LMFDB - Embedded newform 1600.4.a.cw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2106005.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 24x^{2} + 25x - 6 $ x^4 - 2x^3 - 24x^2 + 25*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 800)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.54854$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.0971 q^{3} -10.5923 q^{7} +74.9510 q^{9} +O(q^{10})$	$q+10.0971 q^{3} -10.5923 q^{7} +74.9510 q^{9} -38.9027 q^{11} -68.9510 q^{13} -65.9510 q^{17} +49.4950 q^{19} -106.951 q^{21} -164.524 q^{23} +484.164 q^{27} +170.853 q^{29} +166.505 q^{31} -392.804 q^{33} -384.853 q^{37} -696.203 q^{39} -22.8038 q^{41} -136.709 q^{43} -307.562 q^{47} -230.804 q^{49} -665.912 q^{51} +222.000 q^{53} +499.755 q^{57} -522.765 q^{59} -393.049 q^{61} -793.901 q^{63} +476.845 q^{67} -1661.22 q^{69} -4.26352 q^{71} +601.166 q^{73} +412.068 q^{77} -1075.93 q^{79} +2864.97 q^{81} -1132.74 q^{83} +1725.11 q^{87} +479.755 q^{89} +730.347 q^{91} +1681.22 q^{93} -635.510 q^{97} -2915.80 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 96 q^{9}+O(q^{10}) $ 4 * q + 96 * q^9	$ 4 q + 96 q^{9} - 72 q^{13} - 60 q^{17} - 224 q^{21} + 72 q^{29} - 756 q^{33} - 928 q^{37} + 724 q^{41} - 108 q^{49} + 888 q^{53} + 980 q^{57} - 1776 q^{61} - 3384 q^{69} - 1060 q^{73} - 2224 q^{77} + 7180 q^{81} + 900 q^{89} + 3464 q^{93} - 504 q^{97}+O(q^{100}) $ 4 * q + 96 * q^9 - 72 * q^13 - 60 * q^17 - 224 * q^21 + 72 * q^29 - 756 * q^33 - 928 * q^37 + 724 * q^41 - 108 * q^49 + 888 * q^53 + 980 * q^57 - 1776 * q^61 - 3384 * q^69 - 1060 * q^73 - 2224 * q^77 + 7180 * q^81 + 900 * q^89 + 3464 * q^93 - 504 * q^97

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
$3$	10.0971	1.94318	0.971592	−	0.236664i	$-0.0760539\pi$
$3$	10.0971	1.94318	0.971592	+	0.236664i	$0.0760539\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−10.5923	−0.571929	−0.285964	−	0.958240i	$-0.592314\pi$
$7$	−10.5923	−0.571929	−0.285964	+	0.958240i	$0.592314\pi$
$8$	0	0
$9$	74.9510	2.77596
$10$	0	0
$11$	−38.9027	−1.06633	−0.533164	−	0.846012i	$-0.678997\pi$
$11$	−38.9027	−1.06633	−0.533164	+	0.846012i	$0.678997\pi$
$12$	0	0
$13$	−68.9510	−1.47104	−0.735521	−	0.677502i	$-0.763063\pi$
$13$	−68.9510	−1.47104	−0.735521	+	0.677502i	$0.763063\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−65.9510	−0.940909	−0.470455	−	0.882424i	$-0.655910\pi$
$17$	−65.9510	−0.940909	−0.470455	+	0.882424i	$0.655910\pi$
$18$	0	0
$19$	49.4950	0.597628	0.298814	−	0.954311i	$-0.403409\pi$
$19$	49.4950	0.597628	0.298814	+	0.954311i	$0.403409\pi$
$20$	0	0
$21$	−106.951	−1.11136
$22$	0	0
$23$	−164.524	−1.49155	−0.745776	−	0.666197i	$-0.767921\pi$
$23$	−164.524	−1.49155	−0.745776	+	0.666197i	$0.767921\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	484.164	3.45102
$28$	0	0
$29$	170.853	1.09402	0.547010	−	0.837126i	$-0.315766\pi$
$29$	170.853	1.09402	0.547010	+	0.837126i	$0.315766\pi$
$30$	0	0
$31$	166.505	0.964684	0.482342	−	0.875983i	$-0.339786\pi$
$31$	166.505	0.964684	0.482342	+	0.875983i	$0.339786\pi$
$32$	0	0
$33$	−392.804	−2.07207
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−384.853	−1.70998	−0.854992	−	0.518641i	$-0.826438\pi$
$37$	−384.853	−1.70998	−0.854992	+	0.518641i	$0.826438\pi$
$38$	0	0
$39$	−696.203	−2.85851
$40$	0	0
$41$	−22.8038	−0.0868624	−0.0434312	−	0.999056i	$-0.513829\pi$
$41$	−22.8038	−0.0868624	−0.0434312	+	0.999056i	$0.513829\pi$
$42$	0	0
$43$	−136.709	−0.484836	−0.242418	−	0.970172i	$-0.577941\pi$
$43$	−136.709	−0.484836	−0.242418	+	0.970172i	$0.577941\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−307.562	−0.954523	−0.477261	−	0.878761i	$-0.658370\pi$
$47$	−307.562	−0.954523	−0.477261	+	0.878761i	$0.658370\pi$
$48$	0	0
$49$	−230.804	−0.672897
$50$	0	0
$51$	−665.912	−1.82836
$52$	0	0
$53$	222.000	0.575359	0.287680	−	0.957727i	$-0.407116\pi$
$53$	222.000	0.575359	0.287680	+	0.957727i	$0.407116\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	499.755	1.16130
$58$	0	0
$59$	−522.765	−1.15353	−0.576765	−	0.816910i	$-0.695685\pi$
$59$	−522.765	−1.15353	−0.576765	+	0.816910i	$0.695685\pi$
$60$	0	0
$61$	−393.049	−0.824996	−0.412498	−	0.910958i	$-0.635344\pi$
$61$	−393.049	−0.824996	−0.412498	+	0.910958i	$0.635344\pi$
$62$	0	0
$63$	−793.901	−1.58765
$64$	0	0
$65$	0	0
$66$	0	0
$67$	476.845	0.869492	0.434746	−	0.900553i	$-0.356838\pi$
$67$	476.845	0.869492	0.434746	+	0.900553i	$0.356838\pi$
$68$	0	0
$69$	−1661.22	−2.89836
$70$	0	0
$71$	−4.26352	−0.00712657	−0.00356328	−	0.999994i	$-0.501134\pi$
$71$	−4.26352	−0.00712657	−0.00356328	+	0.999994i	$0.501134\pi$
$72$	0	0
$73$	601.166	0.963852	0.481926	−	0.876212i	$-0.339937\pi$
