LMFDB - Embedded newform 2205.4.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 $ x^5 - x^4 - 37x^3 + 21x^2 + 288*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.84623$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.84623 q^{2} +6.79345 q^{4} -5.00000 q^{5} +4.64067 q^{8} +O(q^{10})$	$q-3.84623 q^{2} +6.79345 q^{4} -5.00000 q^{5} +4.64067 q^{8} +19.2311 q^{10} -23.1120 q^{11} -61.0473 q^{13} -72.1966 q^{16} +0.688309 q^{17} -63.2247 q^{19} -33.9672 q^{20} +88.8938 q^{22} -124.502 q^{23} +25.0000 q^{25} +234.802 q^{26} -104.167 q^{29} -280.022 q^{31} +240.559 q^{32} -2.64739 q^{34} -263.871 q^{37} +243.176 q^{38} -23.2033 q^{40} -243.366 q^{41} +172.541 q^{43} -157.010 q^{44} +478.861 q^{46} -107.381 q^{47} -96.1556 q^{50} -414.722 q^{52} -44.8086 q^{53} +115.560 q^{55} +400.650 q^{58} +457.246 q^{59} -473.802 q^{61} +1077.03 q^{62} -347.672 q^{64} +305.237 q^{65} -229.454 q^{67} +4.67599 q^{68} -407.688 q^{71} +348.475 q^{73} +1014.91 q^{74} -429.514 q^{76} +840.135 q^{79} +360.983 q^{80} +936.040 q^{82} -885.652 q^{83} -3.44154 q^{85} -663.632 q^{86} -107.255 q^{88} -856.096 q^{89} -845.795 q^{92} +413.013 q^{94} +316.124 q^{95} -189.436 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 35 q^{4} - 25 q^{5} - 33 q^{8}+O(q^{10}) $ 5 * q - q^2 + 35 * q^4 - 25 * q^5 - 33 * q^8	$ 5 q - q^{2} + 35 q^{4} - 25 q^{5} - 33 q^{8} + 5 q^{10} - 47 q^{11} - q^{13} + 171 q^{16} - 2 q^{17} + 21 q^{19} - 175 q^{20} + 523 q^{22} - 201 q^{23} + 125 q^{25} - 47 q^{26} - 190 q^{29} - 388 q^{31} + 95 q^{32} - 130 q^{34} - 145 q^{37} + 835 q^{38} + 165 q^{40} - 281 q^{41} + 568 q^{43} - 1091 q^{44} + 337 q^{46} - 473 q^{47} - 25 q^{50} + 379 q^{52} - 351 q^{53} + 235 q^{55} + 1818 q^{58} + 708 q^{59} - 1944 q^{61} + 448 q^{62} - 125 q^{64} + 5 q^{65} + 1118 q^{67} - 3118 q^{68} - 864 q^{71} + 1652 q^{73} + 3285 q^{74} - 691 q^{76} + 218 q^{79} - 855 q^{80} - 1027 q^{82} - 1502 q^{83} + 10 q^{85} + 4264 q^{86} + 2131 q^{88} + 2322 q^{89} + 2957 q^{92} + 2677 q^{94} - 105 q^{95} - 598 q^{97}+O(q^{100}) $ 5 * q - q^2 + 35 * q^4 - 25 * q^5 - 33 * q^8 + 5 * q^10 - 47 * q^11 - q^13 + 171 * q^16 - 2 * q^17 + 21 * q^19 - 175 * q^20 + 523 * q^22 - 201 * q^23 + 125 * q^25 - 47 * q^26 - 190 * q^29 - 388 * q^31 + 95 * q^32 - 130 * q^34 - 145 * q^37 + 835 * q^38 + 165 * q^40 - 281 * q^41 + 568 * q^43 - 1091 * q^44 + 337 * q^46 - 473 * q^47 - 25 * q^50 + 379 * q^52 - 351 * q^53 + 235 * q^55 + 1818 * q^58 + 708 * q^59 - 1944 * q^61 + 448 * q^62 - 125 * q^64 + 5 * q^65 + 1118 * q^67 - 3118 * q^68 - 864 * q^71 + 1652 * q^73 + 3285 * q^74 - 691 * q^76 + 218 * q^79 - 855 * q^80 - 1027 * q^82 - 1502 * q^83 + 10 * q^85 + 4264 * q^86 + 2131 * q^88 + 2322 * q^89 + 2957 * q^92 + 2677 * q^94 - 105 * q^95 - 598 * 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 $

$2$	−3.84623	−1.35985	−0.679923	−	0.733284i	$-0.737987\pi$
$2$	−3.84623	−1.35985	−0.679923	+	0.733284i	$0.737987\pi$
$3$	0	0
$3$	0	0
$4$	6.79345	0.849181
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	4.64067	0.205090
$9$	0	0
$10$	19.2311	0.608142
$11$	−23.1120	−0.633501	−0.316751	−	0.948509i	$-0.602592\pi$
$11$	−23.1120	−0.633501	−0.316751	+	0.948509i	$0.602592\pi$
$12$	0	0
$13$	−61.0473	−1.30242	−0.651211	−	0.758897i	$-0.725739\pi$
$13$	−61.0473	−1.30242	−0.651211	+	0.758897i	$0.725739\pi$
$14$	0	0
$15$	0	0
$16$	−72.1966	−1.12807
$17$	0.688309	0.00981996	0.00490998	−	0.999988i	$-0.498437\pi$
$17$	0.688309	0.00981996	0.00490998	+	0.999988i	$0.498437\pi$
$18$	0	0
$19$	−63.2247	−0.763407	−0.381704	−	0.924285i	$-0.624663\pi$
$19$	−63.2247	−0.763407	−0.381704	+	0.924285i	$0.624663\pi$
$20$	−33.9672	−0.379765
$21$	0	0
$22$	88.8938	0.861464
$23$	−124.502	−1.12871	−0.564356	−	0.825531i	$-0.690875\pi$
$23$	−124.502	−1.12871	−0.564356	+	0.825531i	$0.690875\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	234.802	1.77109
$27$	0	0
$28$	0	0
$29$	−104.167	−0.667011	−0.333506	−	0.942748i	$-0.608232\pi$
$29$	−104.167	−0.667011	−0.333506	+	0.942748i	$0.608232\pi$
$30$	0	0
$31$	−280.022	−1.62237	−0.811185	−	0.584790i	$-0.801177\pi$
$31$	−280.022	−1.62237	−0.811185	+	0.584790i	$0.801177\pi$
$32$	240.559	1.32891
$33$	0	0
$34$	−2.64739	−0.0133536
$35$	0	0
$36$	0	0
$37$	−263.871	−1.17244	−0.586219	−	0.810153i	$-0.699384\pi$
$37$	−263.871	−1.17244	−0.586219	+	0.810153i	$0.699384\pi$
$38$	243.176	1.03812
$39$	0	0
$40$	−23.2033	−0.0917192
$41$	−243.366	−0.927009	−0.463505	−	0.886095i	$-0.653408\pi$
$41$	−243.366	−0.927009	−0.463505	+	0.886095i	$0.653408\pi$
$42$	0	0
$43$	172.541	0.611913	0.305956	−	0.952046i	$-0.401024\pi$
$43$	172.541	0.611913	0.305956	+	0.952046i	$0.401024\pi$
$44$	−157.010	−0.537957
$45$	0	0
$46$	478.861	1.53487
$47$	−107.381	−0.333260	−0.166630	−	0.986020i	$-0.553288\pi$
$47$	−107.381	−0.333260	−0.166630	+	0.986020i	$0.553288\pi$
$48$	0	0
$49$	0	0
$50$	−96.1556	−0.271969
$51$	0	0
$52$	−414.722	−1.10599
$53$	−44.8086	−0.116131	−0.0580654	−	0.998313i	$-0.518493\pi$
$53$	−44.8086	−0.116131	−0.0580654	+	0.998313i	$0.518493\pi$
$54$	0	0
$55$	115.560	0.283310
$56$	0	0
$57$	0	0
$58$	400.650	0.907033
