
$638$	0	0
$639$	−9670.36	−0.598675
$640$	0	0
$641$	20042.5	1.23500	0.617498	−	0.786573i	$-0.288146\pi$
$641$	20042.5	1.23500	0.617498	+	0.786573i	$0.288146\pi$
$642$	0	0
$643$	8072.34	0.495088	0.247544	−	0.968877i	$-0.420376\pi$
$643$	8072.34	0.495088	0.247544	+	0.968877i	$0.420376\pi$
$644$	0	0
$645$	5904.42	0.360444
$646$	0	0
$647$	−25597.0	−1.55537	−0.777683	−	0.628656i	$-0.783605\pi$
$647$	−25597.0	−1.55537	−0.777683	+	0.628656i	$0.783605\pi$
$648$	0	0
$649$	−10000.8	−0.604878
$650$	0	0
$651$	−11069.9	−0.666456
$652$	0	0
$653$	1891.03	0.113326	0.0566630	−	0.998393i	$-0.481954\pi$
$653$	1891.03	0.113326	0.0566630	+	0.998393i	$0.481954\pi$
$654$	0	0
$655$	−5013.92	−0.299099
$656$	0	0
$657$	20483.3	1.21633
$658$	0	0
$659$	−5846.26	−0.345581	−0.172791	−	0.984959i	$-0.555278\pi$
$659$	−5846.26	−0.345581	−0.172791	+	0.984959i	$0.555278\pi$
$660$	0	0
$661$	−22136.1	−1.30256	−0.651281	−	0.758836i	$-0.725768\pi$
$661$	−22136.1	−1.30256	−0.651281	+	0.758836i	$0.725768\pi$
$662$	0	0
$663$	−53229.1	−3.11802
$664$	0	0
$665$	5737.99	0.334601
$666$	0	0
$667$	81.1961	0.00471353
$668$	0	0
$669$	14338.5	0.828636
$670$	0	0
$671$	−1233.03	−0.0709395
$672$	0	0
$673$	−10101.4	−0.578576	−0.289288	−	0.957242i	$-0.593418\pi$
$673$	−10101.4	−0.578576	−0.289288	+	0.957242i	$0.593418\pi$
$674$	0	0
$675$	768.551	0.0438245
$676$	0	0
$677$	−8764.44	−0.497555	−0.248778	−	0.968561i	$-0.580029\pi$
$677$	−8764.44	−0.497555	−0.248778	+	0.968561i	$0.580029\pi$
$678$	0	0
$679$	−9823.58	−0.555220
$680$	0	0
$681$	−18259.9	−1.02749
$682$	0	0
$683$	16379.3	0.917621	0.458811	−	0.888534i	$-0.348276\pi$
$683$	16379.3	0.917621	0.458811	+	0.888534i	$0.348276\pi$
$684$	0	0
$685$	−17469.1	−0.974392
$686$	0	0
$687$	35952.5	1.99661
$688$	0	0
$689$	−16840.7	−0.931175
$690$	0	0
$691$	416.824	0.0229475	0.0114738	−	0.999934i	$-0.496348\pi$
$691$	416.824	0.0229475	0.0114738	+	0.999934i	$0.496348\pi$
$692$	0	0
$693$	3020.57	0.165573
$694$	0	0
$695$	22145.1	1.20865
$696$	0	0
$697$	18969.9	1.03090
$698$	0	0
$699$	−42491.1	−2.29923
$700$	0	0
$701$	8321.81	0.448374	0.224187	−	0.974546i	$-0.428027\pi$
$701$	8321.81	0.448374	0.224187	+	0.974546i	$0.428027\pi$
$702$	0	0
$703$	−31112.3	−1.66916
$704$	0	0
$705$	26991.5	1.44193
$706$	0	0
$707$	12030.7	0.639973
$708$	0	0
$709$	30477.0	1.61437	0.807184	−	0.590300i	$-0.200991\pi$
$709$	30477.0	1.61437	0.807184	+	0.590300i	$0.200991\pi$
$710$	0	0
$711$	−24723.0	−1.30406
$712$	0	0
$713$	−1580.12	−0.0829958
$714$	0	0
$715$	8413.37	0.440059
$716$	0	0
$717$	25289.2	1.31721
$718$	0	0
$719$	22261.8	1.15470	0.577348	−	0.816498i	$-0.304088\pi$
$719$	22261.8	1.15470	0.577348	+	0.816498i	$0.304088\pi$
$720$	0	0
$721$	12088.6	0.624413
$722$	0	0
$723$	−29569.7	−1.52103
$724$	0	0
$725$	742.134	0.0380168
$726$	0	0
$727$	7201.69	0.367395	0.183697	−	0.982983i	$-0.441193\pi$
