LMFDB - Embedded newform 684.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,4,Mod(1,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, 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("684.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	684.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:	$40.3573064439$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.35529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 52x + 4 $ x^3 - x^2 - 52*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.76918$ of defining polynomial
Character	$\chi$	$=$	684.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.0323 q^{5} +5.70454 q^{7} +O(q^{10})$	$q-11.0323 q^{5} +5.70454 q^{7} -59.0565 q^{11} +88.0721 q^{13} +113.193 q^{17} +19.0000 q^{19} -40.1701 q^{23} -3.28800 q^{25} +66.5906 q^{29} -248.695 q^{31} -62.9343 q^{35} -330.857 q^{37} +172.072 q^{41} +56.9507 q^{43} -483.093 q^{47} -310.458 q^{49} +104.380 q^{53} +651.530 q^{55} -579.966 q^{59} +314.576 q^{61} -971.639 q^{65} -12.2803 q^{67} +711.303 q^{71} -704.738 q^{73} -336.890 q^{77} -50.4809 q^{79} -849.747 q^{83} -1248.78 q^{85} -704.355 q^{89} +502.411 q^{91} -209.614 q^{95} -232.096 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{5} + 44 q^{7}+O(q^{10}) $ 3 * q - 9 * q^5 + 44 * q^7	$ 3 q - 9 q^{5} + 44 q^{7} - 79 q^{11} - 11 q^{13} - 82 q^{17} + 57 q^{19} - 103 q^{23} - 210 q^{25} + 93 q^{29} - 116 q^{31} + 93 q^{35} - 466 q^{37} + 188 q^{41} - 11 q^{43} - 163 q^{47} + 69 q^{49} - 197 q^{53} + 231 q^{55} - 1381 q^{59} - 405 q^{61} - 1188 q^{65} + 943 q^{67} - 1052 q^{71} - 580 q^{73} - 1855 q^{77} - 1402 q^{79} - 1802 q^{83} - 1245 q^{85} - 1966 q^{89} - 1723 q^{91} - 171 q^{95} + 48 q^{97}+O(q^{100}) $ 3 * q - 9 * q^5 + 44 * q^7 - 79 * q^11 - 11 * q^13 - 82 * q^17 + 57 * q^19 - 103 * q^23 - 210 * q^25 + 93 * q^29 - 116 * q^31 + 93 * q^35 - 466 * q^37 + 188 * q^41 - 11 * q^43 - 163 * q^47 + 69 * q^49 - 197 * q^53 + 231 * q^55 - 1381 * q^59 - 405 * q^61 - 1188 * q^65 + 943 * q^67 - 1052 * q^71 - 580 * q^73 - 1855 * q^77 - 1402 * q^79 - 1802 * q^83 - 1245 * q^85 - 1966 * q^89 - 1723 * q^91 - 171 * q^95 + 48 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−11.0323	−0.986760	−0.493380	−	0.869814i	$-0.664239\pi$
$5$	−11.0323	−0.986760	−0.493380	+	0.869814i	$0.664239\pi$
$6$	0	0
$7$	5.70454	0.308016	0.154008	−	0.988070i	$-0.450782\pi$
$7$	5.70454	0.308016	0.154008	+	0.988070i	$0.450782\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−59.0565	−1.61874	−0.809372	−	0.587296i	$-0.800193\pi$
$11$	−59.0565	−1.61874	−0.809372	+	0.587296i	$0.800193\pi$
$12$	0	0
$13$	88.0721	1.87898	0.939492	−	0.342570i	$-0.111297\pi$
$13$	88.0721	1.87898	0.939492	+	0.342570i	$0.111297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	113.193	1.61490	0.807452	−	0.589933i	$-0.200846\pi$
$17$	113.193	1.61490	0.807452	+	0.589933i	$0.200846\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−40.1701	−0.364176	−0.182088	−	0.983282i	$-0.558286\pi$
$23$	−40.1701	−0.364176	−0.182088	+	0.983282i	$0.558286\pi$
$24$	0	0
$25$	−3.28800	−0.0263040
$26$	0	0
$27$	0	0
$28$	0	0
$29$	66.5906	0.426399	0.213199	−	0.977009i	$-0.431612\pi$
$29$	66.5906	0.426399	0.213199	+	0.977009i	$0.431612\pi$
$30$	0	0
$31$	−248.695	−1.44087	−0.720433	−	0.693524i	$-0.756057\pi$
$31$	−248.695	−1.44087	−0.720433	+	0.693524i	$0.756057\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−62.9343	−0.303938
$36$	0	0
$37$	−330.857	−1.47007	−0.735034	−	0.678031i	$-0.762834\pi$
$37$	−330.857	−1.47007	−0.735034	+	0.678031i	$0.762834\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	172.072	0.655441	0.327721	−	0.944775i	$-0.393720\pi$
$41$	172.072	0.655441	0.327721	+	0.944775i	$0.393720\pi$
$42$	0	0
$43$	56.9507	0.201974	0.100987	−	0.994888i	$-0.467800\pi$
$43$	56.9507	0.201974	0.100987	+	0.994888i	$0.467800\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−483.093	−1.49929	−0.749643	−	0.661843i	$-0.769775\pi$
$47$	−483.093	−1.49929	−0.749643	+	0.661843i	$0.769775\pi$
$48$	0	0
$49$	−310.458	−0.905126
$50$	0	0
$51$	0	0
$52$	0	0
$53$	104.380	0.270521	0.135261	−	0.990810i	$-0.456813\pi$
$53$	104.380	0.270521	0.135261	+	0.990810i	$0.456813\pi$
$54$	0	0
$55$	651.530	1.59731
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−579.966	−1.27975	−0.639874	−	0.768479i	$-0.721014\pi$
$59$	−579.966	−1.27975	−0.639874	+	0.768479i	$0.721014\pi$
$60$	0	0
$61$	314.576	0.660283	0.330142	−	0.943931i	$-0.392903\pi$
$61$	314.576	0.660283	0.330142	+	0.943931i	$0.392903\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−971.639	−1.85411
$66$	0	0
$67$	−12.2803	−0.0223921	−0.0111961	−	0.999937i	$-0.503564\pi$