$73$	601.166	0.963852	0.481926	+	0.876212i	$0.339937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	412.068	0.609864
$78$	0	0
$79$	−1075.93	−1.53230	−0.766150	−	0.642662i	$-0.777830\pi$
$79$	−1075.93	−1.53230	−0.766150	+	0.642662i	$0.777830\pi$
$80$	0	0
$81$	2864.97	3.93000
$82$	0	0
$83$	−1132.74	−1.49801	−0.749005	−	0.662564i	$-0.769468\pi$
$83$	−1132.74	−1.49801	−0.749005	+	0.662564i	$0.769468\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1725.11	2.12588
$88$	0	0
$89$	479.755	0.571392	0.285696	−	0.958320i	$-0.407775\pi$
$89$	479.755	0.571392	0.285696	+	0.958320i	$0.407775\pi$
$90$	0	0
$91$	730.347	0.841332
$92$	0	0
$93$	1681.22	1.87456
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−635.510	−0.665219	−0.332609	−	0.943065i	$-0.607929\pi$
$97$	−635.510	−0.665219	−0.332609	+	0.943065i	$0.607929\pi$
$98$	0	0
$99$	−2915.80	−2.96009
$100$	0	0
$101$	256.264	0.252468	0.126234	−	0.992001i	$-0.459711\pi$
$101$	256.264	0.252468	0.126234	+	0.992001i	$0.459711\pi$
$102$	0	0
$103$	−81.6825	−0.0781400	−0.0390700	−	0.999236i	$-0.512440\pi$
$103$	−81.6825	−0.0781400	−0.0390700	+	0.999236i	$0.512440\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−742.944	−0.671244	−0.335622	−	0.941997i	$-0.608947\pi$
$107$	−742.944	−0.671244	−0.335622	+	0.941997i	$0.608947\pi$
$108$	0	0
$109$	495.872	0.435742	0.217871	−	0.975978i	$-0.430089\pi$
$109$	495.872	0.435742	0.217871	+	0.975978i	$0.430089\pi$
$110$	0	0
$111$	−3885.89	−3.32281
$112$	0	0
$113$	−1878.61	−1.56393	−0.781967	−	0.623320i	$-0.785783\pi$
$113$	−1878.61	−1.56393	−0.781967	+	0.623320i	$0.785783\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5167.94	−4.08356
$118$	0	0
$119$	698.570	0.538133
$120$	0	0
$121$	182.422	0.137057
$122$	0	0
$123$	−230.252	−0.168790
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2113.24	1.47653	0.738265	−	0.674511i	$-0.235645\pi$
$127$	2113.24	1.47653	0.738265	+	0.674511i	$0.235645\pi$
$128$	0	0
$129$	−1380.36	−0.942125
$130$	0	0
$131$	−1105.42	−0.737263	−0.368631	−	0.929576i	$-0.620173\pi$
$131$	−1105.42	−0.737263	−0.368631	+	0.929576i	$0.620173\pi$
$132$	0	0
$133$	−524.264	−0.341801
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1151.92	0.718360	0.359180	−	0.933268i	$-0.383056\pi$
$137$	1151.92	0.718360	0.359180	+	0.933268i	$0.383056\pi$
$138$	0	0
$139$	354.304	0.216199	0.108099	−	0.994140i	$-0.465524\pi$
$139$	354.304	0.216199	0.108099	+	0.994140i	$0.465524\pi$
$140$	0	0
$141$	−3105.48	−1.85481
$142$	0	0
$143$	2682.38	1.56861
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2330.44	−1.30756
$148$	0	0
$149$	−715.638	−0.393472	−0.196736	−	0.980457i	$-0.563034\pi$
$149$	−715.638	−0.393472	−0.196736	+	0.980457i	$0.563034\pi$
$150$	0	0
$151$	2799.37	1.50867	0.754335	−	0.656490i	$-0.227959\pi$
$151$	2799.37	1.50867	0.754335	+	0.656490i	$0.227959\pi$
$152$	0	0
$153$	−4943.09	−2.61193
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−845.275	−0.429683	−0.214842	−	0.976649i	$-0.568924\pi$
$157$	−845.275	−0.429683	−0.214842	+	0.976649i	$0.568924\pi$
$158$	0	0
$159$	2241.55	1.11803
$160$	0	0
$161$	1742.69	0.853062
$162$	0	0
$163$	539.964	0.259468	0.129734	−	0.991549i	$-0.458588\pi$
$163$	539.964	0.259468	0.129734	+	0.991549i	$0.458588\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1209.20	0.560302	0.280151	−	0.959956i	$-0.409615\pi$
$167$	1209.20	0.560302	0.280151	+	0.959956i	$0.409615\pi$
$168$	0	0
$169$	2557.23	1.16397
$170$	0	0
$171$	3709.70	1.65899
$172$	0	0
$173$	−4061.65	−1.78498	−0.892489	−	0.451069i	$-0.851043\pi$
$173$	−4061.65	−1.78498	−0.892489	+	0.451069i	$0.851043\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5278.40	−2.24152
$178$	0	0
$179$	3113.67	1.30015	0.650074	−	0.759871i	$-0.274738\pi$
$179$	3113.67	1.30015	0.650074	+	0.759871i	$0.274738\pi$
$180$	0	0
$181$	−473.215	−0.194330	−0.0971652	−	0.995268i	$-0.530978\pi$
$181$	−473.215	−0.194330	−0.0971652	+	0.995268i	$0.530978\pi$
$182$	0	0
$183$	−3968.65	−1.60312
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2565.67	1.00332
$188$	0	0
$189$	−5128.40	−1.97374
$190$	0	0
$191$	2250.68	0.852637	0.426318	−	0.904573i	$-0.359810\pi$
$191$	2250.68	0.852637	0.426318	+	0.904573i	$0.359810\pi$
$192$	0	0
$193$	553.461	0.206419	0.103210	−	0.994660i	$-0.467089\pi$
$193$	553.461	0.206419	0.103210	+	0.994660i	$0.467089\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	385.744	0.139508	0.0697541	−	0.997564i	$-0.477779\pi$
$197$	385.744	0.139508	0.0697541	+	0.997564i	$0.477779\pi$
$198$	0	0
$199$	−559.494	−0.199304	−0.0996520	−	0.995022i	$-0.531773\pi$
$199$	−559.494	−0.199304	−0.0996520	+	0.995022i	$0.531773\pi$
