LMFDB - Embedded newform 4332.2.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4332,2,Mod(1,4332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4332, 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("4332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4332 = 2^{2} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4332.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:	$34.5911941556$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.73227321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 18x^{4} + 27x^{3} + 96x^{2} - 48x - 127 $ x^6 - 3x^5 - 18x^4 + 27x^3 + 96x^2 - 48*x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.34215$ of defining polynomial
Character	$\chi$	$=$	4332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.34215 q^{5} -1.58839 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.34215 q^{5} -1.58839 q^{7} +1.00000 q^{9} +5.86457 q^{11} +0.537233 q^{13} +3.34215 q^{15} +0.638297 q^{17} -1.58839 q^{21} -0.472915 q^{23} +6.16998 q^{25} +1.00000 q^{27} +1.53388 q^{29} -3.27283 q^{31} +5.86457 q^{33} -5.30862 q^{35} -8.60894 q^{37} +0.537233 q^{39} +8.40419 q^{41} +11.2265 q^{43} +3.34215 q^{45} +11.4406 q^{47} -4.47703 q^{49} +0.638297 q^{51} +3.62549 q^{53} +19.6003 q^{55} -6.52542 q^{59} -12.0159 q^{61} -1.58839 q^{63} +1.79552 q^{65} -5.30012 q^{67} -0.472915 q^{69} +6.70707 q^{71} -8.32246 q^{73} +6.16998 q^{75} -9.31520 q^{77} -14.4024 q^{79} +1.00000 q^{81} -0.0769880 q^{83} +2.13328 q^{85} +1.53388 q^{87} +16.7634 q^{89} -0.853333 q^{91} -3.27283 q^{93} +11.5592 q^{97} +5.86457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 9 q^{11} + 9 q^{13} + 3 q^{15} + 9 q^{17} + 9 q^{21} + 12 q^{23} + 15 q^{25} + 6 q^{27} + 9 q^{29} - 6 q^{31} + 9 q^{33} - 3 q^{35} - 6 q^{37} + 9 q^{39} - 18 q^{41} + 15 q^{43} + 3 q^{45} + 21 q^{47} + 21 q^{49} + 9 q^{51} + 6 q^{53} + 12 q^{55} + 15 q^{59} - 21 q^{61} + 9 q^{63} - 33 q^{65} + 18 q^{67} + 12 q^{69} - 15 q^{71} + 18 q^{73} + 15 q^{75} - 9 q^{79} + 6 q^{81} - 3 q^{83} + 6 q^{85} + 9 q^{87} + 36 q^{89} + 18 q^{91} - 6 q^{93} + 39 q^{97} + 9 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9 + 9 * q^11 + 9 * q^13 + 3 * q^15 + 9 * q^17 + 9 * q^21 + 12 * q^23 + 15 * q^25 + 6 * q^27 + 9 * q^29 - 6 * q^31 + 9 * q^33 - 3 * q^35 - 6 * q^37 + 9 * q^39 - 18 * q^41 + 15 * q^43 + 3 * q^45 + 21 * q^47 + 21 * q^49 + 9 * q^51 + 6 * q^53 + 12 * q^55 + 15 * q^59 - 21 * q^61 + 9 * q^63 - 33 * q^65 + 18 * q^67 + 12 * q^69 - 15 * q^71 + 18 * q^73 + 15 * q^75 - 9 * q^79 + 6 * q^81 - 3 * q^83 + 6 * q^85 + 9 * q^87 + 36 * q^89 + 18 * q^91 - 6 * q^93 + 39 * q^97 + 9 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.34215	1.49466	0.747328	−	0.664455i	$-0.231336\pi$
$5$	3.34215	1.49466	0.747328	+	0.664455i	$0.231336\pi$
$6$	0	0
$7$	−1.58839	−0.600353	−0.300177	−	0.953884i	$-0.597046\pi$
$7$	−1.58839	−0.600353	−0.300177	+	0.953884i	$0.597046\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.86457	1.76824	0.884118	−	0.467264i	$-0.154760\pi$
$11$	5.86457	1.76824	0.884118	+	0.467264i	$0.154760\pi$
$12$	0	0
$13$	0.537233	0.149002	0.0745008	−	0.997221i	$-0.476264\pi$
$13$	0.537233	0.149002	0.0745008	+	0.997221i	$0.476264\pi$
$14$	0	0
$15$	3.34215	0.862940
$16$	0	0
$17$	0.638297	0.154810	0.0774048	−	0.997000i	$-0.475337\pi$
$17$	0.638297	0.154810	0.0774048	+	0.997000i	$0.475337\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.58839	−0.346614
$22$	0	0
$23$	−0.472915	−0.0986095	−0.0493048	−	0.998784i	$-0.515701\pi$
$23$	−0.472915	−0.0986095	−0.0493048	+	0.998784i	$0.515701\pi$
$24$	0	0
$25$	6.16998	1.23400
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.53388	0.284834	0.142417	−	0.989807i	$-0.454513\pi$
$29$	1.53388	0.284834	0.142417	+	0.989807i	$0.454513\pi$
$30$	0	0
$31$	−3.27283	−0.587818	−0.293909	−	0.955833i	$-0.594956\pi$
$31$	−3.27283	−0.587818	−0.293909	+	0.955833i	$0.594956\pi$
$32$	0	0
$33$	5.86457	1.02089
$34$	0	0
$35$	−5.30862	−0.897321
$36$	0	0
$37$	−8.60894	−1.41530	−0.707650	−	0.706563i	$-0.750245\pi$
$37$	−8.60894	−1.41530	−0.707650	+	0.706563i	$0.750245\pi$
$38$	0	0
$39$	0.537233	0.0860262
$40$	0	0
$41$	8.40419	1.31251	0.656257	−	0.754538i	$-0.272139\pi$
$41$	8.40419	1.31251	0.656257	+	0.754538i	$0.272139\pi$
$42$	0	0
$43$	11.2265	1.71203	0.856016	−	0.516950i	$-0.172933\pi$
$43$	11.2265	1.71203	0.856016	+	0.516950i	$0.172933\pi$
$44$	0	0
$45$	3.34215	0.498219
$46$	0	0
$47$	11.4406	1.66878	0.834388	−	0.551177i	$-0.185821\pi$
$47$	11.4406	1.66878	0.834388	+	0.551177i	$0.185821\pi$
$48$	0	0
$49$	−4.47703	−0.639576
$50$	0	0
$51$	0.638297	0.0893794
$52$	0	0
$53$	3.62549	0.497999	0.248999	−	0.968504i	$-0.419898\pi$
$53$	3.62549	0.497999	0.248999	+	0.968504i	$0.419898\pi$
$54$	0	0
$55$	19.6003	2.64290