$67$	−12.2803	−0.0223921	−0.0111961	+	0.999937i	$0.503564\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	711.303	1.18896	0.594480	−	0.804111i	$-0.297358\pi$
$71$	711.303	1.18896	0.594480	+	0.804111i	$0.297358\pi$
$72$	0	0
$73$	−704.738	−1.12991	−0.564954	−	0.825122i	$-0.691106\pi$
$73$	−704.738	−1.12991	−0.564954	+	0.825122i	$0.691106\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−336.890	−0.498600
$78$	0	0
$79$	−50.4809	−0.0718929	−0.0359465	−	0.999354i	$-0.511445\pi$
$79$	−50.4809	−0.0718929	−0.0359465	+	0.999354i	$0.511445\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−849.747	−1.12376	−0.561878	−	0.827220i	$-0.689921\pi$
$83$	−849.747	−1.12376	−0.561878	+	0.827220i	$0.689921\pi$
$84$	0	0
$85$	−1248.78	−1.59352
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−704.355	−0.838893	−0.419447	−	0.907780i	$-0.637776\pi$
$89$	−704.355	−0.838893	−0.419447	+	0.907780i	$0.637776\pi$
$90$	0	0
$91$	502.411	0.578758
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−209.614	−0.226378
$96$	0	0
$97$	−232.096	−0.242946	−0.121473	−	0.992595i	$-0.538762\pi$
$97$	−232.096	−0.242946	−0.121473	+	0.992595i	$0.538762\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−943.254	−0.929280	−0.464640	−	0.885500i	$-0.653816\pi$
$101$	−943.254	−0.929280	−0.464640	+	0.885500i	$0.653816\pi$
$102$	0	0
$103$	−1796.48	−1.71857	−0.859286	−	0.511496i	$-0.829092\pi$
$103$	−1796.48	−1.71857	−0.859286	+	0.511496i	$0.829092\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1207.11	1.09061	0.545306	−	0.838237i	$-0.316413\pi$
$107$	1207.11	1.09061	0.545306	+	0.838237i	$0.316413\pi$
$108$	0	0
$109$	−640.900	−0.563185	−0.281592	−	0.959534i	$-0.590863\pi$
$109$	−640.900	−0.563185	−0.281592	+	0.959534i	$0.590863\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1217.17	−1.01329	−0.506644	−	0.862155i	$-0.669114\pi$
$113$	−1217.17	−1.01329	−0.506644	+	0.862155i	$0.669114\pi$
$114$	0	0
$115$	443.169	0.359354
$116$	0	0
$117$	0	0
$118$	0	0
$119$	645.715	0.497417
$120$	0	0
$121$	2156.66	1.62033
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1415.31	1.01272
$126$	0	0
$127$	413.264	0.288750	0.144375	−	0.989523i	$-0.453883\pi$
$127$	413.264	0.288750	0.144375	+	0.989523i	$0.453883\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	834.232	0.556391	0.278195	−	0.960525i	$-0.410264\pi$
$131$	834.232	0.556391	0.278195	+	0.960525i	$0.410264\pi$
$132$	0	0
$133$	108.386	0.0706638
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−157.186	−0.0980243	−0.0490122	−	0.998798i	$-0.515607\pi$
$137$	−157.186	−0.0980243	−0.0490122	+	0.998798i	$0.515607\pi$
$138$	0	0
$139$	2450.06	1.49505	0.747524	−	0.664235i	$-0.231242\pi$
$139$	2450.06	1.49505	0.747524	+	0.664235i	$0.231242\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5201.23	−3.04160
$144$	0	0
$145$	−734.648	−0.420753
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−668.114	−0.367343	−0.183671	−	0.982988i	$-0.558798\pi$
$149$	−668.114	−0.367343	−0.183671	+	0.982988i	$0.558798\pi$
$150$	0	0
$151$	−2437.23	−1.31350	−0.656750	−	0.754108i	$-0.728069\pi$
$151$	−2437.23	−1.31350	−0.656750	+	0.754108i	$0.728069\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2743.68	1.42179
$156$	0	0
$157$	−464.281	−0.236011	−0.118005	−	0.993013i	$-0.537650\pi$
$157$	−464.281	−0.236011	−0.118005	+	0.993013i	$0.537650\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−229.152	−0.112172
$162$	0	0
$163$	−1471.23	−0.706968	−0.353484	−	0.935441i	$-0.615003\pi$
$163$	−1471.23	−0.706968	−0.353484	+	0.935441i	$0.615003\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−380.337	−0.176236	−0.0881178	−	0.996110i	$-0.528085\pi$
$167$	−380.337	−0.176236	−0.0881178	+	0.996110i	$0.528085\pi$
$168$	0	0
$169$	5559.69	2.53058
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1315.05	−0.577927	−0.288963	−	0.957340i	$-0.593311\pi$
$173$	−1315.05	−0.577927	−0.288963	+	0.957340i	$0.593311\pi$
$174$	0	0
$175$	−18.7565	−0.00810206
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3965.24	−1.65573	−0.827866	−	0.560926i	$-0.810445\pi$
$179$	−3965.24	−1.65573	−0.827866	+	0.560926i	$0.810445\pi$
$180$	0	0
$181$	102.709	0.0421783	0.0210892	−	0.999778i	$-0.493287\pi$
$181$	102.709	0.0421783	0.0210892	+	0.999778i	$0.493287\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3650.11	1.45060
$186$	0	0
$187$	−6684.79	−2.61412
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3291.55	−1.24695	−0.623477	−	0.781842i	$-0.714281\pi$
$191$	−3291.55	−1.24695	−0.623477	+	0.781842i	$0.714281\pi$
$192$	0	0
$193$	−1106.18	−0.412564	−0.206282	−	0.978493i	$-0.566136\pi$
$193$	−1106.18	−0.412564	−0.206282	+	0.978493i	$0.566136\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1454.31	0.525967	0.262984	−	0.964800i	$-0.415293\pi$
