LMFDB - Embedded newform 10000.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 4x + 16 $ x^4 - x^3 - 9x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.34841$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34841 q^{3} +0.618034 q^{7} +2.51505 q^{9} +O(q^{10})$	$q+2.34841 q^{3} +0.618034 q^{7} +2.51505 q^{9} +4.34841 q^{11} +1.00000 q^{13} +3.96645 q^{17} +6.58448 q^{19} +1.45140 q^{21} -0.245428 q^{23} -1.13887 q^{27} +3.06943 q^{29} -9.43299 q^{31} +10.2119 q^{33} +7.89934 q^{37} +2.34841 q^{39} +3.12749 q^{41} +6.79981 q^{43} -2.84851 q^{47} -6.61803 q^{49} +9.31486 q^{51} -5.43299 q^{53} +15.4631 q^{57} -2.56934 q^{59} -10.3093 q^{61} +1.55439 q^{63} -1.69683 q^{67} -0.576367 q^{69} +15.6898 q^{71} +11.7814 q^{73} +2.68747 q^{77} +4.33346 q^{79} -10.2197 q^{81} -17.4108 q^{83} +7.20830 q^{87} +10.9895 q^{89} +0.618034 q^{91} -22.1526 q^{93} +6.41552 q^{97} +10.9365 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 2 * q^7 + 9 * q^9	$ 4 q + 3 q^{3} - 2 q^{7} + 9 q^{9} + 11 q^{11} + 4 q^{13} + 5 q^{17} + 11 q^{19} - 4 q^{21} + 2 q^{23} + 24 q^{27} - 2 q^{29} + 27 q^{33} - q^{37} + 3 q^{39} - 2 q^{41} + 11 q^{43} + 11 q^{47} - 22 q^{49} + 20 q^{51} + 16 q^{53} + 22 q^{57} - 12 q^{59} - 15 q^{61} - 7 q^{63} + 6 q^{67} - 46 q^{69} + 8 q^{71} + 3 q^{73} - 8 q^{77} - 4 q^{79} + 16 q^{81} - 7 q^{83} - 14 q^{87} - 11 q^{89} - 2 q^{91} - 10 q^{93} + 41 q^{97} + 51 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 2 * q^7 + 9 * q^9 + 11 * q^11 + 4 * q^13 + 5 * q^17 + 11 * q^19 - 4 * q^21 + 2 * q^23 + 24 * q^27 - 2 * q^29 + 27 * q^33 - q^37 + 3 * q^39 - 2 * q^41 + 11 * q^43 + 11 * q^47 - 22 * q^49 + 20 * q^51 + 16 * q^53 + 22 * q^57 - 12 * q^59 - 15 * q^61 - 7 * q^63 + 6 * q^67 - 46 * q^69 + 8 * q^71 + 3 * q^73 - 8 * q^77 - 4 * q^79 + 16 * q^81 - 7 * q^83 - 14 * q^87 - 11 * q^89 - 2 * q^91 - 10 * q^93 + 41 * q^97 + 51 * 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
$3$	2.34841	1.35586	0.677929	−	0.735128i	$-0.262878\pi$
$3$	2.34841	1.35586	0.677929	+	0.735128i	$0.262878\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	0	0
$9$	2.51505	0.838349
$10$	0	0
$11$	4.34841	1.31110	0.655548	−	0.755153i	$-0.272438\pi$
$11$	4.34841	1.31110	0.655548	+	0.755153i	$0.272438\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.96645	0.962005	0.481002	−	0.876719i	$-0.340273\pi$
$17$	3.96645	0.962005	0.481002	+	0.876719i	$0.340273\pi$
$18$	0	0
$19$	6.58448	1.51058	0.755292	−	0.655389i	$-0.227495\pi$
$19$	6.58448	1.51058	0.755292	+	0.655389i	$0.227495\pi$
$20$	0	0
$21$	1.45140	0.316721
$22$	0	0
$23$	−0.245428	−0.0511753	−0.0255877	−	0.999673i	$-0.508146\pi$
$23$	−0.245428	−0.0511753	−0.0255877	+	0.999673i	$0.508146\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.13887	−0.219175
$28$	0	0
$29$	3.06943	0.569980	0.284990	−	0.958531i	$-0.408010\pi$
$29$	3.06943	0.569980	0.284990	+	0.958531i	$0.408010\pi$
$30$	0	0
$31$	−9.43299	−1.69422	−0.847108	−	0.531421i	$-0.821658\pi$
$31$	−9.43299	−1.69422	−0.847108	+	0.531421i	$0.821658\pi$
$32$	0	0
$33$	10.2119	1.77766
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.89934	1.29864	0.649322	−	0.760514i	$-0.275053\pi$
$37$	7.89934	1.29864	0.649322	+	0.760514i	$0.275053\pi$
$38$	0	0
$39$	2.34841	0.376047
$40$	0	0
$41$	3.12749	0.488432	0.244216	−	0.969721i	$-0.421469\pi$
$41$	3.12749	0.488432	0.244216	+	0.969721i	$0.421469\pi$
$42$	0	0
$43$	6.79981	1.03696	0.518481	−	0.855089i	$-0.326498\pi$
$43$	6.79981	1.03696	0.518481	+	0.855089i	$0.326498\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.84851	−0.415498	−0.207749	−	0.978182i	$-0.566614\pi$
$47$	−2.84851	−0.415498	−0.207749	+	0.978182i	$0.566614\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	9.31486	1.30434
$52$	0	0
$53$	−5.43299	−0.746279	−0.373139	−	0.927775i	$-0.621719\pi$
$53$	−5.43299	−0.746279	−0.373139	+	0.927775i	$0.621719\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.4631	2.04814
$58$	0	0
$59$	−2.56934	−0.334499	−0.167250	−	0.985915i	$-0.553489\pi$
$59$	−2.56934	−0.334499	−0.167250	+	0.985915i	$0.553489\pi$
$60$	0	0
$61$	−10.3093	−1.31997	−0.659983	−	0.751280i	$-0.729437\pi$
$61$	−10.3093	−1.31997	−0.659983	+	0.751280i	$0.729437\pi$
$62$	0	0
$63$	1.55439	0.195834
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.69683	−0.207301	−0.103650	−	0.994614i	$-0.533052\pi$
$67$	−1.69683	−0.207301	−0.103650	+	0.994614i	$0.533052\pi$
$68$	0	0
$69$	−0.576367	−0.0693865
$70$	0	0
$71$	15.6898	1.86204	0.931018	−	0.364973i	$-0.118922\pi$
$71$	15.6898	1.86204	0.931018	+	0.364973i	$0.118922\pi$
$72$	0	0
$73$	11.7814	1.37891	0.689455	−	0.724329i	$-0.257850\pi$