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.52542	−0.849538	−0.424769	−	0.905302i	$-0.639645\pi$
$59$	−6.52542	−0.849538	−0.424769	+	0.905302i	$0.639645\pi$
$60$	0	0
$61$	−12.0159	−1.53848	−0.769238	−	0.638962i	$-0.779364\pi$
$61$	−12.0159	−1.53848	−0.769238	+	0.638962i	$0.779364\pi$
$62$	0	0
$63$	−1.58839	−0.200118
$64$	0	0
$65$	1.79552	0.222706
$66$	0	0
$67$	−5.30012	−0.647513	−0.323756	−	0.946140i	$-0.604946\pi$
$67$	−5.30012	−0.647513	−0.323756	+	0.946140i	$0.604946\pi$
$68$	0	0
$69$	−0.472915	−0.0569322
$70$	0	0
$71$	6.70707	0.795983	0.397992	−	0.917389i	$-0.369707\pi$
$71$	6.70707	0.795983	0.397992	+	0.917389i	$0.369707\pi$
$72$	0	0
$73$	−8.32246	−0.974070	−0.487035	−	0.873382i	$-0.661922\pi$
$73$	−8.32246	−0.974070	−0.487035	+	0.873382i	$0.661922\pi$
$74$	0	0
$75$	6.16998	0.712448
$76$	0	0
$77$	−9.31520	−1.06157
$78$	0	0
$79$	−14.4024	−1.62040	−0.810198	−	0.586156i	$-0.800641\pi$
$79$	−14.4024	−1.62040	−0.810198	+	0.586156i	$0.800641\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.0769880	−0.00845053	−0.00422526	−	0.999991i	$-0.501345\pi$
$83$	−0.0769880	−0.00845053	−0.00422526	+	0.999991i	$0.501345\pi$
$84$	0	0
$85$	2.13328	0.231387
$86$	0	0
$87$	1.53388	0.164449
$88$	0	0
$89$	16.7634	1.77691	0.888457	−	0.458960i	$-0.151778\pi$
$89$	16.7634	1.77691	0.888457	+	0.458960i	$0.151778\pi$
$90$	0	0
$91$	−0.853333	−0.0894536
$92$	0	0
$93$	−3.27283	−0.339377
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.5592	1.17366	0.586828	−	0.809712i	$-0.300377\pi$
$97$	11.5592	1.17366	0.586828	+	0.809712i	$0.300377\pi$
$98$	0	0
$99$	5.86457	0.589412
$100$	0	0
$101$	2.16045	0.214973	0.107487	−	0.994207i	$-0.465720\pi$
$101$	2.16045	0.214973	0.107487	+	0.994207i	$0.465720\pi$
$102$	0	0
$103$	−10.3341	−1.01825	−0.509124	−	0.860693i	$-0.670030\pi$
$103$	−10.3341	−1.01825	−0.509124	+	0.860693i	$0.670030\pi$
$104$	0	0
$105$	−5.30862	−0.518069
$106$	0	0
$107$	12.2307	1.18238	0.591191	−	0.806531i	$-0.298658\pi$
$107$	12.2307	1.18238	0.591191	+	0.806531i	$0.298658\pi$
$108$	0	0
$109$	−18.4784	−1.76991	−0.884953	−	0.465679i	$-0.845810\pi$
$109$	−18.4784	−1.76991	−0.884953	+	0.465679i	$0.845810\pi$
$110$	0	0
$111$	−8.60894	−0.817124
$112$	0	0
$113$	−15.3154	−1.44075	−0.720376	−	0.693583i	$-0.756031\pi$
$113$	−15.3154	−1.44075	−0.720376	+	0.693583i	$0.756031\pi$
$114$	0	0
$115$	−1.58055	−0.147387
$116$	0	0
$117$	0.537233	0.0496672
$118$	0	0
$119$	−1.01386	−0.0929405
$120$	0	0
$121$	23.3932	2.12666
$122$	0	0
$123$	8.40419	0.757780
$124$	0	0
$125$	3.91025	0.349743
$126$	0	0
$127$	11.1926	0.993181	0.496591	−	0.867985i	$-0.334585\pi$
$127$	11.1926	0.993181	0.496591	+	0.867985i	$0.334585\pi$
$128$	0	0
$129$	11.2265	0.988442
$130$	0	0
$131$	10.1628	0.887932	0.443966	−	0.896044i	$-0.353571\pi$
$131$	10.1628	0.887932	0.443966	+	0.896044i	$0.353571\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.34215	0.287647
$136$	0	0
$137$	−4.66154	−0.398262	−0.199131	−	0.979973i	$-0.563812\pi$
$137$	−4.66154	−0.398262	−0.199131	+	0.979973i	$0.563812\pi$
$138$	0	0
$139$	1.85157	0.157048	0.0785242	−	0.996912i	$-0.474979\pi$
$139$	1.85157	0.157048	0.0785242	+	0.996912i	$0.474979\pi$
$140$	0	0
$141$	11.4406	0.963468
$142$	0	0
$143$	3.15064	0.263470
$144$	0	0
$145$	5.12645	0.425728
$146$	0	0
$147$	−4.47703	−0.369259
$148$	0	0
$149$	13.3903	1.09697	0.548486	−	0.836159i	$-0.315204\pi$
$149$	13.3903	1.09697	0.548486	+	0.836159i	$0.315204\pi$
$150$	0	0
$151$	−0.201035	−0.0163600	−0.00818000	−	0.999967i	$-0.502604\pi$
$151$	−0.201035	−0.0163600	−0.00818000	+	0.999967i	$0.502604\pi$
$152$	0	0
$153$	0.638297	0.0516032
$154$	0	0
$155$	−10.9383	−0.878585
$156$	0	0
$157$	3.60868	0.288004	0.144002	−	0.989577i	$-0.454003\pi$
$157$	3.60868	0.288004	0.144002	+	0.989577i	$0.454003\pi$
$158$	0	0
$159$	3.62549	0.287520
$160$	0	0
$161$	0.751171	0.0592005
$162$	0	0
$163$	7.47770	0.585699	0.292849	−	0.956159i	$-0.405397\pi$
$163$	7.47770	0.585699	0.292849	+	0.956159i	$0.405397\pi$
$164$	0	0
$165$	19.6003	1.52588
$166$	0	0
$167$	16.6002	1.28457	0.642283	−	0.766468i	$-0.277988\pi$
$167$	16.6002	1.28457	0.642283	+	0.766468i	$0.277988\pi$
$168$	0	0
$169$	−12.7114	−0.977798
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−23.4727	−1.78460	−0.892298	−	0.451447i	$-0.850908\pi$
$173$	−23.4727	−1.78460	−0.892298	+	0.451447i	$0.850908\pi$
$174$	0	0
$175$	−9.80030	−0.740833
$176$	0	0
$177$	−6.52542	−0.490481
$178$	0	0
$179$	4.85641	0.362985	0.181493	−	0.983392i	$-0.441907\pi$
$179$	4.85641	0.362985	0.181493	+	0.983392i	$0.441907\pi$
$180$	0	0
$181$	17.2665	1.28341	0.641704	−	0.766952i	$-0.278228\pi$
$181$	17.2665	1.28341	0.641704	+	0.766952i	$0.278228\pi$
$182$	0	0
$183$	−12.0159	−0.888240
$184$	0	0
$185$	−28.7724	−2.11539
$186$	0	0
$187$	3.74334	0.273740
$188$	0	0
$189$	−1.58839	−0.115538
$190$	0	0
$191$	−7.44491	−0.538695	−0.269347	−	0.963043i	$-0.586808\pi$