$197$	1454.31	0.525967	0.262984	+	0.964800i	$0.415293\pi$
$198$	0	0
$199$	−1796.58	−0.639979	−0.319990	−	0.947421i	$-0.603679\pi$
$199$	−1796.58	−0.639979	−0.319990	+	0.947421i	$0.603679\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	379.869	0.131338
$204$	0	0
$205$	−1898.35	−0.646764
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1122.07	−0.371365
$210$	0	0
$211$	2697.12	0.879987	0.439994	−	0.898001i	$-0.354981\pi$
$211$	2697.12	0.879987	0.439994	+	0.898001i	$0.354981\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−628.298	−0.199300
$216$	0	0
$217$	−1418.69	−0.443811
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9969.16	3.03438
$222$	0	0
$223$	−5405.35	−1.62318	−0.811590	−	0.584228i	$-0.801397\pi$
$223$	−5405.35	−1.62318	−0.811590	+	0.584228i	$0.801397\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−696.832	−0.203746	−0.101873	−	0.994797i	$-0.532484\pi$
$227$	−696.832	−0.203746	−0.101873	+	0.994797i	$0.532484\pi$
$228$	0	0
$229$	5249.49	1.51483	0.757414	−	0.652934i	$-0.226462\pi$
$229$	5249.49	1.51483	0.757414	+	0.652934i	$0.226462\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1209.87	−0.340177	−0.170089	−	0.985429i	$-0.554405\pi$
$233$	−1209.87	−0.340177	−0.170089	+	0.985429i	$0.554405\pi$
$234$	0	0
$235$	5329.64	1.47944
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1186.77	0.321197	0.160598	−	0.987020i	$-0.448658\pi$
$239$	1186.77	0.321197	0.160598	+	0.987020i	$0.448658\pi$
$240$	0	0
$241$	−2595.99	−0.693869	−0.346934	−	0.937889i	$-0.612777\pi$
$241$	−2595.99	−0.693869	−0.346934	+	0.937889i	$0.612777\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3425.07	0.893142
$246$	0	0
$247$	1673.37	0.431069
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1493.59	−0.375596	−0.187798	−	0.982208i	$-0.560135\pi$
$251$	−1493.59	−0.375596	−0.187798	+	0.982208i	$0.560135\pi$
$252$	0	0
$253$	2372.30	0.589508
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3323.84	0.806752	0.403376	−	0.915034i	$-0.367837\pi$
$257$	3323.84	0.806752	0.403376	+	0.915034i	$0.367837\pi$
$258$	0	0
$259$	−1887.39	−0.452805
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4258.10	0.998349	0.499175	−	0.866501i	$-0.333637\pi$
$263$	4258.10	0.998349	0.499175	+	0.866501i	$0.333637\pi$
$264$	0	0
$265$	−1151.55	−0.266940
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5924.75	1.34289	0.671447	−	0.741053i	$-0.265673\pi$
$269$	5924.75	1.34289	0.671447	+	0.741053i	$0.265673\pi$
$270$	0	0
$271$	8022.37	1.79824	0.899122	−	0.437698i	$-0.144206\pi$
$271$	8022.37	1.79824	0.899122	+	0.437698i	$0.144206\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	194.177	0.0425794
$276$	0	0
$277$	932.970	0.202371	0.101185	−	0.994868i	$-0.467736\pi$
$277$	932.970	0.202371	0.101185	+	0.994868i	$0.467736\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−956.475	−0.203055	−0.101528	−	0.994833i	$-0.532373\pi$
$281$	−956.475	−0.203055	−0.101528	+	0.994833i	$0.532373\pi$
$282$	0	0
$283$	4386.74	0.921431	0.460715	−	0.887548i	$-0.347593\pi$
$283$	4386.74	0.921431	0.460715	+	0.887548i	$0.347593\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	981.591	0.201887
$288$	0	0
$289$	7899.70	1.60792
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1357.48	0.270665	0.135332	−	0.990800i	$-0.456790\pi$
$293$	1357.48	0.270665	0.135332	+	0.990800i	$0.456790\pi$
$294$	0	0
$295$	6398.37	1.26281
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3537.86	−0.684281
$300$	0	0
$301$	324.878	0.0622114
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3470.50	−0.651541
$306$	0	0
$307$	−9332.32	−1.73493	−0.867465	−	0.497498i	$-0.834252\pi$
$307$	−9332.32	−1.73493	−0.867465	+	0.497498i	$0.834252\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2944.72	−0.536913	−0.268456	−	0.963292i	$-0.586514\pi$
$311$	−2944.72	−0.536913	−0.268456	+	0.963292i	$0.586514\pi$
$312$	0	0
$313$	2890.66	0.522011	0.261006	−	0.965337i	$-0.415946\pi$
$313$	2890.66	0.522011	0.261006	+	0.965337i	$0.415946\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6749.05	−1.19579	−0.597894	−	0.801575i	$-0.703995\pi$
$317$	−6749.05	−1.19579	−0.597894	+	0.801575i	$0.703995\pi$
$318$	0	0
$319$	−3932.60	−0.690231
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2150.67	0.370485
$324$	0	0
$325$	−289.581	−0.0494248
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2755.83	−0.461804
$330$	0	0
$331$	9193.52	1.52665	0.763325	−	0.646014i	$-0.223565\pi$
$331$	9193.52	1.52665	0.763325	+	0.646014i	$0.223565\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	135.480	0.0220957
$336$	0	0
$337$	−5068.33	−0.819257	−0.409628	−	0.912252i	$-0.634342\pi$