$73$	11.7814	1.37891	0.689455	+	0.724329i	$0.257850\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.68747	0.306265
$78$	0	0
$79$	4.33346	0.487553	0.243776	−	0.969831i	$-0.421614\pi$
$79$	4.33346	0.487553	0.243776	+	0.969831i	$0.421614\pi$
$80$	0	0
$81$	−10.2197	−1.13552
$82$	0	0
$83$	−17.4108	−1.91109	−0.955543	−	0.294853i	$-0.904729\pi$
$83$	−17.4108	−1.91109	−0.955543	+	0.294853i	$0.904729\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.20830	0.772811
$88$	0	0
$89$	10.9895	1.16489	0.582443	−	0.812871i	$-0.302097\pi$
$89$	10.9895	1.16489	0.582443	+	0.812871i	$0.302097\pi$
$90$	0	0
$91$	0.618034	0.0647876
$92$	0	0
$93$	−22.1526	−2.29711
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.41552	0.651397	0.325699	−	0.945474i	$-0.394401\pi$
$97$	6.41552	0.651397	0.325699	+	0.945474i	$0.394401\pi$
$98$	0	0
$99$	10.9365	1.09916
$100$	0	0
$101$	−19.5024	−1.94056	−0.970282	−	0.241977i	$-0.922204\pi$
$101$	−19.5024	−1.94056	−0.970282	+	0.241977i	$0.922204\pi$
$102$	0	0
$103$	−3.62362	−0.357046	−0.178523	−	0.983936i	$-0.557132\pi$
$103$	−3.62362	−0.357046	−0.178523	+	0.983936i	$0.557132\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.1561	1.07851	0.539253	−	0.842144i	$-0.318707\pi$
$107$	11.1561	1.07851	0.539253	+	0.842144i	$0.318707\pi$
$108$	0	0
$109$	13.2927	1.27321	0.636604	−	0.771191i	$-0.280339\pi$
$109$	13.2927	1.27321	0.636604	+	0.771191i	$0.280339\pi$
$110$	0	0
$111$	18.5509	1.76078
$112$	0	0
$113$	−4.49664	−0.423008	−0.211504	−	0.977377i	$-0.567836\pi$
$113$	−4.49664	−0.423008	−0.211504	+	0.977377i	$0.567836\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.51505	0.232516
$118$	0	0
$119$	2.45140	0.224719
$120$	0	0
$121$	7.90870	0.718973
$122$	0	0
$123$	7.34464	0.662244
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.6921	−1.65866	−0.829329	−	0.558761i	$-0.811277\pi$
$127$	−18.6921	−1.65866	−0.829329	+	0.558761i	$0.811277\pi$
$128$	0	0
$129$	15.9688	1.40597
$130$	0	0
$131$	−9.18555	−0.802545	−0.401273	−	0.915959i	$-0.631432\pi$
$131$	−9.18555	−0.802545	−0.401273	+	0.915959i	$0.631432\pi$
$132$	0	0
$133$	4.06943	0.352865
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.9352	1.19057	0.595283	−	0.803516i	$-0.297040\pi$
$137$	13.9352	1.19057	0.595283	+	0.803516i	$0.297040\pi$
$138$	0	0
$139$	−0.263835	−0.0223782	−0.0111891	−	0.999937i	$-0.503562\pi$
$139$	−0.263835	−0.0223782	−0.0111891	+	0.999937i	$0.503562\pi$
$140$	0	0
$141$	−6.68948	−0.563356
$142$	0	0
$143$	4.34841	0.363633
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.5419	−1.28187
$148$	0	0
$149$	15.2086	1.24594	0.622969	−	0.782246i	$-0.285926\pi$
$149$	15.2086	1.24594	0.622969	+	0.782246i	$0.285926\pi$
$150$	0	0
$151$	−1.50802	−0.122721	−0.0613604	−	0.998116i	$-0.519544\pi$
$151$	−1.50802	−0.122721	−0.0613604	+	0.998116i	$0.519544\pi$
$152$	0	0
$153$	9.97581	0.806496
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.29972	−0.183537	−0.0917687	−	0.995780i	$-0.529252\pi$
$157$	−2.29972	−0.183537	−0.0917687	+	0.995780i	$0.529252\pi$
$158$	0	0
$159$	−12.7589	−1.01185
$160$	0	0
$161$	−0.151683	−0.0119543
$162$	0	0
$163$	15.5208	1.21569	0.607843	−	0.794057i	$-0.292035\pi$
$163$	15.5208	1.21569	0.607843	+	0.794057i	$0.292035\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.90280	−0.147243	−0.0736215	−	0.997286i	$-0.523456\pi$
$167$	−1.90280	−0.147243	−0.0736215	+	0.997286i	$0.523456\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	16.5603	1.26640
$172$	0	0
$173$	−14.9352	−1.13550	−0.567752	−	0.823200i	$-0.692187\pi$
$173$	−14.9352	−1.13550	−0.567752	+	0.823200i	$0.692187\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.03387	−0.453533
$178$	0	0
$179$	−3.94986	−0.295227	−0.147613	−	0.989045i	$-0.547159\pi$
$179$	−3.94986	−0.295227	−0.147613	+	0.989045i	$0.547159\pi$
$180$	0	0
$181$	−18.1807	−1.35136	−0.675679	−	0.737196i	$-0.736149\pi$
$181$	−18.1807	−1.35136	−0.675679	+	0.737196i	$0.736149\pi$
$182$	0	0
$183$	−24.2104	−1.78969
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17.2478	1.26128
$188$	0	0
$189$	−0.703859	−0.0511982
$190$	0	0
$191$	−0.864588	−0.0625594	−0.0312797	−	0.999511i	$-0.509958\pi$
$191$	−0.864588	−0.0625594	−0.0312797	+	0.999511i	$0.509958\pi$
$192$	0	0
$193$	−20.9836	−1.51043	−0.755217	−	0.655475i	$-0.772468\pi$
$193$	−20.9836	−1.51043	−0.755217	+	0.655475i	$0.772468\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.2037	−1.72444	−0.862222	−	0.506531i	$-0.830928\pi$
$197$	−24.2037	−1.72444	−0.862222	+	0.506531i	$0.830928\pi$
$198$	0	0
$199$	23.1181	1.63880	0.819400	−	0.573222i	$-0.194307\pi$