$191$	−7.44491	−0.538695	−0.269347	+	0.963043i	$0.586808\pi$
$192$	0	0
$193$	−20.4159	−1.46957	−0.734783	−	0.678302i	$-0.762716\pi$
$193$	−20.4159	−1.46957	−0.734783	+	0.678302i	$0.762716\pi$
$194$	0	0
$195$	1.79552	0.128580
$196$	0	0
$197$	5.34751	0.380994	0.190497	−	0.981688i	$-0.438990\pi$
$197$	5.34751	0.380994	0.190497	+	0.981688i	$0.438990\pi$
$198$	0	0
$199$	−23.6274	−1.67491	−0.837453	−	0.546510i	$-0.815956\pi$
$199$	−23.6274	−1.67491	−0.837453	+	0.546510i	$0.815956\pi$
$200$	0	0
$201$	−5.30012	−0.373842
$202$	0	0
$203$	−2.43638	−0.171001
$204$	0	0
$205$	28.0881	1.96176
$206$	0	0
$207$	−0.472915	−0.0328698
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.6984	−0.736508	−0.368254	−	0.929725i	$-0.620044\pi$
$211$	−10.6984	−0.736508	−0.368254	+	0.929725i	$0.620044\pi$
$212$	0	0
$213$	6.70707	0.459561
$214$	0	0
$215$	37.5208	2.55890
$216$	0	0
$217$	5.19851	0.352898
$218$	0	0
$219$	−8.32246	−0.562380
$220$	0	0
$221$	0.342914	0.0230669
$222$	0	0
$223$	15.2485	1.02111	0.510557	−	0.859844i	$-0.329439\pi$
$223$	15.2485	1.02111	0.510557	+	0.859844i	$0.329439\pi$
$224$	0	0
$225$	6.16998	0.411332
$226$	0	0
$227$	−8.57171	−0.568924	−0.284462	−	0.958687i	$-0.591815\pi$
$227$	−8.57171	−0.568924	−0.284462	+	0.958687i	$0.591815\pi$
$228$	0	0
$229$	−2.84667	−0.188113	−0.0940565	−	0.995567i	$-0.529983\pi$
$229$	−2.84667	−0.188113	−0.0940565	+	0.995567i	$0.529983\pi$
$230$	0	0
$231$	−9.31520	−0.612895
$232$	0	0
$233$	−2.94809	−0.193136	−0.0965678	−	0.995326i	$-0.530786\pi$
$233$	−2.94809	−0.193136	−0.0965678	+	0.995326i	$0.530786\pi$
$234$	0	0
$235$	38.2361	2.49425
$236$	0	0
$237$	−14.4024	−0.935536
$238$	0	0
$239$	2.03210	0.131445	0.0657227	−	0.997838i	$-0.479065\pi$
$239$	2.03210	0.131445	0.0657227	+	0.997838i	$0.479065\pi$
$240$	0	0
$241$	4.03463	0.259893	0.129947	−	0.991521i	$-0.458519\pi$
$241$	4.03463	0.259893	0.129947	+	0.991521i	$0.458519\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.9629	−0.955946
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.0769880	−0.00487892
$250$	0	0
$251$	21.8222	1.37740	0.688702	−	0.725044i	$-0.258181\pi$
$251$	21.8222	1.37740	0.688702	+	0.725044i	$0.258181\pi$
$252$	0	0
$253$	−2.77344	−0.174365
$254$	0	0
$255$	2.13328	0.133591
$256$	0	0
$257$	−7.79706	−0.486367	−0.243184	−	0.969980i	$-0.578192\pi$
$257$	−7.79706	−0.486367	−0.243184	+	0.969980i	$0.578192\pi$
$258$	0	0
$259$	13.6743	0.849680
$260$	0	0
$261$	1.53388	0.0949445
$262$	0	0
$263$	−6.17222	−0.380595	−0.190298	−	0.981726i	$-0.560945\pi$
$263$	−6.17222	−0.380595	−0.190298	+	0.981726i	$0.560945\pi$
$264$	0	0
$265$	12.1169	0.744337
$266$	0	0
$267$	16.7634	1.02590
$268$	0	0
$269$	−20.3242	−1.23919	−0.619594	−	0.784922i	$-0.712703\pi$
$269$	−20.3242	−1.23919	−0.619594	+	0.784922i	$0.712703\pi$
$270$	0	0
$271$	15.1964	0.923116	0.461558	−	0.887110i	$-0.347291\pi$
$271$	15.1964	0.923116	0.461558	+	0.887110i	$0.347291\pi$
$272$	0	0
$273$	−0.853333	−0.0516461
$274$	0	0
$275$	36.1843	2.18200
$276$	0	0
$277$	11.1518	0.670047	0.335024	−	0.942210i	$-0.391256\pi$
$277$	11.1518	0.670047	0.335024	+	0.942210i	$0.391256\pi$
$278$	0	0
$279$	−3.27283	−0.195939
$280$	0	0
$281$	−21.6320	−1.29046	−0.645228	−	0.763990i	$-0.723238\pi$
$281$	−21.6320	−1.29046	−0.645228	+	0.763990i	$0.723238\pi$
$282$	0	0
$283$	9.05165	0.538065	0.269032	−	0.963131i	$-0.413296\pi$
$283$	9.05165	0.538065	0.269032	+	0.963131i	$0.413296\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.3491	−0.787972
$288$	0	0
$289$	−16.5926	−0.976034
$290$	0	0
$291$	11.5592	0.677610
$292$	0	0
$293$	−14.3813	−0.840165	−0.420082	−	0.907486i	$-0.637999\pi$
$293$	−14.3813	−0.840165	−0.420082	+	0.907486i	$0.637999\pi$
$294$	0	0
$295$	−21.8090	−1.26977
$296$	0	0
$297$	5.86457	0.340297
$298$	0	0
$299$	−0.254065	−0.0146930
$300$	0	0
$301$	−17.8321	−1.02782
$302$	0	0
$303$	2.16045	0.124115
$304$	0	0
$305$	−40.1589	−2.29949
$306$	0	0
$307$	9.15685	0.522609	0.261305	−	0.965256i	$-0.415847\pi$
$307$	9.15685	0.522609	0.261305	+	0.965256i	$0.415847\pi$
$308$	0	0
$309$	−10.3341	−0.587885
$310$	0	0
$311$	−19.1788	−1.08753	−0.543766	−	0.839237i	$-0.683002\pi$
$311$	−19.1788	−1.08753	−0.543766	+	0.839237i	$0.683002\pi$
$312$	0	0
$313$	20.5224	1.15999	0.579997	−	0.814618i	$-0.303054\pi$
$313$	20.5224	1.15999	0.579997	+	0.814618i	$0.303054\pi$
$314$	0	0
$315$	−5.30862	−0.299107
$316$	0	0
$317$	4.72386	0.265319	0.132659	−	0.991162i	$-0.457648\pi$
$317$	4.72386	0.265319	0.132659	+	0.991162i	$0.457648\pi$
$318$	0	0
$319$	8.99552	0.503653
$320$	0	0
$321$	12.2307	0.682649
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.31472	0.183867
$326$	0	0
$327$	−18.4784	−1.02186
$328$	0	0
$329$	−18.1720	−1.00185
$330$	0	0
$331$	−11.1030	−0.610275	−0.305137	−	0.952308i	$-0.598702\pi$
$331$	−11.1030	−0.610275	−0.305137	+	0.952308i	$0.598702\pi$