$59$	457.246	1.00895	0.504477	−	0.863425i	$-0.331685\pi$
$59$	457.246	1.00895	0.504477	+	0.863425i	$0.331685\pi$
$60$	0	0
$61$	−473.802	−0.994493	−0.497247	−	0.867609i	$-0.665656\pi$
$61$	−473.802	−0.994493	−0.497247	+	0.867609i	$0.665656\pi$
$62$	1077.03	2.20617
$63$	0	0
$64$	−347.672	−0.679047
$65$	305.237	0.582461
$66$	0	0
$67$	−229.454	−0.418392	−0.209196	−	0.977874i	$-0.567085\pi$
$67$	−229.454	−0.418392	−0.209196	+	0.977874i	$0.567085\pi$
$68$	4.67599	0.00833893
$69$	0	0
$70$	0	0
$71$	−407.688	−0.681460	−0.340730	−	0.940161i	$-0.610674\pi$
$71$	−407.688	−0.681460	−0.340730	+	0.940161i	$0.610674\pi$
$72$	0	0
$73$	348.475	0.558712	0.279356	−	0.960188i	$-0.409879\pi$
$73$	348.475	0.558712	0.279356	+	0.960188i	$0.409879\pi$
$74$	1014.91	1.59433
$75$	0	0
$76$	−429.514	−0.648271
$77$	0	0
$78$	0	0
$79$	840.135	1.19649	0.598244	−	0.801314i	$-0.295865\pi$
$79$	840.135	1.19649	0.598244	+	0.801314i	$0.295865\pi$
$80$	360.983	0.504489
$81$	0	0
$82$	936.040	1.26059
$83$	−885.652	−1.17124	−0.585620	−	0.810586i	$-0.699149\pi$
$83$	−885.652	−1.17124	−0.585620	+	0.810586i	$0.699149\pi$
$84$	0	0
$85$	−3.44154	−0.00439162
$86$	−663.632	−0.832107
$87$	0	0
$88$	−107.255	−0.129925
$89$	−856.096	−1.01962	−0.509809	−	0.860288i	$-0.670284\pi$
$89$	−856.096	−1.01962	−0.509809	+	0.860288i	$0.670284\pi$
$90$	0	0
$91$	0	0
$92$	−845.795	−0.958481
$93$	0	0
$94$	413.013	0.453182
$95$	316.124	0.341406
$96$	0	0
$97$	−189.436	−0.198292	−0.0991459	−	0.995073i	$-0.531611\pi$
$97$	−189.436	−0.198292	−0.0991459	+	0.995073i	$0.531611\pi$
$98$	0	0
$99$	0	0
$100$	169.836	0.169836
$101$	795.803	0.784013	0.392007	−	0.919962i	$-0.371781\pi$
$101$	795.803	0.784013	0.392007	+	0.919962i	$0.371781\pi$
$102$	0	0
$103$	−1940.71	−1.85654	−0.928269	−	0.371909i	$-0.878703\pi$
$103$	−1940.71	−1.85654	−0.928269	+	0.371909i	$0.878703\pi$
$104$	−283.300	−0.267114
$105$	0	0
$106$	172.344	0.157920
$107$	−488.688	−0.441526	−0.220763	−	0.975327i	$-0.570855\pi$
$107$	−488.688	−0.441526	−0.220763	+	0.975327i	$0.570855\pi$
$108$	0	0
$109$	1630.05	1.43239	0.716197	−	0.697899i	$-0.245881\pi$
$109$	1630.05	1.43239	0.716197	+	0.697899i	$0.245881\pi$
$110$	−444.469	−0.385259
$111$	0	0
$112$	0	0
$113$	−345.925	−0.287981	−0.143991	−	0.989579i	$-0.545993\pi$
$113$	−345.925	−0.287981	−0.143991	+	0.989579i	$0.545993\pi$
$114$	0	0
$115$	622.508	0.504775
$116$	−707.653	−0.566414
$117$	0	0
$118$	−1758.67	−1.37202
$119$	0	0
$120$	0	0
$121$	−796.838	−0.598676
$122$	1822.35	1.35236
$123$	0	0
$124$	−1902.32	−1.37769
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1665.24	−1.16351	−0.581757	−	0.813362i	$-0.697635\pi$
$127$	−1665.24	−1.16351	−0.581757	+	0.813362i	$0.697635\pi$
$128$	−587.249	−0.405516
$129$	0	0
$130$	−1174.01	−0.792057
$131$	676.747	0.451356	0.225678	−	0.974202i	$-0.427540\pi$
$131$	676.747	0.451356	0.225678	+	0.974202i	$0.427540\pi$
$132$	0	0
$133$	0	0
$134$	882.531	0.568948
$135$	0	0
$136$	3.19421	0.00201398
$137$	1034.23	0.644967	0.322484	−	0.946575i	$-0.395482\pi$
$137$	1034.23	0.644967	0.322484	+	0.946575i	$0.395482\pi$
$138$	0	0
$139$	−435.826	−0.265944	−0.132972	−	0.991120i	$-0.542452\pi$
$139$	−435.826	−0.265944	−0.132972	+	0.991120i	$0.542452\pi$
$140$	0	0
$141$	0	0
$142$	1568.06	0.926681
$143$	1410.92	0.825086
$144$	0	0
$145$	520.835	0.298297
$146$	−1340.31	−0.759762
$147$	0	0
$148$	−1792.60	−0.995612
$149$	−2119.95	−1.16559	−0.582795	−	0.812619i	$-0.698041\pi$
$149$	−2119.95	−1.16559	−0.582795	+	0.812619i	$0.698041\pi$
$150$	0	0
$151$	−2400.40	−1.29365	−0.646827	−	0.762637i	$-0.723904\pi$
$151$	−2400.40	−1.29365	−0.646827	+	0.762637i	$0.723904\pi$
$152$	−293.405	−0.156568
$153$	0	0
$154$	0	0
$155$	1400.11	0.725546
$156$	0	0
$157$	506.188	0.257313	0.128657	−	0.991689i	$-0.458933\pi$
$157$	506.188	0.257313	0.128657	+	0.991689i	$0.458933\pi$
$158$	−3231.35	−1.62704
$159$	0	0
$160$	−1202.80	−0.594309
$161$	0	0
$162$	0	0
$163$	2600.95	1.24983	0.624913	−	0.780694i	$-0.285134\pi$
$163$	2600.95	1.24983	0.624913	+	0.780694i	$0.285134\pi$
$164$	−1653.29	−0.787199
$165$	0	0
$166$	3406.42	1.59271
$167$	546.077	0.253034	0.126517	−	0.991964i	$-0.459620\pi$
$167$	546.077	0.253034	0.126517	+	0.991964i	$0.459620\pi$
$168$	0	0
$169$	1529.78	0.696303
$170$	13.2370	0.00597193
$171$	0	0
$172$	1172.15	0.519625
$173$	−3020.32	−1.32735	−0.663673	−	0.748023i	$-0.731003\pi$
$173$	−3020.32	−1.32735	−0.663673	+	0.748023i	$0.731003\pi$
$174$	0	0
$175$	0	0
$176$	1668.61	0.714636
$177$	0	0
$178$	3292.74	1.38652
$179$	−1999.17	−0.834775	−0.417387	−	0.908729i	$-0.637054\pi$
$179$	−1999.17	−0.834775	−0.417387	+	0.908729i	$0.637054\pi$
$180$	0	0
$181$	681.145	0.279719	0.139859	−	0.990171i	$-0.455335\pi$
$181$	681.145	0.279719	0.139859	+	0.990171i	$0.455335\pi$
$182$	0	0
$183$	0	0
$184$	−577.770	−0.231488
$185$	1319.36	0.524330
$186$	0	0
$187$	−15.9082	−0.00622096
$188$	−729.491	−0.282998
$189$	0	0
$190$	−1215.88	−0.464260
$191$	1612.76	0.610970	0.305485	−	0.952197i	$-0.401181\pi$
$191$	1612.76	0.610970	0.305485	+	0.952197i	$0.401181\pi$
$192$	0	0
$193$	808.921	0.301697	0.150848	−	0.988557i	$-0.451800\pi$
$193$	808.921	0.301697	0.150848	+	0.988557i	$0.451800\pi$