$199$	23.1181	1.63880	0.819400	+	0.573222i	$0.194307\pi$
$200$	0	0
$201$	−3.98485	−0.281070
$202$	0	0
$203$	1.89701	0.133144
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.617264	−0.0429028
$208$	0	0
$209$	28.6321	1.98052
$210$	0	0
$211$	−5.33234	−0.367093	−0.183547	−	0.983011i	$-0.558758\pi$
$211$	−5.33234	−0.367093	−0.183547	+	0.983011i	$0.558758\pi$
$212$	0	0
$213$	36.8461	2.52466
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.82991	−0.395760
$218$	0	0
$219$	27.6676	1.86960
$220$	0	0
$221$	3.96645	0.266812
$222$	0	0
$223$	26.1899	1.75381	0.876903	−	0.480668i	$-0.159606\pi$
$223$	26.1899	1.75381	0.876903	+	0.480668i	$0.159606\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.0752	1.73067	0.865337	−	0.501191i	$-0.167105\pi$
$227$	26.0752	1.73067	0.865337	+	0.501191i	$0.167105\pi$
$228$	0	0
$229$	−3.63531	−0.240228	−0.120114	−	0.992760i	$-0.538326\pi$
$229$	−3.63531	−0.240228	−0.120114	+	0.992760i	$0.538326\pi$
$230$	0	0
$231$	6.31129	0.415252
$232$	0	0
$233$	−10.0362	−0.657493	−0.328747	−	0.944418i	$-0.606626\pi$
$233$	−10.0362	−0.657493	−0.328747	+	0.944418i	$0.606626\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.1768	0.661052
$238$	0	0
$239$	6.26616	0.405324	0.202662	−	0.979249i	$-0.435041\pi$
$239$	6.26616	0.405324	0.202662	+	0.979249i	$0.435041\pi$
$240$	0	0
$241$	−11.8597	−0.763950	−0.381975	−	0.924173i	$-0.624756\pi$
$241$	−11.8597	−0.763950	−0.381975	+	0.924173i	$0.624756\pi$
$242$	0	0
$243$	−20.5834	−1.32043
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.58448	0.418961
$248$	0	0
$249$	−40.8878	−2.59116
$250$	0	0
$251$	−9.58825	−0.605205	−0.302603	−	0.953117i	$-0.597855\pi$
$251$	−9.58825	−0.605205	−0.302603	+	0.953117i	$0.597855\pi$
$252$	0	0
$253$	−1.06722	−0.0670958
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.46981	0.341197	0.170599	−	0.985341i	$-0.445430\pi$
$257$	5.46981	0.341197	0.170599	+	0.985341i	$0.445430\pi$
$258$	0	0
$259$	4.88206	0.303357
$260$	0	0
$261$	7.71977	0.477842
$262$	0	0
$263$	1.22671	0.0756420	0.0378210	−	0.999285i	$-0.487958\pi$
$263$	1.22671	0.0756420	0.0378210	+	0.999285i	$0.487958\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.8079	1.57942
$268$	0	0
$269$	−11.9364	−0.727772	−0.363886	−	0.931443i	$-0.618550\pi$
$269$	−11.9364	−0.727772	−0.363886	+	0.931443i	$0.618550\pi$
$270$	0	0
$271$	23.6353	1.43574	0.717872	−	0.696176i	$-0.245116\pi$
$271$	23.6353	1.43574	0.717872	+	0.696176i	$0.245116\pi$
$272$	0	0
$273$	1.45140	0.0878427
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.3076	0.679410	0.339705	−	0.940532i	$-0.389673\pi$
$277$	11.3076	0.679410	0.339705	+	0.940532i	$0.389673\pi$
$278$	0	0
$279$	−23.7244	−1.42034
$280$	0	0
$281$	−12.0045	−0.716126	−0.358063	−	0.933697i	$-0.616563\pi$
$281$	−12.0045	−0.716126	−0.358063	+	0.933697i	$0.616563\pi$
$282$	0	0
$283$	1.21910	0.0724680	0.0362340	−	0.999343i	$-0.488464\pi$
$283$	1.21910	0.0724680	0.0362340	+	0.999343i	$0.488464\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.93290	0.114095
$288$	0	0
$289$	−1.26729	−0.0745465
$290$	0	0
$291$	15.0663	0.883202
$292$	0	0
$293$	−12.7403	−0.744297	−0.372149	−	0.928173i	$-0.621379\pi$
$293$	−12.7403	−0.744297	−0.372149	+	0.928173i	$0.621379\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.95227	−0.287360
$298$	0	0
$299$	−0.245428	−0.0141935
$300$	0	0
$301$	4.20252	0.242229
$302$	0	0
$303$	−45.7998	−2.63113
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.80572	0.502569	0.251284	−	0.967913i	$-0.419147\pi$
$307$	8.80572	0.502569	0.251284	+	0.967913i	$0.419147\pi$
$308$	0	0
$309$	−8.50977	−0.484104
$310$	0	0
$311$	−1.20862	−0.0685343	−0.0342671	−	0.999413i	$-0.510910\pi$
$311$	−1.20862	−0.0685343	−0.0342671	+	0.999413i	$0.510910\pi$
$312$	0	0
$313$	6.29148	0.355616	0.177808	−	0.984065i	$-0.443099\pi$
$313$	6.29148	0.355616	0.177808	+	0.984065i	$0.443099\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.0344	1.12525	0.562623	−	0.826713i	$-0.309792\pi$
$317$	20.0344	1.12525	0.562623	+	0.826713i	$0.309792\pi$
$318$	0	0
$319$	13.3472	0.747298
$320$	0	0
$321$	26.1993	1.46230
$322$	0	0
$323$	26.1170	1.45319
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.2167	1.72629
$328$	0	0
$329$	−1.76048	−0.0970582
$330$	0	0
$331$	4.65423	0.255820	0.127910	−	0.991786i	$-0.459173\pi$
$331$	4.65423	0.255820	0.127910	+	0.991786i	$0.459173\pi$
$332$	0	0
$333$	19.8672	1.08872
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.48294	−0.407622	−0.203811	−	0.979010i	$-0.565333\pi$
$337$	−7.48294	−0.407622	−0.203811	+	0.979010i	$0.565333\pi$
$338$	0	0
$339$	−10.5600	−0.573539
$340$	0	0
$341$	−41.0186	−2.22128
$342$	0	0