$332$	0	0
$333$	−8.60894	−0.471767
$334$	0	0
$335$	−17.7138	−0.967809
$336$	0	0
$337$	16.4948	0.898528	0.449264	−	0.893399i	$-0.351686\pi$
$337$	16.4948	0.898528	0.449264	+	0.893399i	$0.351686\pi$
$338$	0	0
$339$	−15.3154	−0.831819
$340$	0	0
$341$	−19.1937	−1.03940
$342$	0	0
$343$	18.2299	0.984325
$344$	0	0
$345$	−1.58055	−0.0850941
$346$	0	0
$347$	32.8945	1.76587	0.882934	−	0.469497i	$-0.155565\pi$
$347$	32.8945	1.76587	0.882934	+	0.469497i	$0.155565\pi$
$348$	0	0
$349$	6.97831	0.373540	0.186770	−	0.982404i	$-0.440198\pi$
$349$	6.97831	0.373540	0.186770	+	0.982404i	$0.440198\pi$
$350$	0	0
$351$	0.537233	0.0286754
$352$	0	0
$353$	−0.602959	−0.0320923	−0.0160461	−	0.999871i	$-0.505108\pi$
$353$	−0.602959	−0.0320923	−0.0160461	+	0.999871i	$0.505108\pi$
$354$	0	0
$355$	22.4161	1.18972
$356$	0	0
$357$	−1.01386	−0.0536592
$358$	0	0
$359$	−12.8153	−0.676368	−0.338184	−	0.941080i	$-0.609813\pi$
$359$	−12.8153	−0.676368	−0.338184	+	0.941080i	$0.609813\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	23.3932	1.22783
$364$	0	0
$365$	−27.8149	−1.45590
$366$	0	0
$367$	1.98490	0.103611	0.0518054	−	0.998657i	$-0.483502\pi$
$367$	1.98490	0.103611	0.0518054	+	0.998657i	$0.483502\pi$
$368$	0	0
$369$	8.40419	0.437504
$370$	0	0
$371$	−5.75867	−0.298975
$372$	0	0
$373$	−16.7321	−0.866356	−0.433178	−	0.901308i	$-0.642608\pi$
$373$	−16.7321	−0.866356	−0.433178	+	0.901308i	$0.642608\pi$
$374$	0	0
$375$	3.91025	0.201924
$376$	0	0
$377$	0.824049	0.0424407
$378$	0	0
$379$	5.98558	0.307459	0.153729	−	0.988113i	$-0.450872\pi$
$379$	5.98558	0.307459	0.153729	+	0.988113i	$0.450872\pi$
$380$	0	0
$381$	11.1926	0.573413
$382$	0	0
$383$	27.1069	1.38510	0.692548	−	0.721372i	$-0.256488\pi$
$383$	27.1069	1.38510	0.692548	+	0.721372i	$0.256488\pi$
$384$	0	0
$385$	−31.1328	−1.58668
$386$	0	0
$387$	11.2265	0.570677
$388$	0	0
$389$	−1.22312	−0.0620145	−0.0310073	−	0.999519i	$-0.509872\pi$
$389$	−1.22312	−0.0620145	−0.0310073	+	0.999519i	$0.509872\pi$
$390$	0	0
$391$	−0.301860	−0.0152657
$392$	0	0
$393$	10.1628	0.512648
$394$	0	0
$395$	−48.1350	−2.42194
$396$	0	0
$397$	−18.1627	−0.911561	−0.455780	−	0.890092i	$-0.650640\pi$
$397$	−18.1627	−0.911561	−0.455780	+	0.890092i	$0.650640\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.4379	−0.571183	−0.285592	−	0.958351i	$-0.592190\pi$
$401$	−11.4379	−0.571183	−0.285592	+	0.958351i	$0.592190\pi$
$402$	0	0
$403$	−1.75827	−0.0875858
$404$	0	0
$405$	3.34215	0.166073
$406$	0	0
$407$	−50.4877	−2.50258
$408$	0	0
$409$	−15.1466	−0.748953	−0.374477	−	0.927236i	$-0.622178\pi$
$409$	−15.1466	−0.748953	−0.374477	+	0.927236i	$0.622178\pi$
$410$	0	0
$411$	−4.66154	−0.229937
$412$	0	0
$413$	10.3649	0.510023
$414$	0	0
$415$	−0.257306	−0.0126306
$416$	0	0
$417$	1.85157	0.0906720
$418$	0	0
$419$	−17.3675	−0.848456	−0.424228	−	0.905555i	$-0.639455\pi$
$419$	−17.3675	−0.848456	−0.424228	+	0.905555i	$0.639455\pi$
$420$	0	0
$421$	11.1207	0.541992	0.270996	−	0.962581i	$-0.412647\pi$
$421$	11.1207	0.541992	0.270996	+	0.962581i	$0.412647\pi$
$422$	0	0
$423$	11.4406	0.556259
$424$	0	0
$425$	3.93828	0.191034
$426$	0	0
$427$	19.0859	0.923629
$428$	0	0
$429$	3.15064	0.152114
$430$	0	0
$431$	−3.66909	−0.176734	−0.0883669	−	0.996088i	$-0.528165\pi$
$431$	−3.66909	−0.176734	−0.0883669	+	0.996088i	$0.528165\pi$
$432$	0	0
$433$	−19.7665	−0.949919	−0.474959	−	0.880008i	$-0.657537\pi$
$433$	−19.7665	−0.949919	−0.474959	+	0.880008i	$0.657537\pi$
$434$	0	0
$435$	5.12645	0.245794
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−3.86931	−0.184672	−0.0923361	−	0.995728i	$-0.529433\pi$
$439$	−3.86931	−0.184672	−0.0923361	+	0.995728i	$0.529433\pi$
$440$	0	0
$441$	−4.47703	−0.213192
$442$	0	0
$443$	26.7081	1.26894	0.634471	−	0.772947i	$-0.281218\pi$
$443$	26.7081	1.26894	0.634471	+	0.772947i	$0.281218\pi$
$444$	0	0
$445$	56.0257	2.65587
$446$	0	0
$447$	13.3903	0.633338
$448$	0	0
$449$	41.5672	1.96168	0.980838	−	0.194827i	$-0.0624145\pi$
$449$	41.5672	1.96168	0.980838	+	0.194827i	$0.0624145\pi$
$450$	0	0
$451$	49.2870	2.32083
$452$	0	0
$453$	−0.201035	−0.00944546
$454$	0	0
$455$	−2.85197	−0.133702
$456$	0	0
$457$	35.4121	1.65651	0.828254	−	0.560354i	$-0.189335\pi$
$457$	35.4121	1.65651	0.828254	+	0.560354i	$0.189335\pi$
$458$	0	0
$459$	0.638297	0.0297931
$460$	0	0
$461$	−9.11914	−0.424721	−0.212360	−	0.977191i	$-0.568115\pi$
$461$	−9.11914	−0.424721	−0.212360	+	0.977191i	$0.568115\pi$
$462$	0	0
$463$	−37.4764	−1.74167	−0.870837	−	0.491571i	$-0.836423\pi$
$463$	−37.4764	−1.74167	−0.870837	+	0.491571i	$0.836423\pi$
$464$	0	0
$465$	−10.9383	−0.507251
$466$	0	0
$467$	12.0288	0.556625	0.278313	−	0.960491i	$-0.410225\pi$
$467$	12.0288	0.556625	0.278313	+	0.960491i	$0.410225\pi$
$468$	0	0
$469$	8.41863	0.388736
$470$	0	0
$471$	3.60868	0.166279
$472$	0	0
$473$	65.8389	3.02727