$727$	7201.69	0.367395	0.183697	+	0.982983i	$0.441193\pi$
$728$	0	0
$729$	−21827.6	−1.10896
$730$	0	0
$731$	−10182.2	−0.515189
$732$	0	0
$733$	37843.4	1.90693	0.953464	−	0.301507i	$-0.0974896\pi$
$733$	37843.4	1.90693	0.953464	+	0.301507i	$0.0974896\pi$
$734$	0	0
$735$	2755.29	0.138273
$736$	0	0
$737$	6884.93	0.344111
$738$	0	0
$739$	17733.5	0.882730	0.441365	−	0.897328i	$-0.354494\pi$
$739$	17733.5	0.882730	0.441365	+	0.897328i	$0.354494\pi$
$740$	0	0
$741$	−59622.5	−2.95586
$742$	0	0
$743$	−18609.0	−0.918840	−0.459420	−	0.888219i	$-0.651943\pi$
$743$	−18609.0	−0.918840	−0.459420	+	0.888219i	$0.651943\pi$
$744$	0	0
$745$	7295.78	0.358787
$746$	0	0
$747$	28995.9	1.42022
$748$	0	0
$749$	14564.7	0.710525
$750$	0	0
$751$	16843.0	0.818389	0.409195	−	0.912447i	$-0.365810\pi$
$751$	16843.0	0.818389	0.409195	+	0.912447i	$0.365810\pi$
$752$	0	0
$753$	−4498.90	−0.217728
$754$	0	0
$755$	−20804.0	−1.00283
$756$	0	0
$757$	32305.6	1.55108	0.775541	−	0.631298i	$-0.217477\pi$
$757$	32305.6	1.55108	0.775541	+	0.631298i	$0.217477\pi$
$758$	0	0
$759$	839.394	0.0401424
$760$	0	0
$761$	3490.47	0.166267	0.0831336	−	0.996538i	$-0.473507\pi$
$761$	3490.47	0.166267	0.0831336	+	0.996538i	$0.473507\pi$
$762$	0	0
$763$	5815.28	0.275921
$764$	0	0
$765$	20868.3	0.986267
$766$	0	0
$767$	48689.4	2.29214
$768$	0	0
$769$	5147.40	0.241378	0.120689	−	0.992690i	$-0.461490\pi$
$769$	5147.40	0.241378	0.120689	+	0.992690i	$0.461490\pi$
$770$	0	0
$771$	−34210.3	−1.59800
$772$	0	0
$773$	11176.9	0.520057	0.260028	−	0.965601i	$-0.416268\pi$
$773$	11176.9	0.520057	0.260028	+	0.965601i	$0.416268\pi$
$774$	0	0
$775$	−14442.3	−0.669399
$776$	0	0
$777$	−14939.6	−0.689776
$778$	0	0
$779$	21248.5	0.977286
$780$	0	0
$781$	−5131.68	−0.235116
$782$	0	0
$783$	123.184	0.00562225
$784$	0	0
$785$	−861.379	−0.0391642
$786$	0	0
$787$	−599.161	−0.0271382	−0.0135691	−	0.999908i	$-0.504319\pi$
$787$	−599.161	−0.0271382	−0.0135691	+	0.999908i	$0.504319\pi$
$788$	0	0
$789$	15328.0	0.691622
$790$	0	0
$791$	−6825.38	−0.306805
$792$	0	0
$793$	6003.05	0.268820
$794$	0	0
$795$	12853.7	0.573426
$796$	0	0
$797$	22245.8	0.988690	0.494345	−	0.869266i	$-0.335408\pi$
$797$	22245.8	0.988690	0.494345	+	0.869266i	$0.335408\pi$
$798$	0	0
$799$	−46547.1	−2.06097
$800$	0	0
$801$	917.047	0.0404523
$802$	0	0
$803$	10869.7	0.477688
$804$	0	0
$805$	393.292	0.0172195
$806$	0	0
$807$	27966.6	1.21991
$808$	0	0
$809$	−11818.2	−0.513605	−0.256803	−	0.966464i	$-0.582669\pi$
$809$	−11818.2	−0.513605	−0.256803	+	0.966464i	$0.582669\pi$
$810$	0	0
$811$	10661.5	0.461623	0.230811	−	0.972999i	$-0.425862\pi$
$811$	10661.5	0.461623	0.230811	+	0.972999i	$0.425862\pi$
$812$	0	0
$813$	−8167.04	−0.352313
$814$	0	0
$815$	18033.0	0.775053
$816$	0	0
$817$	−11405.2	−0.488395
$818$	0	0
$819$	−14705.8	−0.627426
$820$	0	0
$821$	−32522.0	−1.38249	−0.691246	−	0.722620i	$-0.742938\pi$