$200$	0	0
$201$	4814.74	1.68958
$202$	0	0
$203$	−1809.72	−0.625702
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−12331.3	−4.14049
$208$	0	0
$209$	−1925.49	−0.637268
$210$	0	0
$211$	−3721.00	−1.21405	−0.607025	−	0.794683i	$-0.707637\pi$
$211$	−3721.00	−1.21405	−0.607025	+	0.794683i	$0.707637\pi$
$212$	0	0
$213$	−43.0490	−0.0138482
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1763.67	−0.551731
$218$	0	0
$219$	6070.02	1.87294
$220$	0	0
$221$	4547.38	1.38412
$222$	0	0
$223$	5506.41	1.65353	0.826763	−	0.562550i	$-0.190180\pi$
$223$	5506.41	1.65353	0.826763	+	0.562550i	$0.190180\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1139.27	−0.333109	−0.166555	−	0.986032i	$-0.553264\pi$
$227$	−1139.27	−0.333109	−0.166555	+	0.986032i	$0.553264\pi$
$228$	0	0
$229$	−1727.08	−0.498378	−0.249189	−	0.968455i	$-0.580164\pi$
$229$	−1727.08	−0.498378	−0.249189	+	0.968455i	$0.580164\pi$
$230$	0	0
$231$	4160.68	1.18508
$232$	0	0
$233$	1267.84	0.356477	0.178238	−	0.983987i	$-0.442960\pi$
$233$	1267.84	0.356477	0.178238	+	0.983987i	$0.442960\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10863.8	−2.97754
$238$	0	0
$239$	−493.995	−0.133698	−0.0668492	−	0.997763i	$-0.521295\pi$
$239$	−493.995	−0.133698	−0.0668492	+	0.997763i	$0.521295\pi$
$240$	0	0
$241$	−332.676	−0.0889192	−0.0444596	−	0.999011i	$-0.514157\pi$
$241$	−332.676	−0.0889192	−0.0444596	+	0.999011i	$0.514157\pi$
$242$	0	0
$243$	15855.4	4.18569
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3412.73	−0.879136
$248$	0	0
$249$	−11437.4	−2.91091
$250$	0	0
$251$	4365.75	1.09786	0.548932	−	0.835867i	$-0.315035\pi$
$251$	4365.75	1.09786	0.548932	+	0.835867i	$0.315035\pi$
$252$	0	0
$253$	6400.45	1.59048
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3407.19	0.826984	0.413492	−	0.910508i	$-0.364309\pi$
$257$	3407.19	0.826984	0.413492	+	0.910508i	$0.364309\pi$
$258$	0	0
$259$	4076.47	0.977989
$260$	0	0
$261$	12805.6	3.03696
$262$	0	0
$263$	−2388.20	−0.559934	−0.279967	−	0.960010i	$-0.590324\pi$
$263$	−2388.20	−0.559934	−0.279967	+	0.960010i	$0.590324\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4844.12	1.11032
$268$	0	0
$269$	3800.33	0.861377	0.430688	−	0.902501i	$-0.358271\pi$
$269$	3800.33	0.861377	0.430688	+	0.902501i	$0.358271\pi$
$270$	0	0
$271$	2270.39	0.508917	0.254459	−	0.967084i	$-0.418103\pi$
$271$	2270.39	0.508917	0.254459	+	0.967084i	$0.418103\pi$
$272$	0	0
$273$	7374.37	1.63486
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7070.93	1.53376	0.766879	−	0.641792i	$-0.221809\pi$
$277$	7070.93	1.53376	0.766879	+	0.641792i	$0.221809\pi$
$278$	0	0
$279$	12479.7	2.67793
$280$	0	0
$281$	1536.57	0.326206	0.163103	−	0.986609i	$-0.447850\pi$
$281$	1536.57	0.326206	0.163103	+	0.986609i	$0.447850\pi$
$282$	0	0
$283$	−6013.32	−1.26309	−0.631545	−	0.775339i	$-0.717579\pi$
$283$	−6013.32	−1.26309	−0.631545	+	0.775339i	$0.717579\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	241.544	0.0496791
$288$	0	0
$289$	−563.471	−0.114690
$290$	0	0
$291$	−6416.79	−1.29264
$292$	0	0
$293$	4651.03	0.927358	0.463679	−	0.886003i	$-0.346529\pi$
$293$	4651.03	0.927358	0.463679	+	0.886003i	$0.346529\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−18835.3	−3.67992
$298$	0	0
$299$	11344.1	2.19414
$300$	0	0
$301$	1448.06	0.277292
$302$	0	0
$303$	2587.52	0.490591
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7261.88	−1.35002	−0.675012	−	0.737807i	$-0.735861\pi$
$307$	−7261.88	−1.35002	−0.675012	+	0.737807i	$0.735861\pi$
$308$	0	0
$309$	−824.755	−0.151840
$310$	0	0
$311$	6759.67	1.23249	0.616247	−	0.787553i	$-0.288652\pi$
$311$	6759.67	1.23249	0.616247	+	0.787553i	$0.288652\pi$
$312$	0	0
$313$	−1865.55	−0.336891	−0.168446	−	0.985711i	$-0.553875\pi$
$313$	−1865.55	−0.336891	−0.168446	+	0.985711i	$0.553875\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2759.41	−0.488908	−0.244454	−	0.969661i	$-0.578609\pi$
$317$	−2759.41	−0.488908	−0.244454	+	0.969661i	$0.578609\pi$
$318$	0	0
$319$	−6646.64	−1.16658
$320$	0	0
$321$	−7501.57	−1.30435
$322$	0	0
$323$	−3264.24	−0.562314
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5006.86	0.846727
$328$	0	0
$329$	3257.78	0.545919
$330$	0	0
$331$	5123.99	0.850876	0.425438	−	0.904988i	$-0.360120\pi$
$331$	5123.99	0.850876	0.425438	+	0.904988i	$0.360120\pi$
$332$	0	0
$333$	−28845.1	−4.74685
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1475.31	0.238473	0.119236	−	0.992866i	$-0.461955\pi$
$337$	1475.31	0.238473	0.119236	+	0.992866i	$0.461955\pi$
$338$	0	0
$339$	−18968.4	−3.03901
$340$	0	0
$341$	−6477.50	−1.02867
$342$	0	0
$343$	6077.88	0.956778