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.62549	0.166000
$478$	0	0
$479$	−4.86062	−0.222087	−0.111044	−	0.993816i	$-0.535419\pi$
$479$	−4.86062	−0.222087	−0.111044	+	0.993816i	$0.535419\pi$
$480$	0	0
$481$	−4.62501	−0.210882
$482$	0	0
$483$	0.751171	0.0341794
$484$	0	0
$485$	38.6325	1.75421
$486$	0	0
$487$	−42.4663	−1.92433	−0.962167	−	0.272462i	$-0.912162\pi$
$487$	−42.4663	−1.92433	−0.962167	+	0.272462i	$0.912162\pi$
$488$	0	0
$489$	7.47770	0.338153
$490$	0	0
$491$	−3.99260	−0.180183	−0.0900917	−	0.995933i	$-0.528716\pi$
$491$	−3.99260	−0.180183	−0.0900917	+	0.995933i	$0.528716\pi$
$492$	0	0
$493$	0.979067	0.0440950
$494$	0	0
$495$	19.6003	0.880968
$496$	0	0
$497$	−10.6534	−0.477871
$498$	0	0
$499$	2.65497	0.118853	0.0594264	−	0.998233i	$-0.481073\pi$
$499$	2.65497	0.118853	0.0594264	+	0.998233i	$0.481073\pi$
$500$	0	0
$501$	16.6002	0.741644
$502$	0	0
$503$	25.8228	1.15138	0.575691	−	0.817667i	$-0.304733\pi$
$503$	25.8228	1.15138	0.575691	+	0.817667i	$0.304733\pi$
$504$	0	0
$505$	7.22056	0.321311
$506$	0	0
$507$	−12.7114	−0.564532
$508$	0	0
$509$	29.0033	1.28555	0.642775	−	0.766055i	$-0.277783\pi$
$509$	29.0033	1.28555	0.642775	+	0.766055i	$0.277783\pi$
$510$	0	0
$511$	13.2193	0.584786
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−34.5381	−1.52193
$516$	0	0
$517$	67.0940	2.95079
$518$	0	0
$519$	−23.4727	−1.03034
$520$	0	0
$521$	−37.0992	−1.62535	−0.812673	−	0.582720i	$-0.801989\pi$
$521$	−37.0992	−1.62535	−0.812673	+	0.582720i	$0.801989\pi$
$522$	0	0
$523$	−6.11145	−0.267235	−0.133618	−	0.991033i	$-0.542659\pi$
$523$	−6.11145	−0.267235	−0.133618	+	0.991033i	$0.542659\pi$
$524$	0	0
$525$	−9.80030	−0.427720
$526$	0	0
$527$	−2.08904	−0.0909998
$528$	0	0
$529$	−22.7764	−0.990276
$530$	0	0
$531$	−6.52542	−0.283179
$532$	0	0
$533$	4.51501	0.195567
$534$	0	0
$535$	40.8767	1.76725
$536$	0	0
$537$	4.85641	0.209570
$538$	0	0
$539$	−26.2559	−1.13092
$540$	0	0
$541$	−15.9994	−0.687866	−0.343933	−	0.938994i	$-0.611759\pi$
$541$	−15.9994	−0.687866	−0.343933	+	0.938994i	$0.611759\pi$
$542$	0	0
$543$	17.2665	0.740977
$544$	0	0
$545$	−61.7575	−2.64540
$546$	0	0
$547$	21.4059	0.915251	0.457626	−	0.889145i	$-0.348700\pi$
$547$	21.4059	0.915251	0.457626	+	0.889145i	$0.348700\pi$
$548$	0	0
$549$	−12.0159	−0.512826
$550$	0	0
$551$	0	0
$552$	0	0
$553$	22.8766	0.972810
$554$	0	0
$555$	−28.7724	−1.22132
$556$	0	0
$557$	37.3980	1.58461	0.792303	−	0.610128i	$-0.208882\pi$
$557$	37.3980	1.58461	0.792303	+	0.610128i	$0.208882\pi$
$558$	0	0
$559$	6.03127	0.255096
$560$	0	0
$561$	3.74334	0.158044
$562$	0	0
$563$	−15.5372	−0.654814	−0.327407	−	0.944883i	$-0.606175\pi$
$563$	−15.5372	−0.654814	−0.327407	+	0.944883i	$0.606175\pi$
$564$	0	0
$565$	−51.1864	−2.15343
$566$	0	0
$567$	−1.58839	−0.0667059
$568$	0	0
$569$	18.3328	0.768550	0.384275	−	0.923219i	$-0.374451\pi$
$569$	18.3328	0.768550	0.384275	+	0.923219i	$0.374451\pi$
$570$	0	0
$571$	13.4713	0.563758	0.281879	−	0.959450i	$-0.409042\pi$
$571$	13.4713	0.563758	0.281879	+	0.959450i	$0.409042\pi$
$572$	0	0
$573$	−7.44491	−0.311016
$574$	0	0
$575$	−2.91787	−0.121684
$576$	0	0
$577$	−26.2627	−1.09333	−0.546664	−	0.837352i	$-0.684103\pi$
$577$	−26.2627	−1.09333	−0.546664	+	0.837352i	$0.684103\pi$
$578$	0	0
$579$	−20.4159	−0.848454
$580$	0	0
$581$	0.122287	0.00507330
$582$	0	0
$583$	21.2619	0.880579
$584$	0	0
$585$	1.79552	0.0742354
$586$	0	0
$587$	−28.8629	−1.19130	−0.595649	−	0.803245i	$-0.703105\pi$
$587$	−28.8629	−1.19130	−0.595649	+	0.803245i	$0.703105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	5.34751	0.219967
$592$	0	0
$593$	13.4748	0.553344	0.276672	−	0.960964i	$-0.410769\pi$
$593$	13.4748	0.553344	0.276672	+	0.960964i	$0.410769\pi$
$594$	0	0
$595$	−3.38848	−0.138914
$596$	0	0
$597$	−23.6274	−0.967007
$598$	0	0
$599$	30.9345	1.26395	0.631974	−	0.774989i	$-0.282245\pi$
$599$	30.9345	1.26395	0.631974	+	0.774989i	$0.282245\pi$
$600$	0	0
$601$	−2.45446	−0.100120	−0.0500598	−	0.998746i	$-0.515941\pi$
$601$	−2.45446	−0.100120	−0.0500598	+	0.998746i	$0.515941\pi$
$602$	0	0
$603$	−5.30012	−0.215838
$604$	0	0
$605$	78.1837	3.17862
$606$	0	0
$607$	1.40984	0.0572236	0.0286118	−	0.999591i	$-0.490891\pi$
$607$	1.40984	0.0572236	0.0286118	+	0.999591i	$0.490891\pi$
$608$	0	0
$609$	−2.43638	−0.0987273
$610$	0	0
$611$	6.14625	0.248650
$612$	0	0
$613$	11.6066	0.468788	0.234394	−	0.972142i	$-0.424689\pi$
$613$	11.6066	0.468788	0.234394	+	0.972142i	$0.424689\pi$
$614$	0	0
$615$	28.0881	1.13262
$616$	0	0
$617$	−31.3966	−1.26398	−0.631990	−	0.774976i	$-0.717762\pi$
$617$	−31.3966	−1.26398	−0.631990	+	0.774976i	$0.717762\pi$
$618$	0	0
$619$	−35.4366	−1.42432	−0.712159	−	0.702018i	$-0.752283\pi$
$619$	−35.4366	−1.42432	−0.712159	+	0.702018i	$0.752283\pi$
$620$	0	0
$621$	−0.472915	−0.0189774
$622$	0	0