$337$	−5068.33	−0.819257	−0.409628	+	0.912252i	$0.634342\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14687.0	2.33240
$342$	0	0
$343$	−3727.68	−0.586810
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8586.97	1.32845	0.664226	−	0.747532i	$-0.268761\pi$
$347$	8586.97	1.32845	0.664226	+	0.747532i	$0.268761\pi$
$348$	0	0
$349$	−4670.75	−0.716388	−0.358194	−	0.933647i	$-0.616607\pi$
$349$	−4670.75	−0.716388	−0.358194	+	0.933647i	$0.616607\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6183.32	0.932308	0.466154	−	0.884703i	$-0.345639\pi$
$353$	6183.32	0.932308	0.466154	+	0.884703i	$0.345639\pi$
$354$	0	0
$355$	−7847.32	−1.17322
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5485.69	−0.806473	−0.403237	−	0.915096i	$-0.632115\pi$
$359$	−5485.69	−0.806473	−0.403237	+	0.915096i	$0.632115\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7774.89	1.11495
$366$	0	0
$367$	−783.301	−0.111411	−0.0557057	−	0.998447i	$-0.517741\pi$
$367$	−783.301	−0.111411	−0.0557057	+	0.998447i	$0.517741\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	595.438	0.0833250
$372$	0	0
$373$	7453.82	1.03470	0.517351	−	0.855773i	$-0.326918\pi$
$373$	7453.82	1.03470	0.517351	+	0.855773i	$0.326918\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5864.77	0.801197
$378$	0	0
$379$	12884.3	1.74623	0.873116	−	0.487513i	$-0.162096\pi$
$379$	12884.3	1.74623	0.873116	+	0.487513i	$0.162096\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4151.36	−0.553850	−0.276925	−	0.960892i	$-0.589315\pi$
$383$	−4151.36	−0.553850	−0.276925	+	0.960892i	$0.589315\pi$
$384$	0	0
$385$	3716.68	0.491999
$386$	0	0
$387$	0	0
$388$	0	0
$389$	395.512	0.0515508	0.0257754	−	0.999668i	$-0.491795\pi$
$389$	395.512	0.0515508	0.0257754	+	0.999668i	$0.491795\pi$
$390$	0	0
$391$	−4546.98	−0.588109
$392$	0	0
$393$	0	0
$394$	0	0
$395$	556.921	0.0709411
$396$	0	0
$397$	−2749.61	−0.347604	−0.173802	−	0.984781i	$-0.555605\pi$
$397$	−2749.61	−0.347604	−0.173802	+	0.984781i	$0.555605\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5196.07	−0.647080	−0.323540	−	0.946214i	$-0.604873\pi$
$401$	−5196.07	−0.647080	−0.323540	+	0.946214i	$0.604873\pi$
$402$	0	0
$403$	−21903.1	−2.70737
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19539.2	2.37966
$408$	0	0
$409$	−8811.47	−1.06528	−0.532639	−	0.846342i	$-0.678800\pi$
$409$	−8811.47	−1.06528	−0.532639	+	0.846342i	$0.678800\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3308.44	−0.394184
$414$	0	0
$415$	9374.67	1.10888
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9178.16	−1.07012	−0.535062	−	0.844813i	$-0.679712\pi$
$419$	−9178.16	−1.07012	−0.535062	+	0.844813i	$0.679712\pi$
$420$	0	0
$421$	11290.8	1.30708	0.653541	−	0.756891i	$-0.273283\pi$
$421$	11290.8	1.30708	0.653541	+	0.756891i	$0.273283\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−372.179	−0.0424784
$426$	0	0
$427$	1794.51	0.203378
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1907.51	0.213183	0.106591	−	0.994303i	$-0.466006\pi$
$431$	1907.51	0.213183	0.106591	+	0.994303i	$0.466006\pi$
$432$	0	0
$433$	13643.2	1.51421	0.757104	−	0.653294i	$-0.226613\pi$
$433$	13643.2	1.51421	0.757104	+	0.653294i	$0.226613\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−763.232	−0.0835476
$438$	0	0
$439$	−9906.74	−1.07705	−0.538523	−	0.842611i	$-0.681017\pi$
$439$	−9906.74	−1.07705	−0.538523	+	0.842611i	$0.681017\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7580.47	−0.813000	−0.406500	−	0.913651i	$-0.633251\pi$
$443$	−7580.47	−0.813000	−0.406500	+	0.913651i	$0.633251\pi$
$444$	0	0
$445$	7770.67	0.827787
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8593.98	−0.903285	−0.451642	−	0.892199i	$-0.649162\pi$
$449$	−8593.98	−0.903285	−0.451642	+	0.892199i	$0.649162\pi$
$450$	0	0
$451$	−10161.9	−1.06099
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5542.76	−0.571096
$456$	0	0
$457$	−14294.6	−1.46318	−0.731589	−	0.681746i	$-0.761221\pi$
$457$	−14294.6	−1.46318	−0.731589	+	0.681746i	$0.761221\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16465.7	−1.66352	−0.831761	−	0.555134i	$-0.812667\pi$
$461$	−16465.7	−1.66352	−0.831761	+	0.555134i	$0.812667\pi$
$462$	0	0
$463$	15503.4	1.55617	0.778083	−	0.628162i	$-0.216192\pi$
$463$	15503.4	1.55617	0.778083	+	0.628162i	$0.216192\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3263.71	−0.323397	−0.161699	−	0.986840i	$-0.551697\pi$
$467$	−3263.71	−0.323397	−0.161699	+	0.986840i	$0.551697\pi$
$468$	0	0
$469$	−70.0532	−0.00689714
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3363.31	−0.326945
$474$	0	0
$475$	−62.4720	−0.00603455
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1763.00	−0.168170	−0.0840852	−	0.996459i	$-0.526797\pi$