$821$	−32522.0	−1.38249	−0.691246	+	0.722620i	$0.742938\pi$
$822$	0	0
$823$	737.978	0.0312567	0.0156284	−	0.999878i	$-0.495025\pi$
$823$	737.978	0.0312567	0.0156284	+	0.999878i	$0.495025\pi$
$824$	0	0
$825$	7672.08	0.323767
$826$	0	0
$827$	−36205.4	−1.52235	−0.761177	−	0.648544i	$-0.775378\pi$
$827$	−36205.4	−1.52235	−0.761177	+	0.648544i	$0.775378\pi$
$828$	0	0
$829$	−29545.1	−1.23781	−0.618906	−	0.785465i	$-0.712424\pi$
$829$	−29545.1	−1.23781	−0.618906	+	0.785465i	$0.712424\pi$
$830$	0	0
$831$	20937.2	0.874012
$832$	0	0
$833$	−4751.52	−0.197636
$834$	0	0
$835$	−2157.99	−0.0894377
$836$	0	0
$837$	−2397.22	−0.0989965
$838$	0	0
$839$	18067.3	0.743446	0.371723	−	0.928344i	$-0.378767\pi$
$839$	18067.3	0.743446	0.371723	+	0.928344i	$0.378767\pi$
$840$	0	0
$841$	−24270.1	−0.995123
$842$	0	0
$843$	20167.4	0.823967
$844$	0	0
$845$	−24380.6	−0.992566
$846$	0	0
$847$	−7714.10	−0.312940
$848$	0	0
$849$	13223.5	0.534546
$850$	0	0
$851$	−2132.49	−0.0858999
$852$	0	0
$853$	−871.541	−0.0349836	−0.0174918	−	0.999847i	$-0.505568\pi$
$853$	−871.541	−0.0349836	−0.0174918	+	0.999847i	$0.505568\pi$
$854$	0	0
$855$	23374.8	0.934973
$856$	0	0
$857$	8416.92	0.335492	0.167746	−	0.985830i	$-0.446351\pi$
$857$	8416.92	0.335492	0.167746	+	0.985830i	$0.446351\pi$
$858$	0	0
$859$	−14529.1	−0.577097	−0.288548	−	0.957465i	$-0.593173\pi$
$859$	−14529.1	−0.577097	−0.288548	+	0.957465i	$0.593173\pi$
$860$	0	0
$861$	10203.2	0.403860
$862$	0	0
$863$	39811.3	1.57033	0.785164	−	0.619287i	$-0.212578\pi$
$863$	39811.3	1.57033	0.785164	+	0.619287i	$0.212578\pi$
$864$	0	0
$865$	9405.95	0.369725
$866$	0	0
$867$	−33455.7	−1.31051
$868$	0	0
$869$	−13119.5	−0.512139
$870$	0	0
$871$	−33519.6	−1.30398
$872$	0	0
$873$	−40018.3	−1.55145
$874$	0	0
$875$	10198.1	0.394012
$876$	0	0
$877$	18605.5	0.716378	0.358189	−	0.933649i	$-0.383394\pi$
$877$	18605.5	0.716378	0.358189	+	0.933649i	$0.383394\pi$
$878$	0	0
$879$	36537.7	1.40203
$880$	0	0
$881$	24296.6	0.929143	0.464571	−	0.885536i	$-0.346208\pi$
$881$	24296.6	0.929143	0.464571	+	0.885536i	$0.346208\pi$
$882$	0	0
$883$	−14183.7	−0.540565	−0.270282	−	0.962781i	$-0.587117\pi$
$883$	−14183.7	−0.540565	−0.270282	+	0.962781i	$0.587117\pi$
$884$	0	0
$885$	−37162.3	−1.41152
$886$	0	0
$887$	46082.8	1.74443	0.872214	−	0.489125i	$-0.162684\pi$
$887$	46082.8	1.74443	0.872214	+	0.489125i	$0.162684\pi$
$888$	0	0
$889$	−8036.99	−0.303208
$890$	0	0
$891$	−10377.3	−0.390182
$892$	0	0
$893$	−52138.0	−1.95379
$894$	0	0
$895$	−24106.6	−0.900331
$896$	0	0
$897$	−4086.64	−0.152117
$898$	0	0
$899$	−2314.82	−0.0858772
$900$	0	0
$901$	−22166.3	−0.819608
$902$	0	0
$903$	−5476.61	−0.201828
$904$	0	0
$905$	−30160.1	−1.10780
$906$	0	0
$907$	19138.8	0.700656	0.350328	−	0.936627i	$-0.386070\pi$
$907$	19138.8	0.700656	0.350328	+	0.936627i	$0.386070\pi$
$908$	0	0
$909$	49009.4	1.78827