$343$	−8.41641	−0.454443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.8222	0.688331	0.344165	−	0.938909i	$-0.388162\pi$
$347$	12.8222	0.688331	0.344165	+	0.938909i	$0.388162\pi$
$348$	0	0
$349$	30.7341	1.64516	0.822580	−	0.568650i	$-0.192534\pi$
$349$	30.7341	1.64516	0.822580	+	0.568650i	$0.192534\pi$
$350$	0	0
$351$	−1.13887	−0.0607882
$352$	0	0
$353$	32.0965	1.70832	0.854161	−	0.520008i	$-0.174071\pi$
$353$	32.0965	1.70832	0.854161	+	0.520008i	$0.174071\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.75690	0.304688
$358$	0	0
$359$	−20.8830	−1.10216	−0.551080	−	0.834452i	$-0.685784\pi$
$359$	−20.8830	−1.10216	−0.551080	+	0.834452i	$0.685784\pi$
$360$	0	0
$361$	24.3554	1.28186
$362$	0	0
$363$	18.5729	0.974825
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.57858	−0.239000	−0.119500	−	0.992834i	$-0.538129\pi$
$367$	−4.57858	−0.239000	−0.119500	+	0.992834i	$0.538129\pi$
$368$	0	0
$369$	7.86579	0.409477
$370$	0	0
$371$	−3.35777	−0.174327
$372$	0	0
$373$	−26.0165	−1.34709	−0.673543	−	0.739148i	$-0.735228\pi$
$373$	−26.0165	−1.34709	−0.673543	+	0.739148i	$0.735228\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.06943	0.158084
$378$	0	0
$379$	13.4966	0.693276	0.346638	−	0.937999i	$-0.387323\pi$
$379$	13.4966	0.693276	0.346638	+	0.937999i	$0.387323\pi$
$380$	0	0
$381$	−43.8969	−2.24890
$382$	0	0
$383$	11.6113	0.593311	0.296655	−	0.954985i	$-0.404129\pi$
$383$	11.6113	0.593311	0.296655	+	0.954985i	$0.404129\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.1019	0.869336
$388$	0	0
$389$	−1.55614	−0.0788994	−0.0394497	−	0.999222i	$-0.512560\pi$
$389$	−1.55614	−0.0788994	−0.0394497	+	0.999222i	$0.512560\pi$
$390$	0	0
$391$	−0.973479	−0.0492309
$392$	0	0
$393$	−21.5715	−1.08814
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.5508	1.33255	0.666273	−	0.745707i	$-0.267888\pi$
$397$	26.5508	1.33255	0.666273	+	0.745707i	$0.267888\pi$
$398$	0	0
$399$	9.55671	0.478434
$400$	0	0
$401$	16.7895	0.838429	0.419214	−	0.907887i	$-0.362306\pi$
$401$	16.7895	0.838429	0.419214	+	0.907887i	$0.362306\pi$
$402$	0	0
$403$	−9.43299	−0.469891
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.3496	1.70265
$408$	0	0
$409$	13.2966	0.657477	0.328739	−	0.944421i	$-0.393377\pi$
$409$	13.2966	0.657477	0.328739	+	0.944421i	$0.393377\pi$
$410$	0	0
$411$	32.7257	1.61424
$412$	0	0
$413$	−1.58794	−0.0781373
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.619594	−0.0303416
$418$	0	0
$419$	−10.5325	−0.514547	−0.257274	−	0.966339i	$-0.582824\pi$
$419$	−10.5325	−0.514547	−0.257274	+	0.966339i	$0.582824\pi$
$420$	0	0
$421$	−35.1345	−1.71235	−0.856175	−	0.516686i	$-0.827165\pi$
$421$	−35.1345	−1.71235	−0.856175	+	0.516686i	$0.827165\pi$
$422$	0	0
$423$	−7.16414	−0.348333
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.37148	−0.308338
$428$	0	0
$429$	10.2119	0.493034
$430$	0	0
$431$	20.6207	0.993263	0.496632	−	0.867961i	$-0.334570\pi$
$431$	20.6207	0.993263	0.496632	+	0.867961i	$0.334570\pi$
$432$	0	0
$433$	−27.0050	−1.29778	−0.648888	−	0.760884i	$-0.724766\pi$
$433$	−27.0050	−1.29778	−0.648888	+	0.760884i	$0.724766\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.61602	−0.0773046
$438$	0	0
$439$	4.46258	0.212987	0.106494	−	0.994313i	$-0.466038\pi$
$439$	4.46258	0.212987	0.106494	+	0.994313i	$0.466038\pi$
$440$	0	0
$441$	−16.6447	−0.792604
$442$	0	0
$443$	20.2085	0.960135	0.480067	−	0.877232i	$-0.340612\pi$
$443$	20.2085	0.960135	0.480067	+	0.877232i	$0.340612\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.7161	1.68931
$448$	0	0
$449$	−3.62413	−0.171033	−0.0855167	−	0.996337i	$-0.527254\pi$
$449$	−3.62413	−0.171033	−0.0855167	+	0.996337i	$0.527254\pi$
$450$	0	0
$451$	13.5996	0.640381
$452$	0	0
$453$	−3.54145	−0.166392
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.5327	−1.28793	−0.643963	−	0.765057i	$-0.722711\pi$
$457$	−27.5327	−1.28793	−0.643963	+	0.765057i	$0.722711\pi$
$458$	0	0
$459$	−4.51726	−0.210848
$460$	0	0
$461$	26.4838	1.23347	0.616737	−	0.787169i	$-0.288454\pi$
$461$	26.4838	1.23347	0.616737	+	0.787169i	$0.288454\pi$
$462$	0	0
$463$	−9.95150	−0.462485	−0.231243	−	0.972896i	$-0.574279\pi$
$463$	−9.95150	−0.462485	−0.231243	+	0.972896i	$0.574279\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.6260	1.32465	0.662326	−	0.749216i	$-0.269570\pi$
$467$	28.6260	1.32465	0.662326	+	0.749216i	$0.269570\pi$
$468$	0	0
$469$	−1.04870	−0.0484243
$470$	0	0
$471$	−5.40069	−0.248851
$472$	0	0
$473$	29.5684	1.35956
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.6642	−0.625643
$478$	0	0
$479$	30.0965	1.37514	0.687571	−	0.726117i	$-0.258677\pi$