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3478.67	−0.538170	−0.269085	−	0.963116i	$-0.586721\pi$
$347$	−3478.67	−0.538170	−0.269085	+	0.963116i	$0.586721\pi$
$348$	0	0
$349$	−11392.3	−1.74732	−0.873660	−	0.486537i	$-0.838260\pi$
$349$	−11392.3	−1.74732	−0.873660	+	0.486537i	$0.838260\pi$
$350$	0	0
$351$	−33383.6	−5.07660
$352$	0	0
$353$	−2735.39	−0.412437	−0.206218	−	0.978506i	$-0.566116\pi$
$353$	−2735.39	−0.412437	−0.206218	+	0.978506i	$0.566116\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7053.52	1.04569
$358$	0	0
$359$	2390.41	0.351424	0.175712	−	0.984442i	$-0.443777\pi$
$359$	2390.41	0.351424	0.175712	+	0.984442i	$0.443777\pi$
$360$	0	0
$361$	−4409.25	−0.642841
$362$	0	0
$363$	1841.93	0.266326
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9209.84	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$367$	9209.84	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$368$	0	0
$369$	−1709.17	−0.241127
$370$	0	0
$371$	−2351.48	−0.329065
$372$	0	0
$373$	−13768.4	−1.91127	−0.955634	−	0.294556i	$-0.904828\pi$
$373$	−13768.4	−1.91127	−0.955634	+	0.294556i	$0.904828\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11780.5	−1.60935
$378$	0	0
$379$	−8666.50	−1.17459	−0.587293	−	0.809374i	$-0.699807\pi$
$379$	−8666.50	−1.17459	−0.587293	+	0.809374i	$0.699807\pi$
$380$	0	0
$381$	21337.5	2.86917
$382$	0	0
$383$	7434.12	0.991816	0.495908	−	0.868375i	$-0.334835\pi$
$383$	7434.12	0.991816	0.495908	+	0.868375i	$0.334835\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10246.5	−1.34589
$388$	0	0
$389$	7323.06	0.954483	0.477241	−	0.878772i	$-0.341637\pi$
$389$	7323.06	0.954483	0.477241	+	0.878772i	$0.341637\pi$
$390$	0	0
$391$	10850.5	1.40342
$392$	0	0
$393$	−11161.6	−1.43264
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5185.92	−0.655602	−0.327801	−	0.944747i	$-0.606307\pi$
$397$	−5185.92	−0.655602	−0.327801	+	0.944747i	$0.606307\pi$
$398$	0	0
$399$	−5293.54	−0.664181
$400$	0	0
$401$	−9044.81	−1.12637	−0.563187	−	0.826329i	$-0.690425\pi$
$401$	−9044.81	−1.12637	−0.563187	+	0.826329i	$0.690425\pi$
$402$	0	0
$403$	−11480.7	−1.41909
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14971.8	1.82341
$408$	0	0
$409$	−2311.41	−0.279442	−0.139721	−	0.990191i	$-0.544621\pi$
$409$	−2311.41	−0.279442	−0.139721	+	0.990191i	$0.544621\pi$
$410$	0	0
$411$	11631.0	1.39590
$412$	0	0
$413$	5537.27	0.659737
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3577.43	0.420114
$418$	0	0
$419$	−330.860	−0.0385765	−0.0192882	−	0.999814i	$-0.506140\pi$
$419$	−330.860	−0.0385765	−0.0192882	+	0.999814i	$0.506140\pi$
$420$	0	0
$421$	−1933.39	−0.223819	−0.111909	−	0.993718i	$-0.535697\pi$
$421$	−1933.39	−0.223819	−0.111909	+	0.993718i	$0.535697\pi$
$422$	0	0
$423$	−23052.1	−2.64972
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4163.28	0.471839
$428$	0	0
$429$	27084.2	3.04811
$430$	0	0
$431$	8820.32	0.985753	0.492877	−	0.870099i	$-0.335945\pi$
$431$	8820.32	0.985753	0.492877	+	0.870099i	$0.335945\pi$
$432$	0	0
$433$	11269.7	1.25078	0.625390	−	0.780312i	$-0.284940\pi$
$433$	11269.7	1.25078	0.625390	+	0.780312i	$0.284940\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8143.13	−0.891393
$438$	0	0
$439$	2361.79	0.256770	0.128385	−	0.991724i	$-0.459021\pi$
$439$	2361.79	0.256770	0.128385	+	0.991724i	$0.459021\pi$
$440$	0	0
$441$	−17299.0	−1.86794
$442$	0	0
$443$	−15029.6	−1.61192	−0.805959	−	0.591971i	$-0.798350\pi$
$443$	−15029.6	−1.61192	−0.805959	+	0.591971i	$0.798350\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7225.85	−0.764588
$448$	0	0
$449$	−12661.3	−1.33079	−0.665396	−	0.746491i	$-0.731737\pi$
$449$	−12661.3	−1.33079	−0.665396	+	0.746491i	$0.731737\pi$
$450$	0	0
$451$	887.131	0.0926239
$452$	0	0
$453$	28265.4	2.93162
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1620.47	−0.165870	−0.0829348	−	0.996555i	$-0.526429\pi$
$457$	−1620.47	−0.165870	−0.0829348	+	0.996555i	$0.526429\pi$
$458$	0	0
$459$	−31931.1	−3.24710
$460$	0	0
$461$	−6893.84	−0.696482	−0.348241	−	0.937405i	$-0.613221\pi$
$461$	−6893.84	−0.696482	−0.348241	+	0.937405i	$0.613221\pi$
$462$	0	0
$463$	−12022.8	−1.20680	−0.603398	−	0.797440i	$-0.706187\pi$
$463$	−12022.8	−1.20680	−0.603398	+	0.797440i	$0.706187\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7349.60	0.728263	0.364131	−	0.931348i	$-0.381366\pi$
$467$	7349.60	0.728263	0.364131	+	0.931348i	$0.381366\pi$
$468$	0	0
$469$	−5050.87	−0.497287
$470$	0	0
$471$	−8534.81	−0.834954
$472$	0	0
$473$	5318.36	0.516994
$474$	0	0
$475$	0	0
$476$	0	0
$477$	16639.1	1.59718
$478$	0	0
$479$	−12809.4	−1.22187	−0.610933	−	0.791682i	$-0.709206\pi$
$479$	−12809.4	−1.22187	−0.610933	+	0.791682i	$0.709206\pi$