$910$	0	0
$911$	24817.2	0.902559	0.451280	−	0.892383i	$-0.350968\pi$
$911$	24817.2	0.902559	0.451280	+	0.892383i	$0.350968\pi$
$912$	0	0
$913$	15387.0	0.557760
$914$	0	0
$915$	−4581.84	−0.165542
$916$	0	0
$917$	4650.64	0.167478
$918$	0	0
$919$	−6856.74	−0.246119	−0.123059	−	0.992399i	$-0.539271\pi$
$919$	−6856.74	−0.246119	−0.123059	+	0.992399i	$0.539271\pi$
$920$	0	0
$921$	8417.57	0.301160
$922$	0	0
$923$	24983.8	0.890957
$924$	0	0
$925$	−19491.0	−0.692822
$926$	0	0
$927$	49245.1	1.74479
$928$	0	0
$929$	22938.1	0.810091	0.405046	−	0.914296i	$-0.367256\pi$
$929$	22938.1	0.810091	0.405046	+	0.914296i	$0.367256\pi$
$930$	0	0
$931$	−5322.24	−0.187357
$932$	0	0
$933$	9344.76	0.327903
$934$	0	0
$935$	11074.0	0.387334
$936$	0	0
$937$	20977.0	0.731363	0.365682	−	0.930740i	$-0.380836\pi$
$937$	20977.0	0.731363	0.365682	+	0.930740i	$0.380836\pi$
$938$	0	0
$939$	62106.4	2.15843
$940$	0	0
$941$	−13309.6	−0.461086	−0.230543	−	0.973062i	$-0.574050\pi$
$941$	−13309.6	−0.461086	−0.230543	+	0.973062i	$0.574050\pi$
$942$	0	0
$943$	1456.41	0.0502939
$944$	0	0
$945$	596.668	0.0205393
$946$	0	0
$947$	7191.52	0.246772	0.123386	−	0.992359i	$-0.460625\pi$
$947$	7191.52	0.246772	0.123386	+	0.992359i	$0.460625\pi$
$948$	0	0
$949$	−52919.7	−1.81016
$950$	0	0
$951$	996.195	0.0339683
$952$	0	0
$953$	30131.4	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$953$	30131.4	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$954$	0	0
$955$	−23328.6	−0.790467
$956$	0	0
$957$	1229.68	0.0415361
$958$	0	0
$959$	16203.3	0.545603
$960$	0	0
$961$	15256.7	0.512125
$962$	0	0
$963$	59332.2	1.98542
$964$	0	0
$965$	661.007	0.0220503
$966$	0	0
$967$	37316.4	1.24097	0.620483	−	0.784220i	$-0.286936\pi$
$967$	37316.4	1.24097	0.620483	+	0.784220i	$0.286936\pi$
$968$	0	0
$969$	−78477.3	−2.60171
$970$	0	0
$971$	17057.8	0.563760	0.281880	−	0.959450i	$-0.409042\pi$
$971$	17057.8	0.563760	0.281880	+	0.959450i	$0.409042\pi$
$972$	0	0
$973$	−20540.6	−0.676774
$974$	0	0
$975$	−37351.9	−1.22689
$976$	0	0
$977$	−45294.1	−1.48320	−0.741600	−	0.670843i	$-0.765932\pi$
$977$	−45294.1	−1.48320	−0.741600	+	0.670843i	$0.765932\pi$
$978$	0	0
$979$	486.641	0.0158867
$980$	0	0
$981$	23689.7	0.771003
$982$	0	0
$983$	−24614.5	−0.798657	−0.399329	−	0.916808i	$-0.630757\pi$
$983$	−24614.5	−0.798657	−0.399329	+	0.916808i	$0.630757\pi$
$984$	0	0
$985$	14292.5	0.462333
$986$	0	0
$987$	−25035.8	−0.807396
$988$	0	0
$989$	−781.735	−0.0251342
$990$	0	0
$991$	35590.1	1.14082	0.570412	−	0.821359i	$-0.306783\pi$
$991$	35590.1	1.14082	0.570412	+	0.821359i	$0.306783\pi$
$992$	0	0
$993$	−24158.8	−0.772059
$994$	0	0
$995$	21039.7	0.670355
$996$	0	0
$997$	15139.0	0.480900	0.240450	−	0.970661i	$-0.422705\pi$
$997$	15139.0	0.480900	0.240450	+	0.970661i	$0.422705\pi$
$998$	0	0
$999$	−3235.22	−0.102460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.w.1.1		3