$479$	30.0965	1.37514	0.687571	+	0.726117i	$0.258677\pi$
$480$	0	0
$481$	7.89934	0.360179
$482$	0	0
$483$	−0.356215	−0.0162083
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.4970	0.883494	0.441747	−	0.897140i	$-0.354359\pi$
$487$	19.4970	0.883494	0.441747	+	0.897140i	$0.354359\pi$
$488$	0	0
$489$	36.4493	1.64830
$490$	0	0
$491$	−34.8032	−1.57064	−0.785322	−	0.619087i	$-0.787503\pi$
$491$	−34.8032	−1.57064	−0.785322	+	0.619087i	$0.787503\pi$
$492$	0	0
$493$	12.1747	0.548323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.69683	0.434962
$498$	0	0
$499$	−22.0126	−0.985418	−0.492709	−	0.870194i	$-0.663993\pi$
$499$	−22.0126	−0.985418	−0.492709	+	0.870194i	$0.663993\pi$
$500$	0	0
$501$	−4.46856	−0.199641
$502$	0	0
$503$	−0.686028	−0.0305885	−0.0152942	−	0.999883i	$-0.504868\pi$
$503$	−0.686028	−0.0305885	−0.0152942	+	0.999883i	$0.504868\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−28.1810	−1.25156
$508$	0	0
$509$	−29.4378	−1.30481	−0.652403	−	0.757872i	$-0.726239\pi$
$509$	−29.4378	−1.30481	−0.652403	+	0.757872i	$0.726239\pi$
$510$	0	0
$511$	7.28131	0.322106
$512$	0	0
$513$	−7.49885	−0.331082
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.3865	−0.544758
$518$	0	0
$519$	−35.0741	−1.53958
$520$	0	0
$521$	4.33307	0.189835	0.0949177	−	0.995485i	$-0.469741\pi$
$521$	4.33307	0.189835	0.0949177	+	0.995485i	$0.469741\pi$
$522$	0	0
$523$	33.1514	1.44961	0.724806	−	0.688953i	$-0.241929\pi$
$523$	33.1514	1.44961	0.724806	+	0.688953i	$0.241929\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.4155	−1.62984
$528$	0	0
$529$	−22.9398	−0.997381
$530$	0	0
$531$	−6.46201	−0.280427
$532$	0	0
$533$	3.12749	0.135467
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.27591	−0.400285
$538$	0	0
$539$	−28.7780	−1.23955
$540$	0	0
$541$	−33.5235	−1.44129	−0.720643	−	0.693306i	$-0.756153\pi$
$541$	−33.5235	−1.44129	−0.720643	+	0.693306i	$0.756153\pi$
$542$	0	0
$543$	−42.6957	−1.83225
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.2483	0.865753	0.432877	−	0.901453i	$-0.357499\pi$
$547$	20.2483	0.865753	0.432877	+	0.901453i	$0.357499\pi$
$548$	0	0
$549$	−25.9283	−1.10659
$550$	0	0
$551$	20.2106	0.861002
$552$	0	0
$553$	2.67823	0.113890
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.9428	1.43820	0.719102	−	0.694905i	$-0.244553\pi$
$557$	33.9428	1.43820	0.719102	+	0.694905i	$0.244553\pi$
$558$	0	0
$559$	6.79981	0.287601
$560$	0	0
$561$	40.5049	1.71012
$562$	0	0
$563$	11.6644	0.491596	0.245798	−	0.969321i	$-0.420950\pi$
$563$	11.6644	0.491596	0.245798	+	0.969321i	$0.420950\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−6.31611	−0.265252
$568$	0	0
$569$	−42.0676	−1.76356	−0.881782	−	0.471657i	$-0.843656\pi$
$569$	−42.0676	−1.76356	−0.881782	+	0.471657i	$0.843656\pi$
$570$	0	0
$571$	5.75771	0.240953	0.120476	−	0.992716i	$-0.461558\pi$
$571$	5.75771	0.240953	0.120476	+	0.992716i	$0.461558\pi$
$572$	0	0
$573$	−2.03041	−0.0848216
$574$	0	0
$575$	0	0
$576$	0	0
$577$	27.5362	1.14635	0.573173	−	0.819434i	$-0.305712\pi$
$577$	27.5362	1.14635	0.573173	+	0.819434i	$0.305712\pi$
$578$	0	0
$579$	−49.2782	−2.04793
$580$	0	0
$581$	−10.7605	−0.446420
$582$	0	0
$583$	−23.6249	−0.978444
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.8833	−1.64616	−0.823079	−	0.567926i	$-0.807746\pi$
$587$	−39.8833	−1.64616	−0.823079	+	0.567926i	$0.807746\pi$
$588$	0	0
$589$	−62.1114	−2.55925
$590$	0	0
$591$	−56.8404	−2.33810
$592$	0	0
$593$	−4.18267	−0.171762	−0.0858808	−	0.996305i	$-0.527370\pi$
$593$	−4.18267	−0.171762	−0.0858808	+	0.996305i	$0.527370\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	54.2909	2.22198
$598$	0	0
$599$	−11.2795	−0.460867	−0.230434	−	0.973088i	$-0.574014\pi$
$599$	−11.2795	−0.460867	−0.230434	+	0.973088i	$0.574014\pi$
$600$	0	0
$601$	−5.86925	−0.239412	−0.119706	−	0.992809i	$-0.538195\pi$
$601$	−5.86925	−0.239412	−0.119706	+	0.992809i	$0.538195\pi$
$602$	0	0
$603$	−4.26760	−0.173790
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.1210	0.979040	0.489520	−	0.871992i	$-0.337172\pi$
$607$	24.1210	0.979040	0.489520	+	0.871992i	$0.337172\pi$
$608$	0	0
$609$	4.45498	0.180525
$610$	0	0
$611$	−2.84851	−0.115238
$612$	0	0
$613$	25.6577	1.03630	0.518152	−	0.855289i	$-0.326620\pi$
$613$	25.6577	1.03630	0.518152	+	0.855289i	$0.326620\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.73415	0.271107	0.135553	−	0.990770i	$-0.456719\pi$
$617$	6.73415	0.271107	0.135553	+	0.990770i	$0.456719\pi$
$618$	0	0
$619$	17.2239	0.692289	0.346144	−	0.938181i	$-0.387491\pi$
$619$	17.2239	0.692289	0.346144	+	0.938181i	$0.387491\pi$
$620$	0	0
$621$	0.279510	0.0112164