$479$	−1763.00	−0.168170	−0.0840852	+	0.996459i	$0.526797\pi$
$480$	0	0
$481$	−29139.2	−2.76223
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2560.55	0.239729
$486$	0	0
$487$	9574.66	0.890902	0.445451	−	0.895306i	$-0.353043\pi$
$487$	9574.66	0.890902	0.445451	+	0.895306i	$0.353043\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18569.3	−1.70676	−0.853380	−	0.521290i	$-0.825451\pi$
$491$	−18569.3	−1.70676	−0.853380	+	0.521290i	$0.825451\pi$
$492$	0	0
$493$	7537.60	0.688593
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4057.66	0.366219
$498$	0	0
$499$	−10116.0	−0.907523	−0.453761	−	0.891123i	$-0.649918\pi$
$499$	−10116.0	−0.907523	−0.453761	+	0.891123i	$0.649918\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−853.516	−0.0756589	−0.0378294	−	0.999284i	$-0.512044\pi$
$503$	−853.516	−0.0756589	−0.0378294	+	0.999284i	$0.512044\pi$
$504$	0	0
$505$	10406.3	0.916977
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5049.04	0.439676	0.219838	−	0.975536i	$-0.429447\pi$
$509$	5049.04	0.439676	0.219838	+	0.975536i	$0.429447\pi$
$510$	0	0
$511$	−4020.21	−0.348030
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19819.4	1.69582
$516$	0	0
$517$	28529.8	2.42696
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8294.77	0.697506	0.348753	−	0.937215i	$-0.386605\pi$
$521$	8294.77	0.697506	0.348753	+	0.937215i	$0.386605\pi$
$522$	0	0
$523$	−3466.62	−0.289837	−0.144918	−	0.989444i	$-0.546292\pi$
$523$	−3466.62	−0.289837	−0.144918	+	0.989444i	$0.546292\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28150.5	−2.32686
$528$	0	0
$529$	−10553.4	−0.867376
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15154.7	1.23156
$534$	0	0
$535$	−13317.2	−1.07617
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18334.6	1.46517
$540$	0	0
$541$	14784.5	1.17493	0.587464	−	0.809250i	$-0.300126\pi$
$541$	14784.5	1.17493	0.587464	+	0.809250i	$0.300126\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7070.62	0.555728
$546$	0	0
$547$	2531.48	0.197876	0.0989382	−	0.995094i	$-0.468455\pi$
$547$	2531.48	0.197876	0.0989382	+	0.995094i	$0.468455\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1265.22	0.0978226
$552$	0	0
$553$	−287.970	−0.0221442
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12526.9	0.952933	0.476466	−	0.879193i	$-0.341917\pi$
$557$	12526.9	0.952933	0.476466	+	0.879193i	$0.341917\pi$
$558$	0	0
$559$	5015.77	0.379507
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16977.2	−1.27087	−0.635437	−	0.772153i	$-0.719180\pi$
$563$	−16977.2	−1.27087	−0.635437	+	0.772153i	$0.719180\pi$
$564$	0	0
$565$	13428.2	0.999872
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12491.3	0.920317	0.460159	−	0.887837i	$-0.347793\pi$
$569$	12491.3	0.920317	0.460159	+	0.887837i	$0.347793\pi$
$570$	0	0
$571$	3471.26	0.254410	0.127205	−	0.991876i	$-0.459399\pi$
$571$	3471.26	0.254410	0.127205	+	0.991876i	$0.459399\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	132.079	0.00957927
$576$	0	0
$577$	−22314.6	−1.61000	−0.804998	−	0.593277i	$-0.797834\pi$
$577$	−22314.6	−1.61000	−0.804998	+	0.593277i	$0.797834\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4847.42	−0.346135
$582$	0	0
$583$	−6164.29	−0.437905
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−797.363	−0.0560659	−0.0280329	−	0.999607i	$-0.508924\pi$
$587$	−797.363	−0.0560659	−0.0280329	+	0.999607i	$0.508924\pi$
$588$	0	0
$589$	−4725.20	−0.330558
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5229.07	0.362112	0.181056	−	0.983473i	$-0.442048\pi$
$593$	5229.07	0.362112	0.181056	+	0.983473i	$0.442048\pi$
$594$	0	0
$595$	−7123.74	−0.490831
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4520.22	0.308332	0.154166	−	0.988045i	$-0.450731\pi$
$599$	4520.22	0.308332	0.154166	+	0.988045i	$0.450731\pi$
$600$	0	0
$601$	−28191.4	−1.91340	−0.956699	−	0.291080i	$-0.905986\pi$
$601$	−28191.4	−1.91340	−0.956699	+	0.291080i	$0.905986\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23793.0	−1.59888
$606$	0	0
$607$	9824.01	0.656910	0.328455	−	0.944520i	$-0.393472\pi$
$607$	9824.01	0.656910	0.328455	+	0.944520i	$0.393472\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−42547.0	−2.81713
$612$	0	0
$613$	−7236.11	−0.476776	−0.238388	−	0.971170i	$-0.576619\pi$
$613$	−7236.11	−0.476776	−0.238388	+	0.971170i	$0.576619\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20177.0	−1.31653	−0.658263	−	0.752788i	$-0.728708\pi$
$617$	−20177.0	−1.31653	−0.658263	+	0.752788i	$0.728708\pi$
$618$	0	0
$619$	−9836.84	−0.638734	−0.319367	−	0.947631i	$-0.603470\pi$
$619$	−9836.84	−0.638734	−0.319367	+	0.947631i	$0.603470\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4018.02	−0.258393