$480$	0	0
$481$	26536.0	2.51546
$482$	0	0
$483$	17596.0	1.65766
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15782.4	1.46852	0.734260	−	0.678868i	$-0.237529\pi$
$487$	15782.4	1.46852	0.734260	+	0.678868i	$0.237529\pi$
$488$	0	0
$489$	5452.06	0.504193
$490$	0	0
$491$	−5972.10	−0.548914	−0.274457	−	0.961599i	$-0.588498\pi$
$491$	−5972.10	−0.548914	−0.274457	+	0.961599i	$0.588498\pi$
$492$	0	0
$493$	−11267.9	−1.02937
$494$	0	0
$495$	0	0
$496$	0	0
$497$	45.1603	0.00407589
$498$	0	0
$499$	14171.8	1.27138	0.635690	−	0.771944i	$-0.280716\pi$
$499$	14171.8	1.27138	0.635690	+	0.771944i	$0.280716\pi$
$500$	0	0
$501$	12209.4	1.08877
$502$	0	0
$503$	5543.95	0.491436	0.245718	−	0.969341i	$-0.420976\pi$
$503$	5543.95	0.491436	0.245718	+	0.969341i	$0.420976\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	25820.6	2.26180
$508$	0	0
$509$	−13228.9	−1.15199	−0.575994	−	0.817454i	$-0.695385\pi$
$509$	−13228.9	−1.15199	−0.575994	+	0.817454i	$0.695385\pi$
$510$	0	0
$511$	−6367.72	−0.551255
$512$	0	0
$513$	23963.7	2.06243
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11965.0	1.01783
$518$	0	0
$519$	−41010.8	−3.46854
$520$	0	0
$521$	−11092.8	−0.932795	−0.466398	−	0.884575i	$-0.654448\pi$
$521$	−11092.8	−0.932795	−0.466398	+	0.884575i	$0.654448\pi$
$522$	0	0
$523$	−18550.2	−1.55094	−0.775470	−	0.631384i	$-0.782487\pi$
$523$	−18550.2	−1.55094	−0.775470	+	0.631384i	$0.782487\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10981.2	−0.907680
$528$	0	0
$529$	14901.3	1.22473
$530$	0	0
$531$	−39181.8	−3.20215
$532$	0	0
$533$	1572.35	0.127778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	31438.9	2.52643
$538$	0	0
$539$	8978.90	0.717530
$540$	0	0
$541$	−8146.08	−0.647370	−0.323685	−	0.946165i	$-0.604922\pi$
$541$	−8146.08	−0.647370	−0.323685	+	0.946165i	$0.604922\pi$
$542$	0	0
$543$	−4778.09	−0.377620
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18001.4	1.40710	0.703550	−	0.710645i	$-0.251597\pi$
$547$	18001.4	1.40710	0.703550	+	0.710645i	$0.251597\pi$
$548$	0	0
$549$	−29459.4	−2.29016
$550$	0	0
$551$	8456.36	0.653817
$552$	0	0
$553$	11396.5	0.876366
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5968.52	0.454029	0.227014	−	0.973891i	$-0.427104\pi$
$557$	5968.52	0.454029	0.227014	+	0.973891i	$0.427104\pi$
$558$	0	0
$559$	9426.22	0.713214
$560$	0	0
$561$	25905.8	1.94963
$562$	0	0
$563$	−6021.42	−0.450751	−0.225375	−	0.974272i	$-0.572361\pi$
$563$	−6021.42	−0.450751	−0.225375	+	0.974272i	$0.572361\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−30346.5	−2.24768
$568$	0	0
$569$	13499.2	0.994582	0.497291	−	0.867584i	$-0.334328\pi$
$569$	13499.2	0.994582	0.497291	+	0.867584i	$0.334328\pi$
$570$	0	0
$571$	2413.96	0.176920	0.0884599	−	0.996080i	$-0.471805\pi$
$571$	2413.96	0.176920	0.0884599	+	0.996080i	$0.471805\pi$
$572$	0	0
$573$	22725.3	1.65683
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−13549.7	−0.977608	−0.488804	−	0.872394i	$-0.662567\pi$
$577$	−13549.7	−0.977608	−0.488804	+	0.872394i	$0.662567\pi$
$578$	0	0
$579$	5588.33	0.401111
$580$	0	0
$581$	11998.3	0.856755
$582$	0	0
$583$	−8636.41	−0.613522
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19364.2	1.36158	0.680789	−	0.732479i	$-0.261637\pi$
$587$	19364.2	1.36158	0.680789	+	0.732479i	$0.261637\pi$
$588$	0	0
$589$	8241.17	0.576522
$590$	0	0
$591$	3894.89	0.271090
$592$	0	0
$593$	−21130.1	−1.46325	−0.731627	−	0.681705i	$-0.761239\pi$
$593$	−21130.1	−1.46325	−0.731627	+	0.681705i	$0.761239\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5649.25	−0.387284
$598$	0	0
$599$	12938.1	0.882532	0.441266	−	0.897376i	$-0.354529\pi$
$599$	12938.1	0.882532	0.441266	+	0.897376i	$0.354529\pi$
$600$	0	0
$601$	−2173.64	−0.147528	−0.0737641	−	0.997276i	$-0.523501\pi$
$601$	−2173.64	−0.147528	−0.0737641	+	0.997276i	$0.523501\pi$
$602$	0	0
$603$	35740.0	2.41367
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13712.9	0.916951	0.458475	−	0.888707i	$-0.348396\pi$
$607$	13712.9	0.916951	0.458475	+	0.888707i	$0.348396\pi$
$608$	0	0
$609$	−18272.9	−1.21585
$610$	0	0
$611$	21206.7	1.40414
$612$	0	0
$613$	3666.83	0.241602	0.120801	−	0.992677i	$-0.461454\pi$
$613$	3666.83	0.241602	0.120801	+	0.992677i	$0.461454\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26402.5	1.72273	0.861367	−	0.507984i	$-0.169609\pi$
$617$	26402.5	1.72273	0.861367	+	0.507984i	$0.169609\pi$
$618$	0	0
$619$	28554.8	1.85414	0.927072	−	0.374882i	$-0.122317\pi$
$619$	28554.8	1.85414	0.927072	+	0.374882i	$0.122317\pi$
$620$	0	0
$621$	−79656.9	−5.14737
$622$	0	0
$623$	−5081.69	−0.326796
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−19441.8	−1.23833