$622$	0	0
$623$	6.79189	0.272111
$624$	0	0
$625$	0	0
$626$	0	0
$627$	67.2399	2.68530
$628$	0	0
$629$	31.3323	1.24930
$630$	0	0
$631$	−40.9069	−1.62848	−0.814239	−	0.580530i	$-0.802846\pi$
$631$	−40.9069	−1.62848	−0.814239	+	0.580530i	$0.802846\pi$
$632$	0	0
$633$	−12.5225	−0.497726
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.61803	−0.262216
$638$	0	0
$639$	39.4606	1.56104
$640$	0	0
$641$	24.0419	0.949596	0.474798	−	0.880095i	$-0.342521\pi$
$641$	24.0419	0.949596	0.474798	+	0.880095i	$0.342521\pi$
$642$	0	0
$643$	8.82218	0.347913	0.173956	−	0.984753i	$-0.444345\pi$
$643$	8.82218	0.347913	0.173956	+	0.984753i	$0.444345\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0104	1.25846	0.629230	−	0.777219i	$-0.283370\pi$
$647$	32.0104	1.25846	0.629230	+	0.777219i	$0.283370\pi$
$648$	0	0
$649$	−11.1725	−0.438561
$650$	0	0
$651$	−13.6910	−0.536594
$652$	0	0
$653$	−41.9238	−1.64061	−0.820303	−	0.571930i	$-0.806195\pi$
$653$	−41.9238	−1.64061	−0.820303	+	0.571930i	$0.806195\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	29.6308	1.15601
$658$	0	0
$659$	−15.0158	−0.584931	−0.292466	−	0.956276i	$-0.594476\pi$
$659$	−15.0158	−0.584931	−0.292466	+	0.956276i	$0.594476\pi$
$660$	0	0
$661$	−15.1059	−0.587553	−0.293777	−	0.955874i	$-0.594912\pi$
$661$	−15.1059	−0.587553	−0.293777	+	0.955874i	$0.594912\pi$
$662$	0	0
$663$	9.31486	0.361759
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.753326	−0.0291689
$668$	0	0
$669$	61.5047	2.37791
$670$	0	0
$671$	−44.8290	−1.73060
$672$	0	0
$673$	42.4093	1.63476	0.817379	−	0.576100i	$-0.195426\pi$
$673$	42.4093	1.63476	0.817379	+	0.576100i	$0.195426\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.2667	0.932644	0.466322	−	0.884615i	$-0.345579\pi$
$677$	24.2667	0.932644	0.466322	+	0.884615i	$0.345579\pi$
$678$	0	0
$679$	3.96501	0.152163
$680$	0	0
$681$	61.2354	2.34655
$682$	0	0
$683$	21.6165	0.827131	0.413566	−	0.910474i	$-0.364283\pi$
$683$	21.6165	0.827131	0.413566	+	0.910474i	$0.364283\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.53722	−0.325715
$688$	0	0
$689$	−5.43299	−0.206981
$690$	0	0
$691$	1.37393	0.0522666	0.0261333	−	0.999658i	$-0.491681\pi$
$691$	1.37393	0.0522666	0.0261333	+	0.999658i	$0.491681\pi$
$692$	0	0
$693$	6.75911	0.256757
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.4050	0.469874
$698$	0	0
$699$	−23.5691	−0.891467
$700$	0	0
$701$	−0.303800	−0.0114744	−0.00573719	−	0.999984i	$-0.501826\pi$
$701$	−0.303800	−0.0114744	−0.00573719	+	0.999984i	$0.501826\pi$
$702$	0	0
$703$	52.0131	1.96171
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.0532	−0.453306
$708$	0	0
$709$	−36.2801	−1.36253	−0.681263	−	0.732039i	$-0.738569\pi$
$709$	−36.2801	−1.36253	−0.681263	+	0.732039i	$0.738569\pi$
$710$	0	0
$711$	10.8989	0.408739
$712$	0	0
$713$	2.31512	0.0867021
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.7155	0.549562
$718$	0	0
$719$	−24.5470	−0.915450	−0.457725	−	0.889094i	$-0.651336\pi$
$719$	−24.5470	−0.915450	−0.457725	+	0.889094i	$0.651336\pi$
$720$	0	0
$721$	−2.23952	−0.0834042
$722$	0	0
$723$	−27.8515	−1.03581
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.99184	0.259313	0.129657	−	0.991559i	$-0.458613\pi$
$727$	6.99184	0.259313	0.129657	+	0.991559i	$0.458613\pi$
$728$	0	0
$729$	−17.6794	−0.654792
$730$	0	0
$731$	26.9711	0.997562
$732$	0	0
$733$	6.73849	0.248892	0.124446	−	0.992226i	$-0.460285\pi$
$733$	6.73849	0.248892	0.124446	+	0.992226i	$0.460285\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.37851	−0.271791
$738$	0	0
$739$	−48.5073	−1.78437	−0.892185	−	0.451671i	$-0.850828\pi$
$739$	−48.5073	−1.78437	−0.892185	+	0.451671i	$0.850828\pi$
$740$	0	0
$741$	15.4631	0.568051
$742$	0	0
$743$	−30.4013	−1.11531	−0.557657	−	0.830071i	$-0.688300\pi$
$743$	−30.4013	−1.11531	−0.557657	+	0.830071i	$0.688300\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−43.7890	−1.60216
$748$	0	0
$749$	6.89488	0.251933
$750$	0	0
$751$	−36.7809	−1.34215	−0.671076	−	0.741388i	$-0.734168\pi$
$751$	−36.7809	−1.34215	−0.671076	+	0.741388i	$0.734168\pi$
$752$	0	0
$753$	−22.5172	−0.820572
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−43.5105	−1.58142	−0.790709	−	0.612192i	$-0.790288\pi$
$757$	−43.5105	−1.58142	−0.790709	+	0.612192i	$0.790288\pi$
$758$	0	0
$759$	−2.50628	−0.0909723
$760$	0	0
$761$	−27.8125	−1.00820	−0.504101	−	0.863644i	$-0.668176\pi$
$761$	−27.8125	−1.00820	−0.504101	+	0.863644i	$0.668176\pi$
$762$	0	0
$763$	8.21533	0.297415
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.56934	−0.0927734
$768$	0	0
$769$	−37.1951	−1.34129	−0.670644	−	0.741779i	$-0.733982\pi$