$624$	0	0
$625$	−15203.2	−0.973004
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37450.7	−2.37402
$630$	0	0
$631$	5325.24	0.335966	0.167983	−	0.985790i	$-0.446275\pi$
$631$	5325.24	0.335966	0.167983	+	0.985790i	$0.446275\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4559.26	−0.284927
$636$	0	0
$637$	−27342.7	−1.70072
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30186.7	−1.86007	−0.930035	−	0.367472i	$-0.880223\pi$
$641$	−30186.7	−1.86007	−0.930035	+	0.367472i	$0.880223\pi$
$642$	0	0
$643$	6815.72	0.418018	0.209009	−	0.977914i	$-0.432976\pi$
$643$	6815.72	0.418018	0.209009	+	0.977914i	$0.432976\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8525.98	0.518069	0.259035	−	0.965868i	$-0.416596\pi$
$647$	8525.98	0.518069	0.259035	+	0.965868i	$0.416596\pi$
$648$	0	0
$649$	34250.8	2.07159
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13797.4	0.826849	0.413425	−	0.910538i	$-0.364332\pi$
$653$	13797.4	0.826849	0.413425	+	0.910538i	$0.364332\pi$
$654$	0	0
$655$	−9203.51	−0.549024
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16673.7	0.985610	0.492805	−	0.870140i	$-0.335972\pi$
$659$	16673.7	0.985610	0.492805	+	0.870140i	$0.335972\pi$
$660$	0	0
$661$	2752.76	0.161982	0.0809909	−	0.996715i	$-0.474192\pi$
$661$	2752.76	0.161982	0.0809909	+	0.996715i	$0.474192\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1195.75	−0.0697282
$666$	0	0
$667$	−2674.95	−0.155284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−18577.7	−1.06883
$672$	0	0
$673$	−3652.27	−0.209190	−0.104595	−	0.994515i	$-0.533355\pi$
$673$	−3652.27	−0.209190	−0.104595	+	0.994515i	$0.533355\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13175.4	0.747964	0.373982	−	0.927436i	$-0.377992\pi$
$677$	13175.4	0.747964	0.373982	+	0.927436i	$0.377992\pi$
$678$	0	0
$679$	−1324.00	−0.0748313
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−25912.2	−1.45169	−0.725845	−	0.687859i	$-0.758551\pi$
$683$	−25912.2	−1.45169	−0.725845	+	0.687859i	$0.758551\pi$
$684$	0	0
$685$	1734.13	0.0967265
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9192.93	0.508306
$690$	0	0
$691$	−19933.9	−1.09742	−0.548712	−	0.836012i	$-0.684882\pi$
$691$	−19933.9	−1.09742	−0.548712	+	0.836012i	$0.684882\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27029.9	−1.47525
$696$	0	0
$697$	19477.4	1.05848
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10583.0	0.570208	0.285104	−	0.958497i	$-0.407972\pi$
$701$	10583.0	0.570208	0.285104	+	0.958497i	$0.407972\pi$
$702$	0	0
$703$	−6286.27	−0.337257
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5380.84	−0.286234
$708$	0	0
$709$	−28731.2	−1.52189	−0.760947	−	0.648814i	$-0.775265\pi$
$709$	−28731.2	−1.52189	−0.760947	+	0.648814i	$0.775265\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9990.08	0.524729
$714$	0	0
$715$	57381.6	3.00133
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7427.02	0.385231	0.192615	−	0.981274i	$-0.438303\pi$
$719$	7427.02	0.385231	0.192615	+	0.981274i	$0.438303\pi$
$720$	0	0
$721$	−10248.1	−0.529348
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−218.950	−0.0112160
$726$	0	0
$727$	5465.02	0.278798	0.139399	−	0.990236i	$-0.455483\pi$
$727$	5465.02	0.278798	0.139399	+	0.990236i	$0.455483\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6446.43	0.326169
$732$	0	0
$733$	11959.6	0.602644	0.301322	−	0.953522i	$-0.402572\pi$
$733$	11959.6	0.602644	0.301322	+	0.953522i	$0.402572\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	725.228	0.0362471
$738$	0	0
$739$	21411.4	1.06581	0.532903	−	0.846176i	$-0.321101\pi$
$739$	21411.4	1.06581	0.532903	+	0.846176i	$0.321101\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39368.8	1.94388	0.971938	−	0.235238i	$-0.0755871\pi$
$743$	39368.8	1.94388	0.971938	+	0.235238i	$0.0755871\pi$
$744$	0	0
$745$	7370.85	0.362479
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6886.00	0.335927
$750$	0	0
$751$	−928.320	−0.0451063	−0.0225532	−	0.999746i	$-0.507180\pi$
$751$	−928.320	−0.0451063	−0.0225532	+	0.999746i	$0.507180\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	26888.2	1.29611
$756$	0	0
$757$	8174.49	0.392480	0.196240	−	0.980556i	$-0.437127\pi$
$757$	8174.49	0.392480	0.196240	+	0.980556i	$0.437127\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21673.8	1.03242	0.516211	−	0.856462i	$-0.327342\pi$
$761$	21673.8	1.03242	0.516211	+	0.856462i	$0.327342\pi$
$762$	0	0
$763$	−3656.04	−0.173470
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−51078.8	−2.40463
$768$	0	0
$769$	5279.64	0.247580	0.123790	−	0.992308i	$-0.460495\pi$