$628$	0	0
$629$	25381.4	1.60894
$630$	0	0
$631$	29567.8	1.86541	0.932706	−	0.360637i	$-0.117441\pi$
$631$	29567.8	1.86541	0.932706	+	0.360637i	$0.117441\pi$
$632$	0	0
$633$	−37571.2	−2.35912
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15914.1	0.989861
$638$	0	0
$639$	−319.555	−0.0197831
$640$	0	0
$641$	−21701.9	−1.33724	−0.668621	−	0.743604i	$-0.733115\pi$
$641$	−21701.9	−1.33724	−0.668621	+	0.743604i	$0.733115\pi$
$642$	0	0
$643$	−6193.48	−0.379855	−0.189928	−	0.981798i	$-0.560825\pi$
$643$	−6193.48	−0.379855	−0.189928	+	0.981798i	$0.560825\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2208.53	−0.134198	−0.0670991	−	0.997746i	$-0.521374\pi$
$647$	−2208.53	−0.134198	−0.0670991	+	0.997746i	$0.521374\pi$
$648$	0	0
$649$	20337.0	1.23004
$650$	0	0
$651$	−17807.9	−1.07211
$652$	0	0
$653$	−20505.9	−1.22888	−0.614438	−	0.788965i	$-0.710617\pi$
$653$	−20505.9	−1.22888	−0.614438	+	0.788965i	$0.710617\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	45058.0	2.67562
$658$	0	0
$659$	−26030.4	−1.53870	−0.769348	−	0.638829i	$-0.779419\pi$
$659$	−26030.4	−1.53870	−0.769348	+	0.638829i	$0.779419\pi$
$660$	0	0
$661$	6729.32	0.395976	0.197988	−	0.980204i	$-0.436559\pi$
$661$	6729.32	0.395976	0.197988	+	0.980204i	$0.436559\pi$
$662$	0	0
$663$	45915.3	2.68959
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28109.5	−1.63179
$668$	0	0
$669$	55598.6	3.21311
$670$	0	0
$671$	15290.7	0.879717
$672$	0	0
$673$	−8716.54	−0.499254	−0.249627	−	0.968342i	$-0.580308\pi$
$673$	−8716.54	−0.499254	−0.249627	+	0.968342i	$0.580308\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24843.1	−1.41034	−0.705170	−	0.709039i	$-0.749129\pi$
$677$	−24843.1	−1.41034	−0.705170	+	0.709039i	$0.749129\pi$
$678$	0	0
$679$	6731.49	0.380458
$680$	0	0
$681$	−11503.3	−0.647292
$682$	0	0
$683$	20079.1	1.12490	0.562449	−	0.826832i	$-0.309859\pi$
$683$	20079.1	1.12490	0.562449	+	0.826832i	$0.309859\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17438.4	−0.968440
$688$	0	0
$689$	−15307.1	−0.846378
$690$	0	0
$691$	16575.8	0.912550	0.456275	−	0.889839i	$-0.349183\pi$
$691$	16575.8	0.912550	0.456275	+	0.889839i	$0.349183\pi$
$692$	0	0
$693$	30884.9	1.69296
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1503.93	0.0817297
$698$	0	0
$699$	12801.5	0.692700
$700$	0	0
$701$	−29052.6	−1.56534	−0.782668	−	0.622440i	$-0.786142\pi$
$701$	−29052.6	−1.56534	−0.782668	+	0.622440i	$0.786142\pi$
$702$	0	0
$703$	−19048.3	−1.02193
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2714.42	−0.144394
$708$	0	0
$709$	−32100.2	−1.70035	−0.850175	−	0.526499i	$-0.823504\pi$
$709$	−32100.2	−1.70035	−0.850175	+	0.526499i	$0.823504\pi$
$710$	0	0
$711$	−80642.0	−4.25360
$712$	0	0
$713$	−27394.2	−1.43888
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4987.91	−0.259800
$718$	0	0
$719$	17819.4	0.924274	0.462137	−	0.886808i	$-0.347083\pi$
$719$	17819.4	0.924274	0.462137	+	0.886808i	$0.347083\pi$
$720$	0	0
$721$	865.203	0.0446905
$722$	0	0
$723$	−3359.05	−0.172786
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−27816.4	−1.41906	−0.709529	−	0.704677i	$-0.751092\pi$
$727$	−27816.4	−1.41906	−0.709529	+	0.704677i	$0.751092\pi$
$728$	0	0
$729$	82738.8	4.20357
$730$	0	0
$731$	9016.10	0.456187
$732$	0	0
$733$	−2522.56	−0.127112	−0.0635559	−	0.997978i	$-0.520244\pi$
$733$	−2522.56	−0.127112	−0.0635559	+	0.997978i	$0.520244\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18550.6	−0.927164
$738$	0	0
$739$	−3526.91	−0.175561	−0.0877805	−	0.996140i	$-0.527977\pi$
$739$	−3526.91	−0.175561	−0.0877805	+	0.996140i	$0.527977\pi$
$740$	0	0
$741$	−34458.6	−1.70832
$742$	0	0
$743$	−8557.90	−0.422556	−0.211278	−	0.977426i	$-0.567763\pi$
$743$	−8557.90	−0.422556	−0.211278	+	0.977426i	$0.567763\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−84900.3	−4.15842
$748$	0	0
$749$	7869.47	0.383904
$750$	0	0
$751$	806.886	0.0392060	0.0196030	−	0.999808i	$-0.493760\pi$
$751$	806.886	0.0392060	0.0196030	+	0.999808i	$0.493760\pi$
$752$	0	0
$753$	44081.4	2.13335
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12898.6	0.619298	0.309649	−	0.950851i	$-0.399788\pi$
$757$	12898.6	0.619298	0.309649	+	0.950851i	$0.399788\pi$
$758$	0	0
$759$	64625.8	3.09060
$760$	0	0
$761$	23315.9	1.11064	0.555322	−	0.831635i	$-0.312595\pi$
$761$	23315.9	1.11064	0.555322	+	0.831635i	$0.312595\pi$
$762$	0	0
$763$	−5252.41	−0.249214
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36045.2	1.69689
$768$	0	0
$769$	41805.3	1.96039	0.980193	−	0.198045i	$-0.0634591\pi$
$769$	41805.3	1.96039	0.980193	+	0.198045i	$0.0634591\pi$
$770$	0	0
$771$	34402.7	1.60698
$772$	0	0