$194$	728.613	0.269646
$195$	0	0
$196$	0	0
$197$	−3704.23	−1.33967	−0.669836	−	0.742509i	$-0.733636\pi$
$197$	−3704.23	−1.33967	−0.669836	+	0.742509i	$0.733636\pi$
$198$	0	0
$199$	367.928	0.131064	0.0655319	−	0.997850i	$-0.479126\pi$
$199$	367.928	0.131064	0.0655319	+	0.997850i	$0.479126\pi$
$200$	116.017	0.0410181
$201$	0	0
$202$	−3060.84	−1.06614
$203$	0	0
$204$	0	0
$205$	1216.83	0.414571
$206$	7464.39	2.52461
$207$	0	0
$208$	4407.41	1.46923
$209$	1461.25	0.483620
$210$	0	0
$211$	−1073.36	−0.350205	−0.175102	−	0.984550i	$-0.556026\pi$
$211$	−1073.36	−0.350205	−0.175102	+	0.984550i	$0.556026\pi$
$212$	−304.405	−0.0986161
$213$	0	0
$214$	1879.61	0.600407
$215$	−862.705	−0.273656
$216$	0	0
$217$	0	0
$218$	−6269.55	−1.94783
$219$	0	0
$220$	785.049	0.240582
$221$	−42.0194	−0.0127897
$222$	0	0
$223$	1725.16	0.518050	0.259025	−	0.965871i	$-0.416599\pi$
$223$	1725.16	0.518050	0.259025	+	0.965871i	$0.416599\pi$
$224$	0	0
$225$	0	0
$226$	1330.50	0.391610
$227$	−3616.99	−1.05757	−0.528784	−	0.848757i	$-0.677352\pi$
$227$	−3616.99	−1.05757	−0.528784	+	0.848757i	$0.677352\pi$
$228$	0	0
$229$	−1461.79	−0.421823	−0.210912	−	0.977505i	$-0.567643\pi$
$229$	−1461.79	−0.421823	−0.210912	+	0.977505i	$0.567643\pi$
$230$	−2394.31	−0.686417
$231$	0	0
$232$	−483.404	−0.136798
$233$	−2135.32	−0.600383	−0.300192	−	0.953879i	$-0.597051\pi$
$233$	−2135.32	−0.600383	−0.300192	+	0.953879i	$0.597051\pi$
$234$	0	0
$235$	536.907	0.149038
$236$	3106.27	0.856785
$237$	0	0
$238$	0	0
$239$	−5952.35	−1.61099	−0.805493	−	0.592605i	$-0.798100\pi$
$239$	−5952.35	−1.61099	−0.805493	+	0.592605i	$0.798100\pi$
$240$	0	0
$241$	3847.56	1.02839	0.514197	−	0.857672i	$-0.328090\pi$
$241$	3847.56	1.02839	0.514197	+	0.857672i	$0.328090\pi$
$242$	3064.82	0.814107
$243$	0	0
$244$	−3218.75	−0.844505
$245$	0	0
$246$	0	0
$247$	3859.70	0.994279
$248$	−1299.49	−0.332732
$249$	0	0
$250$	480.778	0.121628
$251$	−1731.69	−0.435470	−0.217735	−	0.976008i	$-0.569867\pi$
$251$	−1731.69	−0.435470	−0.217735	+	0.976008i	$0.569867\pi$
$252$	0	0
$253$	2877.47	0.715041
$254$	6404.90	1.58220
$255$	0	0
$256$	5040.07	1.23049
$257$	−4054.43	−0.984078	−0.492039	−	0.870573i	$-0.663748\pi$
$257$	−4054.43	−0.984078	−0.492039	+	0.870573i	$0.663748\pi$
$258$	0	0
$259$	0	0
$260$	2073.61	0.494615
$261$	0	0
$262$	−2602.92	−0.613775
$263$	6335.82	1.48549	0.742744	−	0.669576i	$-0.233524\pi$
$263$	6335.82	1.48549	0.742744	+	0.669576i	$0.233524\pi$
$264$	0	0
$265$	224.043	0.0519353
$266$	0	0
$267$	0	0
$268$	−1558.78	−0.355290
$269$	−1321.63	−0.299559	−0.149780	−	0.988719i	$-0.547856\pi$
$269$	−1321.63	−0.299559	−0.149780	+	0.988719i	$0.547856\pi$
$270$	0	0
$271$	2336.00	0.523624	0.261812	−	0.965119i	$-0.415680\pi$
$271$	2336.00	0.523624	0.261812	+	0.965119i	$0.415680\pi$
$272$	−49.6936	−0.0110776
$273$	0	0
$274$	−3977.90	−0.877056
$275$	−577.799	−0.126700
$276$	0	0
$277$	−7085.23	−1.53686	−0.768430	−	0.639934i	$-0.778962\pi$
$277$	−7085.23	−1.53686	−0.768430	+	0.639934i	$0.778962\pi$
$278$	1676.28	0.361643
$279$	0	0
$280$	0	0
$281$	−2123.17	−0.450738	−0.225369	−	0.974273i	$-0.572359\pi$
$281$	−2123.17	−0.450738	−0.225369	+	0.974273i	$0.572359\pi$
$282$	0	0
$283$	−5891.63	−1.23753	−0.618765	−	0.785576i	$-0.712367\pi$
$283$	−5891.63	−1.23753	−0.618765	+	0.785576i	$0.712367\pi$
$284$	−2769.61	−0.578683
$285$	0	0
$286$	−5426.73	−1.12199
$287$	0	0
$288$	0	0
$289$	−4912.53	−0.999904
$290$	−2003.25	−0.405637
$291$	0	0
$292$	2367.35	0.474448
$293$	−8137.75	−1.62257	−0.811284	−	0.584652i	$-0.801231\pi$
$293$	−8137.75	−1.62257	−0.811284	+	0.584652i	$0.801231\pi$
$294$	0	0
$295$	−2286.23	−0.451218
$296$	−1224.54	−0.240456
$297$	0	0
$298$	8153.80	1.58502
$299$	7600.49	1.47006
$300$	0	0
$301$	0	0
$302$	9232.47	1.75917
$303$	0	0
$304$	4564.61	0.861179
$305$	2369.01	0.444751
$306$	0	0
$307$	4797.24	0.891833	0.445917	−	0.895075i	$-0.352878\pi$
$307$	4797.24	0.891833	0.445917	+	0.895075i	$0.352878\pi$
$308$	0	0
$309$	0	0
$310$	−5385.14	−0.986631
$311$	5189.53	0.946210	0.473105	−	0.881006i	$-0.343133\pi$
$311$	5189.53	0.946210	0.473105	+	0.881006i	$0.343133\pi$
$312$	0	0
$313$	2245.24	0.405459	0.202729	−	0.979235i	$-0.435019\pi$
$313$	2245.24	0.405459	0.202729	+	0.979235i	$0.435019\pi$
$314$	−1946.91	−0.349907
$315$	0	0
$316$	5707.42	1.01604
$317$	9742.27	1.72612	0.863060	−	0.505101i	$-0.168545\pi$
$317$	9742.27	1.72612	0.863060	+	0.505101i	$0.168545\pi$
$318$	0	0
$319$	2407.50	0.422553
$320$	1738.36	0.303679
$321$	0	0
$322$	0	0
$323$	−43.5181	−0.00749663
$324$	0	0
$325$	−1526.18	−0.260484
$326$	−10003.8	−1.69957
$327$	0	0
$328$	−1129.38	−0.190121
$329$	0	0
$330$	0	0
$331$	11330.8	1.88156	0.940778	−	0.339023i	$-0.110096\pi$
$331$	11330.8	1.88156	0.940778	+	0.339023i	$0.110096\pi$
$332$	−6016.63	−0.994595
$333$	0	0
$334$	−2100.33	−0.344087
$335$	1147.27	0.187110
$336$	0	0
$337$	3637.35	0.587950	0.293975	−	0.955813i	$-0.405022\pi$
$337$	3637.35	0.587950	0.293975	+	0.955813i	$0.405022\pi$
$338$	−5883.87	−0.946865
$339$	0	0
$340$	−23.3800	−0.00372928
$341$	6471.86	1.02777
$342$	0	0
$343$	0	0
$344$	800.705	0.125497
$345$	0	0
$346$	11616.8	1.80498