$769$	5279.64	0.247580	0.123790	+	0.992308i	$0.460495\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30853.7	1.43562	0.717809	−	0.696241i	$-0.245145\pi$
$773$	30853.7	1.43562	0.717809	+	0.696241i	$0.245145\pi$
$774$	0	0
$775$	817.707	0.0379005
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3269.36	0.150369
$780$	0	0
$781$	−42007.0	−1.92462
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5122.10	0.232886
$786$	0	0
$787$	−14408.0	−0.652590	−0.326295	−	0.945268i	$-0.605800\pi$
$787$	−14408.0	−0.652590	−0.326295	+	0.945268i	$0.605800\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6943.38	−0.312109
$792$	0	0
$793$	27705.3	1.24066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6512.56	−0.289444	−0.144722	−	0.989472i	$-0.546229\pi$
$797$	−6512.56	−0.289444	−0.144722	+	0.989472i	$0.546229\pi$
$798$	0	0
$799$	−54682.9	−2.42120
$800$	0	0
$801$	0	0
$802$	0	0
$803$	41619.3	1.82903
$804$	0	0
$805$	2528.08	0.110687
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22213.1	0.965354	0.482677	−	0.875799i	$-0.339665\pi$
$809$	22213.1	0.965354	0.482677	+	0.875799i	$0.339665\pi$
$810$	0	0
$811$	2480.93	0.107420	0.0537098	−	0.998557i	$-0.482895\pi$
$811$	2480.93	0.107420	0.0537098	+	0.998557i	$0.482895\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16231.1	0.697608
$816$	0	0
$817$	1082.06	0.0463361
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17912.8	−0.761461	−0.380730	−	0.924686i	$-0.624327\pi$
$821$	−17912.8	−0.761461	−0.380730	+	0.924686i	$0.624327\pi$
$822$	0	0
$823$	−3445.46	−0.145931	−0.0729656	−	0.997334i	$-0.523246\pi$
$823$	−3445.46	−0.145931	−0.0729656	+	0.997334i	$0.523246\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33614.6	−1.41341	−0.706707	−	0.707506i	$-0.749820\pi$
$827$	−33614.6	−1.41341	−0.706707	+	0.707506i	$0.749820\pi$
$828$	0	0
$829$	3539.89	0.148306	0.0741529	−	0.997247i	$-0.476375\pi$
$829$	3539.89	0.148306	0.0741529	+	0.997247i	$0.476375\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−35141.7	−1.46169
$834$	0	0
$835$	4196.00	0.173902
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17069.1	0.702374	0.351187	−	0.936305i	$-0.385778\pi$
$839$	17069.1	0.702374	0.351187	+	0.936305i	$0.385778\pi$
$840$	0	0
$841$	−19954.7	−0.818184
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−61336.3	−2.49708
$846$	0	0
$847$	12302.8	0.499090
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13290.5	0.535363
$852$	0	0
$853$	22496.0	0.902989	0.451495	−	0.892274i	$-0.350891\pi$
$853$	22496.0	0.902989	0.451495	+	0.892274i	$0.350891\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14011.8	0.558498	0.279249	−	0.960219i	$-0.409915\pi$
$857$	14011.8	0.558498	0.279249	+	0.960219i	$0.409915\pi$
$858$	0	0
$859$	−26340.3	−1.04624	−0.523120	−	0.852259i	$-0.675232\pi$
$859$	−26340.3	−1.04624	−0.523120	+	0.852259i	$0.675232\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26311.7	1.03785	0.518924	−	0.854821i	$-0.326333\pi$
$863$	26311.7	1.03785	0.518924	+	0.854821i	$0.326333\pi$
$864$	0	0
$865$	14508.0	0.570275
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2981.22	0.116376
$870$	0	0
$871$	−1081.55	−0.0420744
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8073.72	0.311933
$876$	0	0
$877$	29383.8	1.13138	0.565690	−	0.824618i	$-0.308610\pi$
$877$	29383.8	1.13138	0.565690	+	0.824618i	$0.308610\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6960.95	0.266198	0.133099	−	0.991103i	$-0.457507\pi$
$881$	6960.95	0.266198	0.133099	+	0.991103i	$0.457507\pi$
$882$	0	0
$883$	−31161.5	−1.18762	−0.593810	−	0.804605i	$-0.702377\pi$
$883$	−31161.5	−1.18762	−0.593810	+	0.804605i	$0.702377\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17886.8	0.677091	0.338545	−	0.940950i	$-0.390065\pi$
$887$	17886.8	0.677091	0.338545	+	0.940950i	$0.390065\pi$
$888$	0	0
$889$	2357.48	0.0889398
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9178.78	−0.343960
$894$	0	0
$895$	43745.8	1.63381
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16560.7	−0.614384
$900$	0	0
$901$	11815.1	0.436866
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1133.11	−0.0416199
$906$	0	0
$907$	23803.3	0.871417	0.435709	−	0.900088i	$-0.356498\pi$
$907$	23803.3	0.871417	0.435709	+	0.900088i	$0.356498\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41133.8	−1.49596	−0.747982	−	0.663719i	$-0.768977\pi$
$911$	−41133.8	−1.49596	−0.747982	+	0.663719i	$0.768977\pi$
$912$	0	0
$913$	50183.0	1.81908
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4758.91	0.171377
$918$	0	0
$919$	9314.09	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$919$	9314.09	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	62645.9	2.23404
$924$	0	0
$925$	1087.86	0.0386686