$769$	−37.1951	−1.34129	−0.670644	+	0.741779i	$0.733982\pi$
$770$	0	0
$771$	12.8454	0.462615
$772$	0	0
$773$	38.8423	1.39706	0.698530	−	0.715581i	$-0.253838\pi$
$773$	38.8423	1.39706	0.698530	+	0.715581i	$0.253838\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.4651	0.411308
$778$	0	0
$779$	20.5929	0.737818
$780$	0	0
$781$	68.2257	2.44131
$782$	0	0
$783$	−3.49568	−0.124925
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−23.0053	−0.820050	−0.410025	−	0.912074i	$-0.634480\pi$
$787$	−23.0053	−0.820050	−0.410025	+	0.912074i	$0.634480\pi$
$788$	0	0
$789$	2.88082	0.102560
$790$	0	0
$791$	−2.77908	−0.0988126
$792$	0	0
$793$	−10.3093	−0.366093
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.2622	−0.434349	−0.217175	−	0.976133i	$-0.569684\pi$
$797$	−12.2622	−0.434349	−0.217175	+	0.976133i	$0.569684\pi$
$798$	0	0
$799$	−11.2985	−0.399711
$800$	0	0
$801$	27.6392	0.976582
$802$	0	0
$803$	51.2304	1.80788
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−28.0315	−0.986755
$808$	0	0
$809$	8.90292	0.313010	0.156505	−	0.987677i	$-0.449977\pi$
$809$	8.90292	0.313010	0.156505	+	0.987677i	$0.449977\pi$
$810$	0	0
$811$	39.5305	1.38810	0.694051	−	0.719926i	$-0.255824\pi$
$811$	39.5305	1.38810	0.694051	+	0.719926i	$0.255824\pi$
$812$	0	0
$813$	55.5055	1.94666
$814$	0	0
$815$	0	0
$816$	0	0
$817$	44.7733	1.56642
$818$	0	0
$819$	1.55439	0.0543146
$820$	0	0
$821$	26.6000	0.928347	0.464173	−	0.885744i	$-0.346351\pi$
$821$	26.6000	0.928347	0.464173	+	0.885744i	$0.346351\pi$
$822$	0	0
$823$	−32.3771	−1.12859	−0.564297	−	0.825572i	$-0.690853\pi$
$823$	−32.3771	−1.12859	−0.564297	+	0.825572i	$0.690853\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.6398	−0.926356	−0.463178	−	0.886265i	$-0.653291\pi$
$827$	−26.6398	−0.926356	−0.463178	+	0.886265i	$0.653291\pi$
$828$	0	0
$829$	−25.8735	−0.898624	−0.449312	−	0.893375i	$-0.648331\pi$
$829$	−25.8735	−0.898624	−0.449312	+	0.893375i	$0.648331\pi$
$830$	0	0
$831$	26.5550	0.921183
$832$	0	0
$833$	−26.2501	−0.909512
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.7429	0.371330
$838$	0	0
$839$	10.6020	0.366023	0.183011	−	0.983111i	$-0.441416\pi$
$839$	10.6020	0.366023	0.183011	+	0.983111i	$0.441416\pi$
$840$	0	0
$841$	−19.5786	−0.675123
$842$	0	0
$843$	−28.1915	−0.970965
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.88785	0.167948
$848$	0	0
$849$	2.86295	0.0982563
$850$	0	0
$851$	−1.93872	−0.0664586
$852$	0	0
$853$	−3.76658	−0.128965	−0.0644825	−	0.997919i	$-0.520540\pi$
$853$	−3.76658	−0.128965	−0.0644825	+	0.997919i	$0.520540\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.7966	−1.01783	−0.508915	−	0.860817i	$-0.669953\pi$
$857$	−29.7966	−1.01783	−0.508915	+	0.860817i	$0.669953\pi$
$858$	0	0
$859$	52.2878	1.78404	0.892018	−	0.451999i	$-0.149289\pi$
$859$	52.2878	1.78404	0.892018	+	0.451999i	$0.149289\pi$
$860$	0	0
$861$	4.53924	0.154697
$862$	0	0
$863$	29.0392	0.988507	0.494253	−	0.869318i	$-0.335442\pi$
$863$	29.0392	0.988507	0.494253	+	0.869318i	$0.335442\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.97612	−0.101074
$868$	0	0
$869$	18.8437	0.639228
$870$	0	0
$871$	−1.69683	−0.0574948
$872$	0	0
$873$	16.1353	0.546098
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.4495	0.352853	0.176427	−	0.984314i	$-0.443546\pi$
$877$	10.4495	0.352853	0.176427	+	0.984314i	$0.443546\pi$
$878$	0	0
$879$	−29.9195	−1.00916
$880$	0	0
$881$	16.9437	0.570846	0.285423	−	0.958402i	$-0.407866\pi$
$881$	16.9437	0.570846	0.285423	+	0.958402i	$0.407866\pi$
$882$	0	0
$883$	9.18259	0.309019	0.154509	−	0.987991i	$-0.450620\pi$
$883$	9.18259	0.309019	0.154509	+	0.987991i	$0.450620\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.3291	−1.45485	−0.727424	−	0.686188i	$-0.759283\pi$
$887$	−43.3291	−1.45485	−0.727424	+	0.686188i	$0.759283\pi$
$888$	0	0
$889$	−11.5524	−0.387454
$890$	0	0
$891$	−44.4394	−1.48878
$892$	0	0
$893$	−18.7560	−0.627645
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.576367	−0.0192443
$898$	0	0
$899$	−28.9539	−0.965668
$900$	0	0
$901$	−21.5497	−0.717924
$902$	0	0
$903$	9.86925	0.328428
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.3689	−0.908770	−0.454385	−	0.890805i	$-0.650141\pi$
$907$	−27.3689	−0.908770	−0.454385	+	0.890805i	$0.650141\pi$
$908$	0	0
$909$	−49.0495	−1.62687
$910$	0	0
$911$	46.0713	1.52641	0.763205	−	0.646157i	$-0.223625\pi$
$911$	46.0713	1.52641	0.763205	+	0.646157i	$0.223625\pi$
$912$	0	0
$913$	−75.7094	−2.50562
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.67698	−0.187470
$918$	0	0
$919$	−40.7632	−1.34465	−0.672327	−	0.740254i	$-0.734705\pi$