$347$	4030.12	0.623482	0.311741	−	0.950167i	$-0.399088\pi$
$347$	4030.12	0.623482	0.311741	+	0.950167i	$0.399088\pi$
$348$	0	0
$349$	−2887.69	−0.442906	−0.221453	−	0.975171i	$-0.571080\pi$
$349$	−2887.69	−0.442906	−0.221453	+	0.975171i	$0.571080\pi$
$350$	0	0
$351$	0	0
$352$	−5559.79	−0.841869
$353$	−524.918	−0.0791461	−0.0395731	−	0.999217i	$-0.512600\pi$
$353$	−524.918	−0.0791461	−0.0395731	+	0.999217i	$0.512600\pi$
$354$	0	0
$355$	2038.44	0.304758
$356$	−5815.84	−0.865840
$357$	0	0
$358$	7689.25	1.13517
$359$	3285.31	0.482987	0.241493	−	0.970402i	$-0.422363\pi$
$359$	3285.31	0.482987	0.241493	+	0.970402i	$0.422363\pi$
$360$	0	0
$361$	−2861.64	−0.417209
$362$	−2619.84	−0.380374
$363$	0	0
$364$	0	0
$365$	−1742.38	−0.249864
$366$	0	0
$367$	−9053.95	−1.28777	−0.643886	−	0.765121i	$-0.722679\pi$
$367$	−9053.95	−1.28777	−0.643886	+	0.765121i	$0.722679\pi$
$368$	8988.60	1.27327
$369$	0	0
$370$	−5074.54	−0.713008
$371$	0	0
$372$	0	0
$373$	−5209.10	−0.723102	−0.361551	−	0.932352i	$-0.617753\pi$
$373$	−5209.10	−0.723102	−0.361551	+	0.932352i	$0.617753\pi$
$374$	61.1864	0.00845955
$375$	0	0
$376$	−498.322	−0.0683483
$377$	6359.12	0.868730
$378$	0	0
$379$	3200.68	0.433794	0.216897	−	0.976194i	$-0.430406\pi$
$379$	3200.68	0.433794	0.216897	+	0.976194i	$0.430406\pi$
$380$	2147.57	0.289916
$381$	0	0
$382$	−6203.04	−0.830825
$383$	−2202.81	−0.293886	−0.146943	−	0.989145i	$-0.546943\pi$
$383$	−2202.81	−0.293886	−0.146943	+	0.989145i	$0.546943\pi$
$384$	0	0
$385$	0	0
$386$	−3111.29	−0.410261
$387$	0	0
$388$	−1286.92	−0.168386
$389$	6241.59	0.813525	0.406762	−	0.913534i	$-0.366658\pi$
$389$	6241.59	0.813525	0.406762	+	0.913534i	$0.366658\pi$
$390$	0	0
$391$	−85.6955	−0.0110839
$392$	0	0
$393$	0	0
$394$	14247.3	1.82175
$395$	−4200.68	−0.535086
$396$	0	0
$397$	−10407.2	−1.31567	−0.657834	−	0.753163i	$-0.728527\pi$
$397$	−10407.2	−1.31567	−0.657834	+	0.753163i	$0.728527\pi$
$398$	−1415.13	−0.178227
$399$	0	0
$400$	−1804.92	−0.225615
$401$	14973.3	1.86466	0.932330	−	0.361608i	$-0.117772\pi$
$401$	14973.3	1.86466	0.932330	+	0.361608i	$0.117772\pi$
$402$	0	0
$403$	17094.6	2.11301
$404$	5406.25	0.665769
$405$	0	0
$406$	0	0
$407$	6098.58	0.742741
$408$	0	0
$409$	7059.10	0.853422	0.426711	−	0.904388i	$-0.359672\pi$
$409$	7059.10	0.853422	0.426711	+	0.904388i	$0.359672\pi$
$410$	−4680.20	−0.563753
$411$	0	0
$412$	−13184.1	−1.57654
$413$	0	0
$414$	0	0
$415$	4428.26	0.523795
$416$	−14685.5	−1.73081
$417$	0	0
$418$	−5620.28	−0.657648
$419$	928.543	0.108263	0.0541316	−	0.998534i	$-0.482761\pi$
$419$	928.543	0.108263	0.0541316	+	0.998534i	$0.482761\pi$
$420$	0	0
$421$	7105.06	0.822517	0.411258	−	0.911519i	$-0.365089\pi$
$421$	7105.06	0.822517	0.411258	+	0.911519i	$0.365089\pi$
$422$	4128.39	0.476225
$423$	0	0
$424$	−207.942	−0.0238173
$425$	17.2077	0.00196399
$426$	0	0
$427$	0	0
$428$	−3319.88	−0.374936
$429$	0	0
$430$	3318.16	0.372130
$431$	4319.27	0.482719	0.241360	−	0.970436i	$-0.422407\pi$
$431$	4319.27	0.482719	0.241360	+	0.970436i	$0.422407\pi$
$432$	0	0
$433$	−4600.60	−0.510602	−0.255301	−	0.966862i	$-0.582175\pi$
$433$	−4600.60	−0.510602	−0.255301	+	0.966862i	$0.582175\pi$
$434$	0	0
$435$	0	0
$436$	11073.7	1.21636
$437$	7871.58	0.861667
$438$	0	0
$439$	−1923.74	−0.209146	−0.104573	−	0.994517i	$-0.533348\pi$
$439$	−1923.74	−0.209146	−0.104573	+	0.994517i	$0.533348\pi$
$440$	536.274	0.0581042
$441$	0	0
$442$	161.616	0.0173921
$443$	4534.87	0.486362	0.243181	−	0.969981i	$-0.421809\pi$
$443$	4534.87	0.486362	0.243181	+	0.969981i	$0.421809\pi$
$444$	0	0
$445$	4280.48	0.455987
$446$	−6635.35	−0.704468
$447$	0	0
$448$	0	0
$449$	17431.9	1.83221	0.916105	−	0.400937i	$-0.131316\pi$
$449$	17431.9	1.83221	0.916105	+	0.400937i	$0.131316\pi$
$450$	0	0
$451$	5624.66	0.587262
$452$	−2350.02	−0.244548
$453$	0	0
$454$	13911.7	1.43813
$455$	0	0
$456$	0	0
$457$	−1962.32	−0.200861	−0.100431	−	0.994944i	$-0.532022\pi$
$457$	−1962.32	−0.200861	−0.100431	+	0.994944i	$0.532022\pi$
$458$	5622.36	0.573614
$459$	0	0
$460$	4228.98	0.428646
$461$	14344.4	1.44920	0.724602	−	0.689167i	$-0.242024\pi$
$461$	14344.4	1.44920	0.724602	+	0.689167i	$0.242024\pi$
$462$	0	0
$463$	11657.8	1.17016	0.585081	−	0.810975i	$-0.301063\pi$
$463$	11657.8	1.17016	0.585081	+	0.810975i	$0.301063\pi$
$464$	7520.51	0.752437
$465$	0	0
$466$	8212.91	0.816429
$467$	−16513.8	−1.63633	−0.818164	−	0.574985i	$-0.805008\pi$
$467$	−16513.8	−1.63633	−0.818164	+	0.574985i	$0.805008\pi$
$468$	0	0
$469$	0	0
$470$	−2065.07	−0.202669
$471$	0	0
$472$	2121.92	0.206927
$473$	−3987.76	−0.387648
$474$	0	0
$475$	−1580.62	−0.152681
$476$	0	0
$477$	0	0
$478$	22894.1	2.19069
$479$	−15620.1	−1.48998	−0.744988	−	0.667077i	$-0.767545\pi$
$479$	−15620.1	−1.48998	−0.744988	+	0.667077i	$0.767545\pi$
$480$	0	0
$481$	16108.6	1.52701
$482$	−14798.6	−1.39846
$483$	0	0
$484$	−5413.28	−0.508384
$485$	947.180	0.0886788
$486$	0	0
$487$	1901.38	0.176920	0.0884598	−	0.996080i	$-0.471806\pi$
$487$	1901.38	0.176920	0.0884598	+	0.996080i	$0.471806\pi$
$488$	−2198.76	−0.203961
$489$	0	0
$490$	0	0
$491$	15964.1	1.46731	0.733654	−	0.679524i	$-0.237814\pi$
$491$	15964.1	1.46731	0.733654	+	0.679524i	$0.237814\pi$