$623$	−26.6267	−1.06678
$624$	0	0
$625$	−17.7812	−0.711250
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.49505	−0.219102
$630$	0	0
$631$	11.5892	0.461360	0.230680	−	0.973030i	$-0.425905\pi$
$631$	11.5892	0.461360	0.230680	+	0.973030i	$0.425905\pi$
$632$	0	0
$633$	−10.6984	−0.425223
$634$	0	0
$635$	37.4073	1.48446
$636$	0	0
$637$	−2.40521	−0.0952979
$638$	0	0
$639$	6.70707	0.265328
$640$	0	0
$641$	−7.45377	−0.294406	−0.147203	−	0.989106i	$-0.547027\pi$
$641$	−7.45377	−0.294406	−0.147203	+	0.989106i	$0.547027\pi$
$642$	0	0
$643$	15.0670	0.594183	0.297092	−	0.954849i	$-0.403983\pi$
$643$	15.0670	0.594183	0.297092	+	0.954849i	$0.403983\pi$
$644$	0	0
$645$	37.5208	1.47738
$646$	0	0
$647$	−21.3426	−0.839064	−0.419532	−	0.907741i	$-0.637806\pi$
$647$	−21.3426	−0.839064	−0.419532	+	0.907741i	$0.637806\pi$
$648$	0	0
$649$	−38.2688	−1.50218
$650$	0	0
$651$	5.19851	0.203746
$652$	0	0
$653$	−23.9209	−0.936099	−0.468049	−	0.883702i	$-0.655043\pi$
$653$	−23.9209	−0.936099	−0.468049	+	0.883702i	$0.655043\pi$
$654$	0	0
$655$	33.9658	1.32715
$656$	0	0
$657$	−8.32246	−0.324690
$658$	0	0
$659$	−19.7081	−0.767720	−0.383860	−	0.923391i	$-0.625405\pi$
$659$	−19.7081	−0.767720	−0.383860	+	0.923391i	$0.625405\pi$
$660$	0	0
$661$	29.4888	1.14698	0.573492	−	0.819211i	$-0.305589\pi$
$661$	29.4888	1.14698	0.573492	+	0.819211i	$0.305589\pi$
$662$	0	0
$663$	0.342914	0.0133177
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.725392	−0.0280873
$668$	0	0
$669$	15.2485	0.589540
$670$	0	0
$671$	−70.4680	−2.72039
$672$	0	0
$673$	24.6137	0.948787	0.474394	−	0.880313i	$-0.342667\pi$
$673$	24.6137	0.948787	0.474394	+	0.880313i	$0.342667\pi$
$674$	0	0
$675$	6.16998	0.237483
$676$	0	0
$677$	24.1310	0.927431	0.463716	−	0.885984i	$-0.346516\pi$
$677$	24.1310	0.927431	0.463716	+	0.885984i	$0.346516\pi$
$678$	0	0
$679$	−18.3604	−0.704608
$680$	0	0
$681$	−8.57171	−0.328469
$682$	0	0
$683$	−43.2768	−1.65594	−0.827970	−	0.560772i	$-0.810504\pi$
$683$	−43.2768	−1.65594	−0.827970	+	0.560772i	$0.810504\pi$
$684$	0	0
$685$	−15.5796	−0.595264
$686$	0	0
$687$	−2.84667	−0.108607
$688$	0	0
$689$	1.94773	0.0742027
$690$	0	0
$691$	−20.0400	−0.762359	−0.381179	−	0.924501i	$-0.624482\pi$
$691$	−20.0400	−0.762359	−0.381179	+	0.924501i	$0.624482\pi$
$692$	0	0
$693$	−9.31520	−0.353855
$694$	0	0
$695$	6.18824	0.234733
$696$	0	0
$697$	5.36436	0.203190
$698$	0	0
$699$	−2.94809	−0.111507
$700$	0	0
$701$	−39.4126	−1.48860	−0.744298	−	0.667848i	$-0.767216\pi$
$701$	−39.4126	−1.48860	−0.744298	+	0.667848i	$0.767216\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	38.2361	1.44005
$706$	0	0
$707$	−3.43163	−0.129060
$708$	0	0
$709$	−26.5353	−0.996553	−0.498277	−	0.867018i	$-0.666034\pi$
$709$	−26.5353	−0.996553	−0.498277	+	0.867018i	$0.666034\pi$
$710$	0	0
$711$	−14.4024	−0.540132
$712$	0	0
$713$	1.54777	0.0579644
$714$	0	0
$715$	10.5299	0.393797
$716$	0	0
$717$	2.03210	0.0758901
$718$	0	0
$719$	−13.2132	−0.492768	−0.246384	−	0.969172i	$-0.579242\pi$
$719$	−13.2132	−0.492768	−0.246384	+	0.969172i	$0.579242\pi$
$720$	0	0
$721$	16.4145	0.611308
$722$	0	0
$723$	4.03463	0.150049
$724$	0	0
$725$	9.46398	0.351483
$726$	0	0
$727$	−29.6763	−1.10063	−0.550317	−	0.834956i	$-0.685493\pi$
$727$	−29.6763	−1.10063	−0.550317	+	0.834956i	$0.685493\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.16586	0.265039
$732$	0	0
$733$	−11.0238	−0.407173	−0.203587	−	0.979057i	$-0.565260\pi$
$733$	−11.0238	−0.407173	−0.203587	+	0.979057i	$0.565260\pi$
$734$	0	0
$735$	−14.9629	−0.551916
$736$	0	0
$737$	−31.0830	−1.14496
$738$	0	0
$739$	1.75749	0.0646503	0.0323251	−	0.999477i	$-0.489709\pi$
$739$	1.75749	0.0646503	0.0323251	+	0.999477i	$0.489709\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.54857	0.203557	0.101779	−	0.994807i	$-0.467547\pi$
$743$	5.54857	0.203557	0.101779	+	0.994807i	$0.467547\pi$
$744$	0	0
$745$	44.7523	1.63960
$746$	0	0
$747$	−0.0769880	−0.00281684
$748$	0	0
$749$	−19.4270	−0.709847
$750$	0	0
$751$	−1.72698	−0.0630184	−0.0315092	−	0.999503i	$-0.510031\pi$
$751$	−1.72698	−0.0630184	−0.0315092	+	0.999503i	$0.510031\pi$
$752$	0	0
$753$	21.8222	0.795245
$754$	0	0
$755$	−0.671890	−0.0244526
$756$	0	0
$757$	16.1494	0.586958	0.293479	−	0.955965i	$-0.405187\pi$
$757$	16.1494	0.586958	0.293479	+	0.955965i	$0.405187\pi$
$758$	0	0
$759$	−2.77344	−0.100670
$760$	0	0
$761$	−37.8068	−1.37049	−0.685247	−	0.728310i	$-0.740306\pi$
$761$	−37.8068	−1.37049	−0.685247	+	0.728310i	$0.740306\pi$
$762$	0	0
$763$	29.3508	1.06257
$764$	0	0
$765$	2.13328	0.0771291
$766$	0	0
$767$	−3.50567	−0.126583
$768$	0	0
$769$	−53.0966	−1.91471	−0.957355	−	0.288913i	$-0.906706\pi$
$769$	−53.0966	−1.91471	−0.957355	+	0.288913i	$0.906706\pi$
$770$	0	0
$771$	−7.79706	−0.280804
$772$	0	0
$773$	−39.5388	−1.42211	−0.711056	−	0.703136i	$-0.751783\pi$