4.3	odd	2		448.4.a.v.1.3		3
8.3	odd	2		224.4.a.f.1.1	✓	3
8.5	even	2		224.4.a.g.1.3	yes	3
24.5	odd	2		2016.4.a.x.1.1		3
24.11	even	2		2016.4.a.w.1.1		3
56.13	odd	2		1568.4.a.x.1.1		3
56.27	even	2		1568.4.a.w.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.f.1.1	✓	3	8.3	odd	2
224.4.a.g.1.3	yes	3	8.5	even	2
448.4.a.v.1.3		3	4.3	odd	2
448.4.a.w.1.1		3	1.1	even	1	trivial
1568.4.a.w.1.3		3	56.27	even	2
1568.4.a.x.1.1		3	56.13	odd	2
2016.4.a.w.1.1		3	24.11	even	2
2016.4.a.x.1.1		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)

\(f(q)\)	\(=\)	\(q-7.45090 q^{3} -7.54680 q^{5} +7.00000 q^{7} +28.5159 q^{9} +O(q^{10})\)	\(q-7.45090 q^{3} -7.54680 q^{5} +7.00000 q^{7} +28.5159 q^{9} +15.1323 q^{11} -73.6722 q^{13} +56.2305 q^{15} -96.9699 q^{17} -108.617 q^{19} -52.1563 q^{21} -7.44482 q^{23} -68.0458 q^{25} -11.2946 q^{27} -10.9064 q^{29} +212.244 q^{31} -112.749 q^{33} -52.8276 q^{35} +286.440 q^{37} +548.924 q^{39} -195.627 q^{41} +105.004 q^{43} -215.204 q^{45} +480.016 q^{47} +49.0000 q^{49} +722.513 q^{51} +228.590 q^{53} -114.200 q^{55} +809.296 q^{57} -660.893 q^{59} -81.4833 q^{61} +199.611 q^{63} +555.989 q^{65} +454.984 q^{67} +55.4706 q^{69} -339.122 q^{71} +718.313 q^{73} +507.002 q^{75} +105.926 q^{77} -866.989 q^{79} -685.774 q^{81} +1016.83 q^{83} +731.813 q^{85} +81.2624 q^{87} +32.1592 q^{89} -515.705 q^{91} -1581.41 q^{93} +819.713 q^{95} -1403.37 q^{97} +431.509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9}+O(q^{10}) \) 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9	\( 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9} + 6 q^{13} + 168 q^{15} - 66 q^{17} - 168 q^{19} + 336 q^{23} + 69 q^{25} - 168 q^{27} - 90 q^{29} + 504 q^{31} + 120 q^{33} + 42 q^{35} - 18 q^{37} + 840 q^{39} - 450 q^{41} + 150 q^{45} + 504 q^{47} + 147 q^{49} + 336 q^{51} + 78 q^{53} + 1176 q^{55} + 48 q^{57} + 504 q^{59} - 498 q^{61} + 105 q^{63} + 1068 q^{65} + 1008 q^{67} + 1224 q^{69} + 504 q^{71} - 234 q^{73} + 1848 q^{75} + 168 q^{79} - 981 q^{81} + 3024 q^{83} - 1476 q^{85} - 336 q^{87} + 246 q^{89} + 42 q^{91} - 1200 q^{93} - 2184 q^{95} - 2514 q^{97} + 2688 q^{99}+O(q^{100}) \) 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9 + 6 * q^13 + 168 * q^15 - 66 * q^17 - 168 * q^19 + 336 * q^23 + 69 * q^25 - 168 * q^27 - 90 * q^29 + 504 * q^31 + 120 * q^33 + 42 * q^35 - 18 * q^37 + 840 * q^39 - 450 * q^41 + 150 * q^45 + 504 * q^47 + 147 * q^49 + 336 * q^51 + 78 * q^53 + 1176 * q^55 + 48 * q^57 + 504 * q^59 - 498 * q^61 + 105 * q^63 + 1068 * q^65 + 1008 * q^67 + 1224 * q^69 + 504 * q^71 - 234 * q^73 + 1848 * q^75 + 168 * q^79 - 981 * q^81 + 3024 * q^83 - 1476 * q^85 - 336 * q^87 + 246 * q^89 + 42 * q^91 - 1200 * q^93 - 2184 * q^95 - 2514 * q^97 + 2688 * q^99

Introduction

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