$492$	0	0
$493$	−71.6991	−0.00655003
$494$	−14845.3	−1.35207
$495$	0	0
$496$	20216.7	1.83015
$497$	0	0
$498$	0	0
$499$	−6849.18	−0.614451	−0.307226	−	0.951637i	$-0.599401\pi$
$499$	−6849.18	−0.614451	−0.307226	+	0.951637i	$0.599401\pi$
$500$	−849.181	−0.0759531
$501$	0	0
$502$	6660.45	0.592173
$503$	−766.190	−0.0679179	−0.0339590	−	0.999423i	$-0.510812\pi$
$503$	−766.190	−0.0679179	−0.0339590	+	0.999423i	$0.510812\pi$
$504$	0	0
$505$	−3979.02	−0.350621
$506$	−11067.4	−0.972345
$507$	0	0
$508$	−11312.7	−0.988035
$509$	13604.7	1.18471	0.592356	−	0.805676i	$-0.298198\pi$
$509$	13604.7	1.18471	0.592356	+	0.805676i	$0.298198\pi$
$510$	0	0
$511$	0	0
$512$	−14687.2	−1.26776
$513$	0	0
$514$	15594.2	1.33820
$515$	9703.53	0.830269
$516$	0	0
$517$	2481.80	0.211120
$518$	0	0
$519$	0	0
$520$	1416.50	0.119457
$521$	18428.3	1.54963	0.774817	−	0.632185i	$-0.217842\pi$
$521$	18428.3	1.54963	0.774817	+	0.632185i	$0.217842\pi$
$522$	0	0
$523$	−4340.13	−0.362869	−0.181435	−	0.983403i	$-0.558074\pi$
$523$	−4340.13	−0.362869	−0.181435	+	0.983403i	$0.558074\pi$
$524$	4597.45	0.383283
$525$	0	0
$526$	−24369.0	−2.02003
$527$	−192.742	−0.0159316
$528$	0	0
$529$	3333.65	0.273991
$530$	−861.720	−0.0706240
$531$	0	0
$532$	0	0
$533$	14856.8	1.20736
$534$	0	0
$535$	2443.44	0.197456
$536$	−1064.82	−0.0858081
$537$	0	0
$538$	5083.30	0.407354
$539$	0	0
$540$	0	0
$541$	12616.2	1.00262	0.501308	−	0.865269i	$-0.332852\pi$
$541$	12616.2	1.00262	0.501308	+	0.865269i	$0.332852\pi$
$542$	−8984.79	−0.712047
$543$	0	0
$544$	165.579	0.0130499
$545$	−8150.27	−0.640586
$546$	0	0
$547$	20677.8	1.61630	0.808151	−	0.588975i	$-0.200468\pi$
$547$	20677.8	1.61630	0.808151	+	0.588975i	$0.200468\pi$
$548$	7026.01	0.547694
$549$	0	0
$550$	2222.34	0.172293
$551$	6585.93	0.509201
$552$	0	0
$553$	0	0
$554$	27251.4	2.08989
$555$	0	0
$556$	−2960.76	−0.225835
$557$	1015.17	0.0772248	0.0386124	−	0.999254i	$-0.487706\pi$
$557$	1015.17	0.0772248	0.0386124	+	0.999254i	$0.487706\pi$
$558$	0	0
$559$	−10533.2	−0.796969
$560$	0	0
$561$	0	0
$562$	8166.17	0.612935
$563$	5296.64	0.396495	0.198248	−	0.980152i	$-0.436475\pi$
$563$	5296.64	0.396495	0.198248	+	0.980152i	$0.436475\pi$
$564$	0	0
$565$	1729.62	0.128789
$566$	22660.5	1.68285
$567$	0	0
$568$	−1891.94	−0.139761
$569$	−14741.2	−1.08609	−0.543044	−	0.839704i	$-0.682728\pi$
$569$	−14741.2	−1.08609	−0.543044	+	0.839704i	$0.682728\pi$
$570$	0	0
$571$	13661.4	1.00125	0.500624	−	0.865665i	$-0.333104\pi$
$571$	13661.4	1.00125	0.500624	+	0.865665i	$0.333104\pi$
$572$	9585.04	0.700648
$573$	0	0
$574$	0	0
$575$	−3112.54	−0.225742
$576$	0	0
$577$	−1078.61	−0.0778220	−0.0389110	−	0.999243i	$-0.512389\pi$
$577$	−1078.61	−0.0778220	−0.0389110	+	0.999243i	$0.512389\pi$
$578$	18894.7	1.35971
$579$	0	0
$580$	3538.27	0.253308
$581$	0	0
$582$	0	0
$583$	1035.61	0.0735690
$584$	1617.16	0.114586
$585$	0	0
$586$	31299.6	2.20644
$587$	3863.36	0.271649	0.135824	−	0.990733i	$-0.456632\pi$
$587$	3863.36	0.271649	0.135824	+	0.990733i	$0.456632\pi$
$588$	0	0
$589$	17704.3	1.23853
$590$	8793.35	0.613587
$591$	0	0
$592$	19050.6	1.32259
$593$	−9440.81	−0.653774	−0.326887	−	0.945063i	$-0.606000\pi$
$593$	−9440.81	−0.653774	−0.326887	+	0.945063i	$0.606000\pi$
$594$	0	0
$595$	0	0
$596$	−14401.8	−0.989797
$597$	0	0
$598$	−29233.2	−1.99905
$599$	21368.9	1.45761	0.728807	−	0.684719i	$-0.240075\pi$
$599$	21368.9	1.45761	0.728807	+	0.684719i	$0.240075\pi$
$600$	0	0
$601$	−5801.37	−0.393748	−0.196874	−	0.980429i	$-0.563079\pi$
$601$	−5801.37	−0.393748	−0.196874	+	0.980429i	$0.563079\pi$
$602$	0	0
$603$	0	0
$604$	−16307.0	−1.09855
$605$	3984.19	0.267736
$606$	0	0
$607$	−26740.0	−1.78805	−0.894023	−	0.448021i	$-0.852129\pi$
$607$	−26740.0	−1.78805	−0.894023	+	0.448021i	$0.852129\pi$
$608$	−15209.3	−1.01450
$609$	0	0
$610$	−9111.74	−0.604793
$611$	6555.36	0.434045
$612$	0	0
$613$	−8339.75	−0.549493	−0.274746	−	0.961517i	$-0.588594\pi$
$613$	−8339.75	−0.549493	−0.274746	+	0.961517i	$0.588594\pi$
$614$	−18451.3	−1.21276
$615$	0	0
$616$	0	0
$617$	26639.2	1.73818	0.869088	−	0.494658i	$-0.164707\pi$
$617$	26639.2	1.73818	0.869088	+	0.494658i	$0.164707\pi$
$618$	0	0
$619$	−27860.3	−1.80905	−0.904523	−	0.426424i	$-0.859773\pi$
$619$	−27860.3	−1.80905	−0.904523	+	0.426424i	$0.859773\pi$
$620$	9511.58	0.616120
$621$	0	0
$622$	−19960.1	−1.28670
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−8635.71	−0.551361
$627$	0	0
$628$	3438.76	0.218506
$629$	−181.625	−0.0115133
$630$	0	0
$631$	−10886.7	−0.686833	−0.343417	−	0.939183i	$-0.611584\pi$
$631$	−10886.7	−0.686833	−0.343417	+	0.939183i	$0.611584\pi$
$632$	3898.79	0.245388
$633$	0	0
$634$	−37470.9	−2.34726
$635$	8326.21	0.520340
$636$	0	0
$637$	0	0
$638$	−9259.80	−0.574607
$639$	0	0
$640$	2936.25	0.181352
$641$	16851.5	1.03837	0.519183	−	0.854663i	$-0.326236\pi$
$641$	16851.5	1.03837	0.519183	+	0.854663i	$0.326236\pi$
$642$	0	0
$643$	−4793.72	−0.294006	−0.147003	−	0.989136i	$-0.546963\pi$
$643$	−4793.72	−0.294006	−0.147003	+	0.989136i	$0.546963\pi$
$644$	0	0
$645$	0	0
$646$	167.380	0.0101943
$647$	−26684.5	−1.62145	−0.810725	−	0.585428i	$-0.800927\pi$