$773$	−39.5388	−1.42211	−0.711056	+	0.703136i	$0.751783\pi$
$774$	0	0
$775$	−20.1933	−0.725365
$776$	0	0
$777$	13.6743	0.490563
$778$	0	0
$779$	0	0
$780$	0	0
$781$	39.3341	1.40749
$782$	0	0
$783$	1.53388	0.0548162
$784$	0	0
$785$	12.0608	0.430467
$786$	0	0
$787$	23.3478	0.832259	0.416129	−	0.909305i	$-0.363386\pi$
$787$	23.3478	0.832259	0.416129	+	0.909305i	$0.363386\pi$
$788$	0	0
$789$	−6.17222	−0.219737
$790$	0	0
$791$	24.3268	0.864961
$792$	0	0
$793$	−6.45533	−0.229236
$794$	0	0
$795$	12.1169	0.429743
$796$	0	0
$797$	−26.0433	−0.922502	−0.461251	−	0.887270i	$-0.652599\pi$
$797$	−26.0433	−0.922502	−0.461251	+	0.887270i	$0.652599\pi$
$798$	0	0
$799$	7.30247	0.258343
$800$	0	0
$801$	16.7634	0.592305
$802$	0	0
$803$	−48.8077	−1.72239
$804$	0	0
$805$	2.51053	0.0884844
$806$	0	0
$807$	−20.3242	−0.715446
$808$	0	0
$809$	−24.6685	−0.867297	−0.433648	−	0.901082i	$-0.642774\pi$
$809$	−24.6685	−0.867297	−0.433648	+	0.901082i	$0.642774\pi$
$810$	0	0
$811$	−30.2847	−1.06344	−0.531720	−	0.846920i	$-0.678454\pi$
$811$	−30.2847	−1.06344	−0.531720	+	0.846920i	$0.678454\pi$
$812$	0	0
$813$	15.1964	0.532961
$814$	0	0
$815$	24.9916	0.875418
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−0.853333	−0.0298179
$820$	0	0
$821$	−9.60084	−0.335072	−0.167536	−	0.985866i	$-0.553581\pi$
$821$	−9.60084	−0.335072	−0.167536	+	0.985866i	$0.553581\pi$
$822$	0	0
$823$	−25.9671	−0.905158	−0.452579	−	0.891724i	$-0.649496\pi$
$823$	−25.9671	−0.905158	−0.452579	+	0.891724i	$0.649496\pi$
$824$	0	0
$825$	36.1843	1.25978
$826$	0	0
$827$	26.7786	0.931184	0.465592	−	0.884999i	$-0.345841\pi$
$827$	26.7786	0.931184	0.465592	+	0.884999i	$0.345841\pi$
$828$	0	0
$829$	12.3949	0.430494	0.215247	−	0.976560i	$-0.430944\pi$
$829$	12.3949	0.430494	0.215247	+	0.976560i	$0.430944\pi$
$830$	0	0
$831$	11.1518	0.386852
$832$	0	0
$833$	−2.85767	−0.0990126
$834$	0	0
$835$	55.4805	1.91998
$836$	0	0
$837$	−3.27283	−0.113126
$838$	0	0
$839$	34.8085	1.20172	0.600862	−	0.799353i	$-0.294824\pi$
$839$	34.8085	1.20172	0.600862	+	0.799353i	$0.294824\pi$
$840$	0	0
$841$	−26.6472	−0.918870
$842$	0	0
$843$	−21.6320	−0.745045
$844$	0	0
$845$	−42.4834	−1.46147
$846$	0	0
$847$	−37.1574	−1.27674
$848$	0	0
$849$	9.05165	0.310652
$850$	0	0
$851$	4.07129	0.139562
$852$	0	0
$853$	−13.0391	−0.446449	−0.223224	−	0.974767i	$-0.571658\pi$
$853$	−13.0391	−0.446449	−0.223224	+	0.974767i	$0.571658\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.6779	1.04794	0.523969	−	0.851737i	$-0.324451\pi$
$857$	30.6779	1.04794	0.523969	+	0.851737i	$0.324451\pi$
$858$	0	0
$859$	−40.5487	−1.38350	−0.691751	−	0.722136i	$-0.743161\pi$
$859$	−40.5487	−1.38350	−0.691751	+	0.722136i	$0.743161\pi$
$860$	0	0
$861$	−13.3491	−0.454936
$862$	0	0
$863$	19.6237	0.667999	0.334000	−	0.942573i	$-0.391602\pi$
$863$	19.6237	0.667999	0.334000	+	0.942573i	$0.391602\pi$
$864$	0	0
$865$	−78.4493	−2.66736
$866$	0	0
$867$	−16.5926	−0.563513
$868$	0	0
$869$	−84.4639	−2.86524
$870$	0	0
$871$	−2.84740	−0.0964805
$872$	0	0
$873$	11.5592	0.391218
$874$	0	0
$875$	−6.21098	−0.209970
$876$	0	0
$877$	18.5653	0.626905	0.313453	−	0.949604i	$-0.398514\pi$
$877$	18.5653	0.626905	0.313453	+	0.949604i	$0.398514\pi$
$878$	0	0
$879$	−14.3813	−0.485069
$880$	0	0
$881$	−11.4633	−0.386208	−0.193104	−	0.981178i	$-0.561856\pi$
$881$	−11.4633	−0.386208	−0.193104	+	0.981178i	$0.561856\pi$
$882$	0	0
$883$	−41.2618	−1.38857	−0.694285	−	0.719700i	$-0.744279\pi$
$883$	−41.2618	−1.38857	−0.694285	+	0.719700i	$0.744279\pi$
$884$	0	0
$885$	−21.8090	−0.733100
$886$	0	0
$887$	−17.9315	−0.602082	−0.301041	−	0.953611i	$-0.597334\pi$
$887$	−17.9315	−0.602082	−0.301041	+	0.953611i	$0.597334\pi$
$888$	0	0
$889$	−17.7781	−0.596259
$890$	0	0
$891$	5.86457	0.196471
$892$	0	0
$893$	0	0
$894$	0	0
$895$	16.2309	0.542538
$896$	0	0
$897$	−0.254065	−0.00848300
$898$	0	0
$899$	−5.02011	−0.167430
$900$	0	0
$901$	2.31414	0.0770950
$902$	0	0
$903$	−17.8321	−0.593414
$904$	0	0
$905$	57.7073	1.91825
$906$	0	0
$907$	40.3583	1.34008	0.670038	−	0.742327i	$-0.266278\pi$
$907$	40.3583	1.34008	0.670038	+	0.742327i	$0.266278\pi$
$908$	0	0
$909$	2.16045	0.0716577
$910$	0	0
$911$	11.9120	0.394663	0.197331	−	0.980337i	$-0.436772\pi$
$911$	11.9120	0.394663	0.197331	+	0.980337i	$0.436772\pi$
$912$	0	0
$913$	−0.451502	−0.0149425
$914$	0	0
$915$	−40.1589	−1.32761
$916$	0	0
$917$	−16.1425	−0.533073
$918$	0	0
$919$	−4.91204	−0.162033	−0.0810166	−	0.996713i	$-0.525817\pi$
$919$	−4.91204	−0.162033	−0.0810166	+	0.996713i	$0.525817\pi$
$920$	0	0
$921$	9.15685	0.301729
$922$	0	0
$923$	3.60326	0.118603
$924$	0	0
$925$	−53.1170	−1.74647
$926$	0	0
$927$	−10.3341	−0.339416
$928$	0	0
$929$	−24.9634	−0.819023	−0.409512	−	0.912305i	$-0.634301\pi$
$929$	−24.9634	−0.819023	−0.409512	+	0.912305i	$0.634301\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−19.1788	−0.627887