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29566.7	1.04419	0.522095	−	0.852887i	$-0.325151\pi$
$929$	29566.7	1.04419	0.522095	+	0.852887i	$0.325151\pi$
$930$	0	0
$931$	−5898.71	−0.207650
$932$	0	0
$933$	0	0
$934$	0	0
$935$	73748.7	2.57951
$936$	0	0
$937$	53866.5	1.87806	0.939029	−	0.343837i	$-0.111727\pi$
$937$	53866.5	1.87806	0.939029	+	0.343837i	$0.111727\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47012.9	1.62867	0.814334	−	0.580396i	$-0.197102\pi$
$941$	47012.9	1.62867	0.814334	+	0.580396i	$0.197102\pi$
$942$	0	0
$943$	−6912.14	−0.238696
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24305.9	−0.834038	−0.417019	−	0.908898i	$-0.636925\pi$
$947$	−24305.9	−0.834038	−0.417019	+	0.908898i	$0.636925\pi$
$948$	0	0
$949$	−62067.7	−2.12308
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29214.7	−0.993028	−0.496514	−	0.868029i	$-0.665387\pi$
$953$	−29214.7	−0.993028	−0.496514	+	0.868029i	$0.665387\pi$
$954$	0	0
$955$	36313.4	1.23045
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−896.676	−0.0301931
$960$	0	0
$961$	32058.0	1.07610
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12203.8	0.407102
$966$	0	0
$967$	−32370.0	−1.07647	−0.538236	−	0.842794i	$-0.680909\pi$
$967$	−32370.0	−1.07647	−0.538236	+	0.842794i	$0.680909\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−44819.7	−1.48129	−0.740644	−	0.671897i	$-0.765480\pi$
$971$	−44819.7	−1.48129	−0.740644	+	0.671897i	$0.765480\pi$
$972$	0	0
$973$	13976.5	0.460499
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44799.8	−1.46701	−0.733507	−	0.679682i	$-0.762118\pi$
$977$	−44799.8	−1.46701	−0.733507	+	0.679682i	$0.762118\pi$
$978$	0	0
$979$	41596.7	1.35795
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29289.2	0.950335	0.475167	−	0.879895i	$-0.342388\pi$
$983$	29289.2	0.950335	0.475167	+	0.879895i	$0.342388\pi$
$984$	0	0
$985$	−16044.4	−0.519004
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2287.71	−0.0735542
$990$	0	0
$991$	−23400.6	−0.750096	−0.375048	−	0.927005i	$-0.622374\pi$
$991$	−23400.6	−0.750096	−0.375048	+	0.927005i	$0.622374\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	19820.4	0.631506
$996$	0	0
$997$	−59994.4	−1.90576	−0.952880	−	0.303349i	$-0.901895\pi$
$997$	−59994.4	−1.90576	−0.952880	+	0.303349i	$0.901895\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.4.a.i.1.1		3
3.2	odd	2		76.4.a.b.1.2	✓	3
12.11	even	2		304.4.a.h.1.2		3
15.2	even	4		1900.4.c.c.1749.3		6
15.8	even	4		1900.4.c.c.1749.4		6
15.14	odd	2		1900.4.a.c.1.2		3
24.5	odd	2		1216.4.a.t.1.2		3
24.11	even	2		1216.4.a.v.1.2		3
57.56	even	2		1444.4.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.a.b.1.2	✓	3	3.2	odd	2
304.4.a.h.1.2		3	12.11	even	2
684.4.a.i.1.1		3	1.1	even	1	trivial
1216.4.a.t.1.2		3	24.5	odd	2
1216.4.a.v.1.2		3	24.11	even	2
1444.4.a.e.1.2		3	57.56	even	2
1900.4.a.c.1.2		3	15.14	odd	2
1900.4.c.c.1749.3		6	15.2	even	4
1900.4.c.c.1749.4		6	15.8	even	4

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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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	684.a (trivial)

\(f(q)\)	\(=\)	\(q-11.0323 q^{5} +5.70454 q^{7} +O(q^{10})\)	\(q-11.0323 q^{5} +5.70454 q^{7} -59.0565 q^{11} +88.0721 q^{13} +113.193 q^{17} +19.0000 q^{19} -40.1701 q^{23} -3.28800 q^{25} +66.5906 q^{29} -248.695 q^{31} -62.9343 q^{35} -330.857 q^{37} +172.072 q^{41} +56.9507 q^{43} -483.093 q^{47} -310.458 q^{49} +104.380 q^{53} +651.530 q^{55} -579.966 q^{59} +314.576 q^{61} -971.639 q^{65} -12.2803 q^{67} +711.303 q^{71} -704.738 q^{73} -336.890 q^{77} -50.4809 q^{79} -849.747 q^{83} -1248.78 q^{85} -704.355 q^{89} +502.411 q^{91} -209.614 q^{95} -232.096 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{5} + 44 q^{7}+O(q^{10}) \) 3 * q - 9 * q^5 + 44 * q^7	\( 3 q - 9 q^{5} + 44 q^{7} - 79 q^{11} - 11 q^{13} - 82 q^{17} + 57 q^{19} - 103 q^{23} - 210 q^{25} + 93 q^{29} - 116 q^{31} + 93 q^{35} - 466 q^{37} + 188 q^{41} - 11 q^{43} - 163 q^{47} + 69 q^{49} - 197 q^{53} + 231 q^{55} - 1381 q^{59} - 405 q^{61} - 1188 q^{65} + 943 q^{67} - 1052 q^{71} - 580 q^{73} - 1855 q^{77} - 1402 q^{79} - 1802 q^{83} - 1245 q^{85} - 1966 q^{89} - 1723 q^{91} - 171 q^{95} + 48 q^{97}+O(q^{100}) \) 3 * q - 9 * q^5 + 44 * q^7 - 79 * q^11 - 11 * q^13 - 82 * q^17 + 57 * q^19 - 103 * q^23 - 210 * q^25 + 93 * q^29 - 116 * q^31 + 93 * q^35 - 466 * q^37 + 188 * q^41 - 11 * q^43 - 163 * q^47 + 69 * q^49 - 197 * q^53 + 231 * q^55 - 1381 * q^59 - 405 * q^61 - 1188 * q^65 + 943 * q^67 - 1052 * q^71 - 580 * q^73 - 1855 * q^77 - 1402 * q^79 - 1802 * q^83 - 1245 * q^85 - 1966 * q^89 - 1723 * q^91 - 171 * q^95 + 48 * q^97

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