$647$	−26684.5	−1.62145	−0.810725	+	0.585428i	$0.800927\pi$
$648$	0	0
$649$	−10567.8	−0.639174
$650$	5870.05	0.354219
$651$	0	0
$652$	17669.4	1.06133
$653$	−17541.9	−1.05125	−0.525625	−	0.850716i	$-0.676169\pi$
$653$	−17541.9	−1.05125	−0.525625	+	0.850716i	$0.676169\pi$
$654$	0	0
$655$	−3383.74	−0.201853
$656$	17570.2	1.04573
$657$	0	0
$658$	0	0
$659$	26285.3	1.55377	0.776883	−	0.629645i	$-0.216800\pi$
$659$	26285.3	1.55377	0.776883	+	0.629645i	$0.216800\pi$
$660$	0	0
$661$	−25044.3	−1.47369	−0.736845	−	0.676062i	$-0.763685\pi$
$661$	−25044.3	−1.47369	−0.736845	+	0.676062i	$0.763685\pi$
$662$	−43580.7	−2.55863
$663$	0	0
$664$	−4110.02	−0.240210
$665$	0	0
$666$	0	0
$667$	12969.0	0.752864
$668$	3709.75	0.214872
$669$	0	0
$670$	−4412.66	−0.254441
$671$	10950.5	0.630013
$672$	0	0
$673$	20799.4	1.19132	0.595658	−	0.803238i	$-0.296891\pi$
$673$	20799.4	1.19132	0.595658	+	0.803238i	$0.296891\pi$
$674$	−13990.1	−0.799521
$675$	0	0
$676$	10392.5	0.591288
$677$	29772.5	1.69018	0.845089	−	0.534625i	$-0.179547\pi$
$677$	29772.5	1.69018	0.845089	+	0.534625i	$0.179547\pi$
$678$	0	0
$679$	0	0
$680$	−15.9711	−0.000900679 0
$681$	0	0
$682$	−24892.2	−1.39761
$683$	−23890.3	−1.33842	−0.669208	−	0.743075i	$-0.733367\pi$
$683$	−23890.3	−1.33842	−0.669208	+	0.743075i	$0.733367\pi$
$684$	0	0
$685$	−5171.17	−0.288438
$686$	0	0
$687$	0	0
$688$	−12456.9	−0.690282
$689$	2735.45	0.151251
$690$	0	0
$691$	25297.6	1.39271	0.696356	−	0.717696i	$-0.254803\pi$
$691$	25297.6	1.39271	0.696356	+	0.717696i	$0.254803\pi$
$692$	−20518.4	−1.12716
$693$	0	0
$694$	−15500.8	−0.847839
$695$	2179.13	0.118934
$696$	0	0
$697$	−167.511	−0.00910320
$698$	11106.7	0.602284
$699$	0	0
$700$	0	0
$701$	−18659.1	−1.00534	−0.502670	−	0.864478i	$-0.667649\pi$
$701$	−18659.1	−1.00534	−0.502670	+	0.864478i	$0.667649\pi$
$702$	0	0
$703$	16683.2	0.895047
$704$	8035.38	0.430177
$705$	0	0
$706$	2018.95	0.107627
$707$	0	0
$708$	0	0
$709$	−25694.8	−1.36106	−0.680528	−	0.732722i	$-0.738249\pi$
$709$	−25694.8	−1.36106	−0.680528	+	0.732722i	$0.738249\pi$
$710$	−7840.30	−0.414424
$711$	0	0
$712$	−3972.85	−0.209114
$713$	34863.2	1.83119
$714$	0	0
$715$	−7054.62	−0.368990
$716$	−13581.2	−0.708875
$717$	0	0
$718$	−12636.1	−0.656788
$719$	−27254.7	−1.41367	−0.706834	−	0.707379i	$-0.749877\pi$
$719$	−27254.7	−1.41367	−0.706834	+	0.707379i	$0.749877\pi$
$720$	0	0
$721$	0	0
$722$	11006.5	0.567340
$723$	0	0
$724$	4627.32	0.237532
$725$	−2604.18	−0.133402
$726$	0	0
$727$	13194.3	0.673110	0.336555	−	0.941664i	$-0.390738\pi$
$727$	13194.3	0.673110	0.336555	+	0.941664i	$0.390738\pi$
$728$	0	0
$729$	0	0
$730$	6701.57	0.339776
$731$	118.761	0.00600896
$732$	0	0
$733$	−5808.95	−0.292713	−0.146356	−	0.989232i	$-0.546755\pi$
$733$	−5808.95	−0.292713	−0.146356	+	0.989232i	$0.546755\pi$
$734$	34823.5	1.75117
$735$	0	0
$736$	−29950.0	−1.49996
$737$	5303.13	0.265052
$738$	0	0
$739$	13936.7	0.693734	0.346867	−	0.937914i	$-0.387246\pi$
$739$	13936.7	0.693734	0.346867	+	0.937914i	$0.387246\pi$
$740$	8962.98	0.445251
$741$	0	0
$742$	0	0
$743$	−32424.2	−1.60098	−0.800489	−	0.599347i	$-0.795427\pi$
$743$	−32424.2	−1.60098	−0.800489	+	0.599347i	$0.795427\pi$
$744$	0	0
$745$	10599.7	0.521268
$746$	20035.4	0.983308
$747$	0	0
$748$	−108.071	−0.00528272
$749$	0	0
$750$	0	0
$751$	−35288.3	−1.71463	−0.857316	−	0.514791i	$-0.827870\pi$
$751$	−35288.3	−1.71463	−0.857316	+	0.514791i	$0.827870\pi$
$752$	7752.58	0.375941
$753$	0	0
$754$	−24458.6	−1.18134
$755$	12002.0	0.578539
$756$	0	0
$757$	2079.86	0.0998595	0.0499298	−	0.998753i	$-0.484100\pi$
$757$	2079.86	0.0998595	0.0499298	+	0.998753i	$0.484100\pi$
$758$	−12310.5	−0.589893
$759$	0	0
$760$	1467.02	0.0700191
$761$	12251.4	0.583593	0.291797	−	0.956480i	$-0.405747\pi$
$761$	12251.4	0.583593	0.291797	+	0.956480i	$0.405747\pi$
$762$	0	0
$763$	0	0
$764$	10956.2	0.518824
$765$	0	0
$766$	8472.50	0.399639
$767$	−27913.6	−1.31408
$768$	0	0
$769$	−4335.87	−0.203323	−0.101661	−	0.994819i	$-0.532416\pi$
$769$	−4335.87	−0.203323	−0.101661	+	0.994819i	$0.532416\pi$
$770$	0	0
$771$	0	0
$772$	5495.37	0.256195
$773$	30399.5	1.41448	0.707240	−	0.706973i	$-0.249940\pi$
$773$	30399.5	1.41448	0.707240	+	0.706973i	$0.249940\pi$
$774$	0	0
$775$	−7000.55	−0.324474
$776$	−879.109	−0.0406677
$777$	0	0
$778$	−24006.6	−1.10627
$779$	15386.7	0.707686
$780$	0	0
$781$	9422.47	0.431706
$782$	329.604	0.0150724
$783$	0	0
$784$	0	0
$785$	−2530.94	−0.115074
$786$	0	0
$787$	−35274.2	−1.59770	−0.798850	−	0.601530i	$-0.794558\pi$
$787$	−35274.2	−1.59770	−0.798850	+	0.601530i	$0.794558\pi$
$788$	−25164.5	−1.13762
$789$	0	0
$790$	16156.8	0.727635
$791$	0	0
$792$	0	0
$793$	28924.3	1.29525
$794$	40028.3	1.78911
$795$	0	0
$796$	2499.50	0.111297
$797$	−27524.0	−1.22328	−0.611638	−	0.791138i	$-0.709489\pi$
$797$	−27524.0	−1.22328	−0.611638	+	0.791138i	$0.709489\pi$
$798$	0	0
$799$	−73.9116	−0.00327260
$800$	6013.98	0.265783
$801$	0	0
$802$	−57590.5	−2.53565
$803$	−8053.95	−0.353945
$804$	0	0
$805$	0	0
$806$	−65749.7	−2.87337
$807$	0	0
$808$	3693.06	0.160794
$809$	7067.35	0.307138	0.153569	−	0.988138i	$-0.450923\pi$
$809$	7067.35	0.307138	0.153569	+	0.988138i	$0.450923\pi$