$934$	0	0
$935$	12.5108	0.409147
$936$	0	0
$937$	−44.9795	−1.46942	−0.734708	−	0.678383i	$-0.762681\pi$
$937$	−44.9795	−1.46942	−0.734708	+	0.678383i	$0.762681\pi$
$938$	0	0
$939$	20.5224	0.669723
$940$	0	0
$941$	6.38637	0.208190	0.104095	−	0.994567i	$-0.466805\pi$
$941$	6.38637	0.208190	0.104095	+	0.994567i	$0.466805\pi$
$942$	0	0
$943$	−3.97446	−0.129426
$944$	0	0
$945$	−5.30862	−0.172690
$946$	0	0
$947$	−3.31495	−0.107721	−0.0538607	−	0.998548i	$-0.517153\pi$
$947$	−3.31495	−0.107721	−0.0538607	+	0.998548i	$0.517153\pi$
$948$	0	0
$949$	−4.47110	−0.145138
$950$	0	0
$951$	4.72386	0.153182
$952$	0	0
$953$	−9.00158	−0.291590	−0.145795	−	0.989315i	$-0.546574\pi$
$953$	−9.00158	−0.291590	−0.145795	+	0.989315i	$0.546574\pi$
$954$	0	0
$955$	−24.8820	−0.805163
$956$	0	0
$957$	8.99552	0.290784
$958$	0	0
$959$	7.40431	0.239098
$960$	0	0
$961$	−20.2886	−0.654470
$962$	0	0
$963$	12.2307	0.394127
$964$	0	0
$965$	−68.2329	−2.19649
$966$	0	0
$967$	−20.8312	−0.669885	−0.334943	−	0.942239i	$-0.608717\pi$
$967$	−20.8312	−0.669885	−0.334943	+	0.942239i	$0.608717\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.1308	0.325113	0.162557	−	0.986699i	$-0.448026\pi$
$971$	10.1308	0.325113	0.162557	+	0.986699i	$0.448026\pi$
$972$	0	0
$973$	−2.94101	−0.0942845
$974$	0	0
$975$	3.31472	0.106156
$976$	0	0
$977$	−10.9950	−0.351762	−0.175881	−	0.984411i	$-0.556277\pi$
$977$	−10.9950	−0.351762	−0.175881	+	0.984411i	$0.556277\pi$
$978$	0	0
$979$	98.3100	3.14200
$980$	0	0
$981$	−18.4784	−0.589969
$982$	0	0
$983$	37.5261	1.19690	0.598449	−	0.801161i	$-0.295784\pi$
$983$	37.5261	1.19690	0.598449	+	0.801161i	$0.295784\pi$
$984$	0	0
$985$	17.8722	0.569456
$986$	0	0
$987$	−18.1720	−0.578421
$988$	0	0
$989$	−5.30920	−0.168823
$990$	0	0
$991$	−20.7636	−0.659578	−0.329789	−	0.944055i	$-0.606978\pi$
$991$	−20.7636	−0.659578	−0.329789	+	0.944055i	$0.606978\pi$
$992$	0	0
$993$	−11.1030	−0.352342
$994$	0	0
$995$	−78.9665	−2.50341
$996$	0	0
$997$	−23.9975	−0.760009	−0.380005	−	0.924985i	$-0.624078\pi$
$997$	−23.9975	−0.760009	−0.380005	+	0.924985i	$0.624078\pi$
$998$	0	0
$999$	−8.60894	−0.272375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4332.2.a.u.1.6		6
19.3	odd	18		228.2.q.b.85.1	✓	12
19.13	odd	18		228.2.q.b.169.1	yes	12
19.18	odd	2		4332.2.a.t.1.6		6
57.32	even	18		684.2.bo.e.397.2		12
57.41	even	18		684.2.bo.e.541.2		12
76.3	even	18		912.2.bo.g.769.1		12
76.51	even	18		912.2.bo.g.625.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.q.b.85.1	✓	12	19.3	odd	18
228.2.q.b.169.1	yes	12	19.13	odd	18
684.2.bo.e.397.2		12	57.32	even	18
684.2.bo.e.541.2		12	57.41	even	18
912.2.bo.g.625.1		12	76.51	even	18
912.2.bo.g.769.1		12	76.3	even	18
4332.2.a.t.1.6		6	19.18	odd	2
4332.2.a.u.1.6		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4332.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.34215 q^{5} -1.58839 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.34215 q^{5} -1.58839 q^{7} +1.00000 q^{9} +5.86457 q^{11} +0.537233 q^{13} +3.34215 q^{15} +0.638297 q^{17} -1.58839 q^{21} -0.472915 q^{23} +6.16998 q^{25} +1.00000 q^{27} +1.53388 q^{29} -3.27283 q^{31} +5.86457 q^{33} -5.30862 q^{35} -8.60894 q^{37} +0.537233 q^{39} +8.40419 q^{41} +11.2265 q^{43} +3.34215 q^{45} +11.4406 q^{47} -4.47703 q^{49} +0.638297 q^{51} +3.62549 q^{53} +19.6003 q^{55} -6.52542 q^{59} -12.0159 q^{61} -1.58839 q^{63} +1.79552 q^{65} -5.30012 q^{67} -0.472915 q^{69} +6.70707 q^{71} -8.32246 q^{73} +6.16998 q^{75} -9.31520 q^{77} -14.4024 q^{79} +1.00000 q^{81} -0.0769880 q^{83} +2.13328 q^{85} +1.53388 q^{87} +16.7634 q^{89} -0.853333 q^{91} -3.27283 q^{93} +11.5592 q^{97} +5.86457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 9 q^{11} + 9 q^{13} + 3 q^{15} + 9 q^{17} + 9 q^{21} + 12 q^{23} + 15 q^{25} + 6 q^{27} + 9 q^{29} - 6 q^{31} + 9 q^{33} - 3 q^{35} - 6 q^{37} + 9 q^{39} - 18 q^{41} + 15 q^{43} + 3 q^{45} + 21 q^{47} + 21 q^{49} + 9 q^{51} + 6 q^{53} + 12 q^{55} + 15 q^{59} - 21 q^{61} + 9 q^{63} - 33 q^{65} + 18 q^{67} + 12 q^{69} - 15 q^{71} + 18 q^{73} + 15 q^{75} - 9 q^{79} + 6 q^{81} - 3 q^{83} + 6 q^{85} + 9 q^{87} + 36 q^{89} + 18 q^{91} - 6 q^{93} + 39 q^{97} + 9 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9 + 9 * q^11 + 9 * q^13 + 3 * q^15 + 9 * q^17 + 9 * q^21 + 12 * q^23 + 15 * q^25 + 6 * q^27 + 9 * q^29 - 6 * q^31 + 9 * q^33 - 3 * q^35 - 6 * q^37 + 9 * q^39 - 18 * q^41 + 15 * q^43 + 3 * q^45 + 21 * q^47 + 21 * q^49 + 9 * q^51 + 6 * q^53 + 12 * q^55 + 15 * q^59 - 21 * q^61 + 9 * q^63 - 33 * q^65 + 18 * q^67 + 12 * q^69 - 15 * q^71 + 18 * q^73 + 15 * q^75 - 9 * q^79 + 6 * q^81 - 3 * q^83 + 6 * q^85 + 9 * q^87 + 36 * q^89 + 18 * q^91 - 6 * q^93 + 39 * q^97 + 9 * q^99

Introduction

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