$773$	10587.9	0.492654	0.246327	−	0.969187i	$-0.420776\pi$
$773$	10587.9	0.492654	0.246327	+	0.969187i	$0.420776\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	41160.4	1.90041
$778$	0	0
$779$	−1128.68	−0.0519114
$780$	0	0
$781$	165.862	0.00759926
$782$	0	0
$783$	82720.9	3.77548
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16085.8	0.728584	0.364292	−	0.931285i	$-0.381311\pi$
$787$	16085.8	0.728584	0.364292	+	0.931285i	$0.381311\pi$
$788$	0	0
$789$	−24113.8	−1.08805
$790$	0	0
$791$	19898.7	0.894459
$792$	0	0
$793$	27101.1	1.21360
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13813.3	−0.613919	−0.306959	−	0.951723i	$-0.599312\pi$
$797$	−13813.3	−0.613919	−0.306959	+	0.951723i	$0.599312\pi$
$798$	0	0
$799$	20284.0	0.898119
$800$	0	0
$801$	35958.1	1.58616
$802$	0	0
$803$	−23387.0	−1.02778
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38372.2	1.67381
$808$	0	0
$809$	−14893.1	−0.647234	−0.323617	−	0.946188i	$-0.604899\pi$
$809$	−14893.1	−0.647234	−0.323617	+	0.946188i	$0.604899\pi$
$810$	0	0
$811$	24638.7	1.06681	0.533405	−	0.845860i	$-0.320912\pi$
$811$	24638.7	1.06681	0.533405	+	0.845860i	$0.320912\pi$
$812$	0	0
$813$	22924.3	0.988919
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6766.42	−0.289752
$818$	0	0
$819$	54740.2	2.33550
$820$	0	0
$821$	11953.6	0.508142	0.254071	−	0.967186i	$-0.418230\pi$
$821$	11953.6	0.508142	0.254071	+	0.967186i	$0.418230\pi$
$822$	0	0
$823$	4320.27	0.182983	0.0914916	−	0.995806i	$-0.470837\pi$
$823$	4320.27	0.182983	0.0914916	+	0.995806i	$0.470837\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17980.9	−0.756055	−0.378028	−	0.925794i	$-0.623398\pi$
$827$	−17980.9	−0.756055	−0.378028	+	0.925794i	$0.623398\pi$
$828$	0	0
$829$	−29347.5	−1.22953	−0.614766	−	0.788710i	$-0.710750\pi$
$829$	−29347.5	−1.22953	−0.614766	+	0.788710i	$0.710750\pi$
$830$	0	0
$831$	71395.7	2.98037
$832$	0	0
$833$	15221.7	0.633135
$834$	0	0
$835$	0	0
$836$	0	0
$837$	80615.9	3.32914
$838$	0	0
$839$	−20819.5	−0.856696	−0.428348	−	0.903614i	$-0.640904\pi$
$839$	−20819.5	−0.856696	−0.428348	+	0.903614i	$0.640904\pi$
$840$	0	0
$841$	4801.70	0.196880
$842$	0	0
$843$	15514.8	0.633878
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1932.27	−0.0783866
$848$	0	0
$849$	−60716.9	−2.45442
$850$	0	0
$851$	63317.7	2.55053
$852$	0	0
$853$	16835.3	0.675766	0.337883	−	0.941188i	$-0.390289\pi$
$853$	16835.3	0.675766	0.337883	+	0.941188i	$0.390289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2703.39	−0.107755	−0.0538775	−	0.998548i	$-0.517158\pi$
$857$	−2703.39	−0.107755	−0.0538775	+	0.998548i	$0.517158\pi$
$858$	0	0
$859$	−19064.6	−0.757247	−0.378624	−	0.925551i	$-0.623603\pi$
$859$	−19064.6	−0.757247	−0.378624	+	0.925551i	$0.623603\pi$
$860$	0	0
$861$	2438.89	0.0965356
$862$	0	0
$863$	−41930.0	−1.65390	−0.826949	−	0.562278i	$-0.809925\pi$
$863$	−41930.0	−1.65390	−0.826949	+	0.562278i	$0.809925\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5689.41	−0.222863
$868$	0	0
$869$	41856.6	1.63393
$870$	0	0
$871$	−32878.9	−1.27906
$872$	0	0
$873$	−47632.1	−1.84662
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7701.66	0.296541	0.148270	−	0.988947i	$-0.452629\pi$
$877$	7701.66	0.296541	0.148270	+	0.988947i	$0.452629\pi$
$878$	0	0
$879$	46961.8	1.80203
$880$	0	0
$881$	7799.94	0.298282	0.149141	−	0.988816i	$-0.452349\pi$
$881$	7799.94	0.298282	0.149141	+	0.988816i	$0.452349\pi$
$882$	0	0
$883$	40922.8	1.55964	0.779820	−	0.626003i	$-0.215310\pi$
$883$	40922.8	1.55964	0.779820	+	0.626003i	$0.215310\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9632.80	−0.364642	−0.182321	−	0.983239i	$-0.558361\pi$
$887$	−9632.80	−0.364642	−0.182321	+	0.983239i	$0.558361\pi$
$888$	0	0
$889$	−22384.0	−0.844470
$890$	0	0
$891$	−111455.	−4.19067
$892$	0	0
$893$	−15222.8	−0.570449
$894$	0	0
$895$	0	0
$896$	0	0
$897$	114542.	4.26361
$898$	0	0
$899$	28447.9	1.05538
$900$	0	0
$901$	−14641.1	−0.541361
$902$	0	0
$903$	14621.2	0.538828
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27797.0	1.01762	0.508812	−	0.860878i	$-0.330085\pi$
$907$	27797.0	1.01762	0.508812	+	0.860878i	$0.330085\pi$
$908$	0	0
$909$	19207.3	0.700841
$910$	0	0
$911$	−20735.4	−0.754110	−0.377055	−	0.926191i	$-0.623063\pi$
$911$	−20735.4	−0.754110	−0.377055	+	0.926191i	$0.623063\pi$
$912$	0	0
$913$	44066.9	1.59737
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11709.0	0.421662
$918$	0	0
$919$	−6063.46	−0.217644	−0.108822	−	0.994061i	$-0.534708\pi$
$919$	−6063.46	−0.217644	−0.108822	+	0.994061i	$0.534708\pi$
$920$	0	0
$921$	−73323.7	−2.62334
$922$	0	0
$923$	293.973	0.0104835
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6122.18	−0.216914