$919$	−40.7632	−1.34465	−0.672327	+	0.740254i	$0.734705\pi$
$920$	0	0
$921$	20.6795	0.681412
$922$	0	0
$923$	15.6898	0.516436
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−9.11359	−0.299330
$928$	0	0
$929$	−59.5844	−1.95490	−0.977451	−	0.211162i	$-0.932275\pi$
$929$	−59.5844	−1.95490	−0.977451	+	0.211162i	$0.932275\pi$
$930$	0	0
$931$	−43.5763	−1.42816
$932$	0	0
$933$	−2.83833	−0.0929227
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.5964	0.836197	0.418098	−	0.908402i	$-0.362697\pi$
$937$	25.5964	0.836197	0.418098	+	0.908402i	$0.362697\pi$
$938$	0	0
$939$	14.7750	0.482164
$940$	0	0
$941$	−2.30166	−0.0750321	−0.0375161	−	0.999296i	$-0.511945\pi$
$941$	−2.30166	−0.0750321	−0.0375161	+	0.999296i	$0.511945\pi$
$942$	0	0
$943$	−0.767575	−0.0249957
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.5912	−0.669123	−0.334561	−	0.942374i	$-0.608588\pi$
$947$	−20.5912	−0.669123	−0.334561	+	0.942374i	$0.608588\pi$
$948$	0	0
$949$	11.7814	0.382441
$950$	0	0
$951$	47.0492	1.52567
$952$	0	0
$953$	−9.07646	−0.294016	−0.147008	−	0.989135i	$-0.546964\pi$
$953$	−9.07646	−0.294016	−0.147008	+	0.989135i	$0.546964\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	31.3447	1.01323
$958$	0	0
$959$	8.61244	0.278110
$960$	0	0
$961$	57.9814	1.87037
$962$	0	0
$963$	28.0583	0.904165
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−44.1418	−1.41951	−0.709753	−	0.704451i	$-0.751193\pi$
$967$	−44.1418	−1.41951	−0.709753	+	0.704451i	$0.751193\pi$
$968$	0	0
$969$	61.3335	1.97032
$970$	0	0
$971$	−38.3756	−1.23153	−0.615766	−	0.787929i	$-0.711153\pi$
$971$	−38.3756	−1.23153	−0.615766	+	0.787929i	$0.711153\pi$
$972$	0	0
$973$	−0.163059	−0.00522743
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.6199	0.499724	0.249862	−	0.968282i	$-0.419615\pi$
$977$	15.6199	0.499724	0.249862	+	0.968282i	$0.419615\pi$
$978$	0	0
$979$	47.7870	1.52728
$980$	0	0
$981$	33.4317	1.06739
$982$	0	0
$983$	−42.0550	−1.34135	−0.670673	−	0.741753i	$-0.733995\pi$
$983$	−42.0550	−1.34135	−0.670673	+	0.741753i	$0.733995\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.13433	−0.131597
$988$	0	0
$989$	−1.66887	−0.0530669
$990$	0	0
$991$	−15.1453	−0.481106	−0.240553	−	0.970636i	$-0.577329\pi$
$991$	−15.1453	−0.481106	−0.240553	+	0.970636i	$0.577329\pi$
$992$	0	0
$993$	10.9301	0.346855
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.7989	0.437017	0.218509	−	0.975835i	$-0.429881\pi$
$997$	13.7989	0.437017	0.218509	+	0.975835i	$0.429881\pi$
$998$	0	0
$999$	−8.99630	−0.284630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.ba.1.3		4
4.3	odd	2		5000.2.a.e.1.2	✓	4
5.4	even	2		10000.2.a.p.1.2		4
20.19	odd	2		5000.2.a.j.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.e.1.2	✓	4	4.3	odd	2
5000.2.a.j.1.3	yes	4	20.19	odd	2
10000.2.a.p.1.2		4	5.4	even	2
10000.2.a.ba.1.3		4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34841 q^{3} +0.618034 q^{7} +2.51505 q^{9} +O(q^{10})\)	\(q+2.34841 q^{3} +0.618034 q^{7} +2.51505 q^{9} +4.34841 q^{11} +1.00000 q^{13} +3.96645 q^{17} +6.58448 q^{19} +1.45140 q^{21} -0.245428 q^{23} -1.13887 q^{27} +3.06943 q^{29} -9.43299 q^{31} +10.2119 q^{33} +7.89934 q^{37} +2.34841 q^{39} +3.12749 q^{41} +6.79981 q^{43} -2.84851 q^{47} -6.61803 q^{49} +9.31486 q^{51} -5.43299 q^{53} +15.4631 q^{57} -2.56934 q^{59} -10.3093 q^{61} +1.55439 q^{63} -1.69683 q^{67} -0.576367 q^{69} +15.6898 q^{71} +11.7814 q^{73} +2.68747 q^{77} +4.33346 q^{79} -10.2197 q^{81} -17.4108 q^{83} +7.20830 q^{87} +10.9895 q^{89} +0.618034 q^{91} -22.1526 q^{93} +6.41552 q^{97} +10.9365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 2 * q^7 + 9 * q^9	\( 4 q + 3 q^{3} - 2 q^{7} + 9 q^{9} + 11 q^{11} + 4 q^{13} + 5 q^{17} + 11 q^{19} - 4 q^{21} + 2 q^{23} + 24 q^{27} - 2 q^{29} + 27 q^{33} - q^{37} + 3 q^{39} - 2 q^{41} + 11 q^{43} + 11 q^{47} - 22 q^{49} + 20 q^{51} + 16 q^{53} + 22 q^{57} - 12 q^{59} - 15 q^{61} - 7 q^{63} + 6 q^{67} - 46 q^{69} + 8 q^{71} + 3 q^{73} - 8 q^{77} - 4 q^{79} + 16 q^{81} - 7 q^{83} - 14 q^{87} - 11 q^{89} - 2 q^{91} - 10 q^{93} + 41 q^{97} + 51 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 2 * q^7 + 9 * q^9 + 11 * q^11 + 4 * q^13 + 5 * q^17 + 11 * q^19 - 4 * q^21 + 2 * q^23 + 24 * q^27 - 2 * q^29 + 27 * q^33 - q^37 + 3 * q^39 - 2 * q^41 + 11 * q^43 + 11 * q^47 - 22 * q^49 + 20 * q^51 + 16 * q^53 + 22 * q^57 - 12 * q^59 - 15 * q^61 - 7 * q^63 + 6 * q^67 - 46 * q^69 + 8 * q^71 + 3 * q^73 - 8 * q^77 - 4 * q^79 + 16 * q^81 - 7 * q^83 - 14 * q^87 - 11 * q^89 - 2 * q^91 - 10 * q^93 + 41 * q^97 + 51 * q^99

Introduction

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