$810$	0	0
$811$	37757.5	1.63483	0.817413	−	0.576052i	$-0.195407\pi$
$811$	37757.5	1.63483	0.817413	+	0.576052i	$0.195407\pi$
$812$	0	0
$813$	0	0
$814$	−23456.5	−1.01001
$815$	−13004.7	−0.558940
$816$	0	0
$817$	−10908.9	−0.467139
$818$	−27150.9	−1.16052
$819$	0	0
$820$	8266.47	0.352046
$821$	25307.4	1.07580	0.537902	−	0.843008i	$-0.319217\pi$
$821$	25307.4	1.07580	0.537902	+	0.843008i	$0.319217\pi$
$822$	0	0
$823$	−25484.8	−1.07940	−0.539699	−	0.841858i	$-0.681462\pi$
$823$	−25484.8	−1.07940	−0.539699	+	0.841858i	$0.681462\pi$
$824$	−9006.17	−0.380758
$825$	0	0
$826$	0	0
$827$	−34985.7	−1.47107	−0.735534	−	0.677488i	$-0.763069\pi$
$827$	−34985.7	−1.47107	−0.735534	+	0.677488i	$0.763069\pi$
$828$	0	0
$829$	−3898.09	−0.163313	−0.0816564	−	0.996661i	$-0.526021\pi$
$829$	−3898.09	−0.163313	−0.0816564	+	0.996661i	$0.526021\pi$
$830$	−17032.1	−0.712280
$831$	0	0
$832$	21224.4	0.884405
$833$	0	0
$834$	0	0
$835$	−2730.38	−0.113160
$836$	9926.90	0.410681
$837$	0	0
$838$	−3571.38	−0.147221
$839$	22118.8	0.910161	0.455081	−	0.890450i	$-0.349610\pi$
$839$	22118.8	0.910161	0.455081	+	0.890450i	$0.349610\pi$
$840$	0	0
$841$	−13538.2	−0.555096
$842$	−27327.7	−1.11850
$843$	0	0
$844$	−7291.83	−0.297387
$845$	−7648.89	−0.311396
$846$	0	0
$847$	0	0
$848$	3235.03	0.131004
$849$	0	0
$850$	−66.1848	−0.00267073
$851$	32852.4	1.32334
$852$	0	0
$853$	−33152.4	−1.33074	−0.665368	−	0.746516i	$-0.731725\pi$
$853$	−33152.4	−1.33074	−0.665368	+	0.746516i	$0.731725\pi$
$854$	0	0
$855$	0	0
$856$	−2267.84	−0.0905527
$857$	−10346.2	−0.412393	−0.206197	−	0.978511i	$-0.566109\pi$
$857$	−10346.2	−0.412393	−0.206197	+	0.978511i	$0.566109\pi$
$858$	0	0
$859$	11905.6	0.472893	0.236447	−	0.971644i	$-0.424017\pi$
$859$	11905.6	0.472893	0.236447	+	0.971644i	$0.424017\pi$
$860$	−5860.74	−0.232383
$861$	0	0
$862$	−16612.9	−0.656424
$863$	−30802.3	−1.21497	−0.607487	−	0.794329i	$-0.707823\pi$
$863$	−30802.3	−1.21497	−0.607487	+	0.794329i	$0.707823\pi$
$864$	0	0
$865$	15101.6	0.593607
$866$	17694.9	0.694340
$867$	0	0
$868$	0	0
$869$	−19417.2	−0.757977
$870$	0	0
$871$	14007.5	0.544923
$872$	7564.53	0.293770
$873$	0	0
$874$	−30275.9	−1.17173
$875$	0	0
$876$	0	0
$877$	−21693.3	−0.835268	−0.417634	−	0.908615i	$-0.637141\pi$
$877$	−21693.3	−0.835268	−0.417634	+	0.908615i	$0.637141\pi$
$878$	7399.13	0.284406
$879$	0	0
$880$	−8343.03	−0.319595
$881$	−29963.1	−1.14584	−0.572918	−	0.819612i	$-0.694189\pi$
$881$	−29963.1	−1.14584	−0.572918	+	0.819612i	$0.694189\pi$
$882$	0	0
$883$	−4090.61	−0.155900	−0.0779501	−	0.996957i	$-0.524837\pi$
$883$	−4090.61	−0.155900	−0.0779501	+	0.996957i	$0.524837\pi$
$884$	−285.457	−0.0108608
$885$	0	0
$886$	−17442.1	−0.661377
$887$	−28819.4	−1.09094	−0.545468	−	0.838132i	$-0.683648\pi$
$887$	−28819.4	−1.09094	−0.545468	+	0.838132i	$0.683648\pi$
$888$	0	0
$889$	0	0
$890$	−16463.7	−0.620072
$891$	0	0
$892$	11719.8	0.439918
$893$	6789.16	0.254413
$894$	0	0
$895$	9995.83	0.373323
$896$	0	0
$897$	0	0
$898$	−67047.1	−2.49152
$899$	29169.1	1.08214
$900$	0	0
$901$	−30.8421	−0.00114040
$902$	−21633.7	−0.798585
$903$	0	0
$904$	−1605.32	−0.0590621
$905$	−3405.72	−0.125094
$906$	0	0
$907$	−20638.8	−0.755567	−0.377784	−	0.925894i	$-0.623314\pi$
$907$	−20638.8	−0.755567	−0.377784	+	0.925894i	$0.623314\pi$
$908$	−24571.8	−0.898067
$909$	0	0
$910$	0	0
$911$	15080.1	0.548435	0.274218	−	0.961668i	$-0.411581\pi$
$911$	15080.1	0.548435	0.274218	+	0.961668i	$0.411581\pi$
$912$	0	0
$913$	20469.1	0.741982
$914$	7547.54	0.273141
$915$	0	0
$916$	−9930.56	−0.358204
$917$	0	0
$918$	0	0
$919$	−2164.29	−0.0776858	−0.0388429	−	0.999245i	$-0.512367\pi$
$919$	−2164.29	−0.0776858	−0.0388429	+	0.999245i	$0.512367\pi$
$920$	2888.85	0.103525
$921$	0	0
$922$	−55171.6	−1.97069
$923$	24888.3	0.887549
$924$	0	0
$925$	−6596.78	−0.234487
$926$	−44838.6	−1.59124
$927$	0	0
$928$	−25058.3	−0.886401
$929$	13639.1	0.481684	0.240842	−	0.970564i	$-0.422576\pi$
$929$	13639.1	0.481684	0.240842	+	0.970564i	$0.422576\pi$
$930$	0	0
$931$	0	0
$932$	−14506.2	−0.509834
$933$	0	0
$934$	63515.6	2.22515
$935$	79.5408	0.00278210
$936$	0	0
$937$	−6568.49	−0.229011	−0.114505	−	0.993423i	$-0.536528\pi$
$937$	−6568.49	−0.229011	−0.114505	+	0.993423i	$0.536528\pi$
$938$	0	0
$939$	0	0
$940$	3647.45	0.126560
$941$	9454.51	0.327533	0.163766	−	0.986499i	$-0.447636\pi$
$941$	9454.51	0.327533	0.163766	+	0.986499i	$0.447636\pi$
$942$	0	0
$943$	30299.4	1.04633
$944$	−33011.6	−1.13817
$945$	0	0
$946$	15337.8	0.527141
$947$	−34076.7	−1.16932	−0.584660	−	0.811279i	$-0.698772\pi$
$947$	−34076.7	−1.16932	−0.584660	+	0.811279i	$0.698772\pi$
$948$	0	0
$949$	−21273.5	−0.727679
$950$	6079.41	0.207623
$951$	0	0
$952$	0	0
$953$	32862.3	1.11702	0.558508	−	0.829499i	$-0.311374\pi$
$953$	32862.3	1.11702	0.558508	+	0.829499i	$0.311374\pi$
$954$	0	0
$955$	−8063.81	−0.273234
$956$	−40437.0	−1.36802
$957$	0	0
$958$	60078.3	2.02614
$959$	0	0
$960$	0	0
$961$	48621.4	1.63208
$962$	−61957.5	−2.07650
$963$	0	0
$964$	26138.2	0.873292
$965$	−4044.61	−0.134923
$966$	0	0
$967$	57597.8	1.91543	0.957715	−	0.287720i	$-0.0928972\pi$
$967$	57597.8	1.91543	0.957715	+	0.287720i	$0.0928972\pi$