$928$	0	0
$929$	−694.043	−0.0245111	−0.0122556	−	0.999925i	$-0.503901\pi$
$929$	−694.043	−0.0245111	−0.0122556	+	0.999925i	$0.503901\pi$
$930$	0	0
$931$	−11423.6	−0.402142
$932$	0	0
$933$	68252.9	2.39496
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44470.5	−1.55047	−0.775234	−	0.631674i	$-0.782368\pi$
$937$	−44470.5	−1.55047	−0.775234	+	0.631674i	$0.782368\pi$
$938$	0	0
$939$	−18836.6	−0.654642
$940$	0	0
$941$	34075.3	1.18047	0.590235	−	0.807232i	$-0.299035\pi$
$941$	34075.3	1.18047	0.590235	+	0.807232i	$0.299035\pi$
$942$	0	0
$943$	3751.79	0.129560
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34495.9	1.18370	0.591851	−	0.806048i	$-0.298397\pi$
$947$	34495.9	1.18370	0.591851	+	0.806048i	$0.298397\pi$
$948$	0	0
$949$	−41451.0	−1.41787
$950$	0	0
$951$	−27862.0	−0.950039
$952$	0	0
$953$	−23975.9	−0.814960	−0.407480	−	0.913214i	$-0.633592\pi$
$953$	−23975.9	−0.814960	−0.407480	+	0.913214i	$0.633592\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−67111.7	−2.26689
$958$	0	0
$959$	−12201.5	−0.410851
$960$	0	0
$961$	−2067.03	−0.0693845
$962$	0	0
$963$	−55684.4	−1.86335
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27578.9	−0.917144	−0.458572	−	0.888657i	$-0.651639\pi$
$967$	−27578.9	−0.917144	−0.458572	+	0.888657i	$0.651639\pi$
$968$	0	0
$969$	−32959.3	−1.09268
$970$	0	0
$971$	−20834.3	−0.688574	−0.344287	−	0.938864i	$-0.611879\pi$
$971$	−20834.3	−0.688574	−0.344287	+	0.938864i	$0.611879\pi$
$972$	0	0
$973$	−3752.88	−0.123650
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41636.7	1.36344	0.681718	−	0.731615i	$-0.261233\pi$
$977$	41636.7	1.36344	0.681718	+	0.731615i	$0.261233\pi$
$978$	0	0
$979$	−18663.8	−0.609292
$980$	0	0
$981$	37166.1	1.20960
$982$	0	0
$983$	−18099.2	−0.587259	−0.293629	−	0.955919i	$-0.594863\pi$
$983$	−18099.2	−0.587259	−0.293629	+	0.955919i	$0.594863\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	32894.1	1.06082
$988$	0	0
$989$	22492.0	0.723158
$990$	0	0
$991$	55904.9	1.79201	0.896003	−	0.444049i	$-0.146458\pi$
$991$	55904.9	1.79201	0.896003	+	0.444049i	$0.146458\pi$
$992$	0	0
$993$	51737.3	1.65341
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15808.7	0.502174	0.251087	−	0.967965i	$-0.419212\pi$
$997$	15808.7	0.502174	0.251087	+	0.967965i	$0.419212\pi$
$998$	0	0
$999$	−186332.	−5.90119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.cw.1.4		4
4.3	odd	2	inner	1600.4.a.cw.1.1		4
5.4	even	2		1600.4.a.cx.1.1		4
8.3	odd	2		800.4.a.bb.1.4	yes	4
8.5	even	2		800.4.a.bb.1.1	yes	4
20.19	odd	2		1600.4.a.cx.1.4		4
40.3	even	4		800.4.c.o.449.7		8
40.13	odd	4		800.4.c.o.449.2		8
40.19	odd	2		800.4.a.ba.1.1	✓	4
40.27	even	4		800.4.c.o.449.1		8
40.29	even	2		800.4.a.ba.1.4	yes	4
40.37	odd	4		800.4.c.o.449.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.ba.1.1	✓	4	40.19	odd	2
800.4.a.ba.1.4	yes	4	40.29	even	2
800.4.a.bb.1.1	yes	4	8.5	even	2
800.4.a.bb.1.4	yes	4	8.3	odd	2
800.4.c.o.449.1		8	40.27	even	4
800.4.c.o.449.2		8	40.13	odd	4
800.4.c.o.449.7		8	40.3	even	4
800.4.c.o.449.8		8	40.37	odd	4
1600.4.a.cw.1.1		4	4.3	odd	2	inner
1600.4.a.cw.1.4		4	1.1	even	1	trivial
1600.4.a.cx.1.1		4	5.4	even	2
1600.4.a.cx.1.4		4	20.19	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

\(f(q)\)	\(=\)	\(q+10.0971 q^{3} -10.5923 q^{7} +74.9510 q^{9} +O(q^{10})\)	\(q+10.0971 q^{3} -10.5923 q^{7} +74.9510 q^{9} -38.9027 q^{11} -68.9510 q^{13} -65.9510 q^{17} +49.4950 q^{19} -106.951 q^{21} -164.524 q^{23} +484.164 q^{27} +170.853 q^{29} +166.505 q^{31} -392.804 q^{33} -384.853 q^{37} -696.203 q^{39} -22.8038 q^{41} -136.709 q^{43} -307.562 q^{47} -230.804 q^{49} -665.912 q^{51} +222.000 q^{53} +499.755 q^{57} -522.765 q^{59} -393.049 q^{61} -793.901 q^{63} +476.845 q^{67} -1661.22 q^{69} -4.26352 q^{71} +601.166 q^{73} +412.068 q^{77} -1075.93 q^{79} +2864.97 q^{81} -1132.74 q^{83} +1725.11 q^{87} +479.755 q^{89} +730.347 q^{91} +1681.22 q^{93} -635.510 q^{97} -2915.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 96 q^{9}+O(q^{10}) \) 4 * q + 96 * q^9	\( 4 q + 96 q^{9} - 72 q^{13} - 60 q^{17} - 224 q^{21} + 72 q^{29} - 756 q^{33} - 928 q^{37} + 724 q^{41} - 108 q^{49} + 888 q^{53} + 980 q^{57} - 1776 q^{61} - 3384 q^{69} - 1060 q^{73} - 2224 q^{77} + 7180 q^{81} + 900 q^{89} + 3464 q^{93} - 504 q^{97}+O(q^{100}) \) 4 * q + 96 * q^9 - 72 * q^13 - 60 * q^17 - 224 * q^21 + 72 * q^29 - 756 * q^33 - 928 * q^37 + 724 * q^41 - 108 * q^49 + 888 * q^53 + 980 * q^57 - 1776 * q^61 - 3384 * q^69 - 1060 * q^73 - 2224 * q^77 + 7180 * q^81 + 900 * q^89 + 3464 * q^93 - 504 * q^97

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