$968$	−3697.86	−0.122783
$969$	0	0
$970$	−3643.07	−0.120590
$971$	−28185.7	−0.931536	−0.465768	−	0.884907i	$-0.654222\pi$
$971$	−28185.7	−0.931536	−0.465768	+	0.884907i	$0.654222\pi$
$972$	0	0
$973$	0	0
$974$	−7313.15	−0.240583
$975$	0	0
$976$	34206.9	1.12186
$977$	−44263.6	−1.44945	−0.724727	−	0.689036i	$-0.758034\pi$
$977$	−44263.6	−1.44945	−0.724727	+	0.689036i	$0.758034\pi$
$978$	0	0
$979$	19786.0	0.645929
$980$	0	0
$981$	0	0
$982$	−61401.4	−1.99531
$983$	147.400	0.00478262	0.00239131	−	0.999997i	$-0.499239\pi$
$983$	147.400	0.00478262	0.00239131	+	0.999997i	$0.499239\pi$
$984$	0	0
$985$	18521.1	0.599119
$986$	275.771	0.00890703
$987$	0	0
$988$	26220.7	0.844323
$989$	−21481.6	−0.690674
$990$	0	0
$991$	12486.8	0.400260	0.200130	−	0.979769i	$-0.435864\pi$
$991$	12486.8	0.400260	0.200130	+	0.979769i	$0.435864\pi$
$992$	−67361.9	−2.15599
$993$	0	0
$994$	0	0
$995$	−1839.64	−0.0586135
$996$	0	0
$997$	6621.15	0.210325	0.105162	−	0.994455i	$-0.466464\pi$
$997$	6621.15	0.210325	0.105162	+	0.994455i	$0.466464\pi$
$998$	26343.5	0.835559
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bt.1.2		5
3.2	odd	2		245.4.a.n.1.4		5
7.3	odd	6		315.4.j.g.226.4		10
7.5	odd	6		315.4.j.g.46.4		10
7.6	odd	2		2205.4.a.bu.1.2		5
15.14	odd	2		1225.4.a.bf.1.2		5
21.2	odd	6		245.4.e.o.116.2		10
21.5	even	6		35.4.e.c.11.2	✓	10
21.11	odd	6		245.4.e.o.226.2		10
21.17	even	6		35.4.e.c.16.2	yes	10
21.20	even	2		245.4.a.m.1.4		5
84.47	odd	6		560.4.q.n.81.1		10
84.59	odd	6		560.4.q.n.401.1		10
105.17	odd	12		175.4.k.d.149.3		20
105.38	odd	12		175.4.k.d.149.8		20
105.47	odd	12		175.4.k.d.74.8		20
105.59	even	6		175.4.e.d.51.4		10
105.68	odd	12		175.4.k.d.74.3		20
105.89	even	6		175.4.e.d.151.4		10
105.104	even	2		1225.4.a.bg.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.e.c.11.2	✓	10	21.5	even	6
35.4.e.c.16.2	yes	10	21.17	even	6
175.4.e.d.51.4		10	105.59	even	6
175.4.e.d.151.4		10	105.89	even	6
175.4.k.d.74.3		20	105.68	odd	12
175.4.k.d.74.8		20	105.47	odd	12
175.4.k.d.149.3		20	105.17	odd	12
175.4.k.d.149.8		20	105.38	odd	12
245.4.a.m.1.4		5	21.20	even	2
245.4.a.n.1.4		5	3.2	odd	2
245.4.e.o.116.2		10	21.2	odd	6
245.4.e.o.226.2		10	21.11	odd	6
315.4.j.g.46.4		10	7.5	odd	6
315.4.j.g.226.4		10	7.3	odd	6
560.4.q.n.81.1		10	84.47	odd	6
560.4.q.n.401.1		10	84.59	odd	6
1225.4.a.bf.1.2		5	15.14	odd	2
1225.4.a.bg.1.2		5	105.104	even	2
2205.4.a.bt.1.2		5	1.1	even	1	trivial
2205.4.a.bu.1.2		5	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-3.84623 q^{2} +6.79345 q^{4} -5.00000 q^{5} +4.64067 q^{8} +O(q^{10})\)	\(q-3.84623 q^{2} +6.79345 q^{4} -5.00000 q^{5} +4.64067 q^{8} +19.2311 q^{10} -23.1120 q^{11} -61.0473 q^{13} -72.1966 q^{16} +0.688309 q^{17} -63.2247 q^{19} -33.9672 q^{20} +88.8938 q^{22} -124.502 q^{23} +25.0000 q^{25} +234.802 q^{26} -104.167 q^{29} -280.022 q^{31} +240.559 q^{32} -2.64739 q^{34} -263.871 q^{37} +243.176 q^{38} -23.2033 q^{40} -243.366 q^{41} +172.541 q^{43} -157.010 q^{44} +478.861 q^{46} -107.381 q^{47} -96.1556 q^{50} -414.722 q^{52} -44.8086 q^{53} +115.560 q^{55} +400.650 q^{58} +457.246 q^{59} -473.802 q^{61} +1077.03 q^{62} -347.672 q^{64} +305.237 q^{65} -229.454 q^{67} +4.67599 q^{68} -407.688 q^{71} +348.475 q^{73} +1014.91 q^{74} -429.514 q^{76} +840.135 q^{79} +360.983 q^{80} +936.040 q^{82} -885.652 q^{83} -3.44154 q^{85} -663.632 q^{86} -107.255 q^{88} -856.096 q^{89} -845.795 q^{92} +413.013 q^{94} +316.124 q^{95} -189.436 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 35 q^{4} - 25 q^{5} - 33 q^{8}+O(q^{10}) \) 5 * q - q^2 + 35 * q^4 - 25 * q^5 - 33 * q^8	\( 5 q - q^{2} + 35 q^{4} - 25 q^{5} - 33 q^{8} + 5 q^{10} - 47 q^{11} - q^{13} + 171 q^{16} - 2 q^{17} + 21 q^{19} - 175 q^{20} + 523 q^{22} - 201 q^{23} + 125 q^{25} - 47 q^{26} - 190 q^{29} - 388 q^{31} + 95 q^{32} - 130 q^{34} - 145 q^{37} + 835 q^{38} + 165 q^{40} - 281 q^{41} + 568 q^{43} - 1091 q^{44} + 337 q^{46} - 473 q^{47} - 25 q^{50} + 379 q^{52} - 351 q^{53} + 235 q^{55} + 1818 q^{58} + 708 q^{59} - 1944 q^{61} + 448 q^{62} - 125 q^{64} + 5 q^{65} + 1118 q^{67} - 3118 q^{68} - 864 q^{71} + 1652 q^{73} + 3285 q^{74} - 691 q^{76} + 218 q^{79} - 855 q^{80} - 1027 q^{82} - 1502 q^{83} + 10 q^{85} + 4264 q^{86} + 2131 q^{88} + 2322 q^{89} + 2957 q^{92} + 2677 q^{94} - 105 q^{95} - 598 q^{97}+O(q^{100}) \) 5 * q - q^2 + 35 * q^4 - 25 * q^5 - 33 * q^8 + 5 * q^10 - 47 * q^11 - q^13 + 171 * q^16 - 2 * q^17 + 21 * q^19 - 175 * q^20 + 523 * q^22 - 201 * q^23 + 125 * q^25 - 47 * q^26 - 190 * q^29 - 388 * q^31 + 95 * q^32 - 130 * q^34 - 145 * q^37 + 835 * q^38 + 165 * q^40 - 281 * q^41 + 568 * q^43 - 1091 * q^44 + 337 * q^46 - 473 * q^47 - 25 * q^50 + 379 * q^52 - 351 * q^53 + 235 * q^55 + 1818 * q^58 + 708 * q^59 - 1944 * q^61 + 448 * q^62 - 125 * q^64 + 5 * q^65 + 1118 * q^67 - 3118 * q^68 - 864 * q^71 + 1652 * q^73 + 3285 * q^74 - 691 * q^76 + 218 * q^79 - 855 * q^80 - 1027 * q^82 - 1502 * q^83 + 10 * q^85 + 4264 * q^86 + 2131 * q^88 + 2322 * q^89 + 2957 * q^92 + 2677 * q^94 - 105 * q^95 - 598 * q^97

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