LMFDB - Embedded newform 2205.4.a.ca.1.4

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:	$6$
Coefficient field:	6.6.1163891200.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.04490$ 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)

$f(q)$	$=$	$q+0.369315 q^{2} -7.86361 q^{4} +5.00000 q^{5} -5.85867 q^{8} +O(q^{10})$	$q+0.369315 q^{2} -7.86361 q^{4} +5.00000 q^{5} -5.85867 q^{8} +1.84657 q^{10} -30.6361 q^{11} -36.4622 q^{13} +60.7452 q^{16} -79.7341 q^{17} +152.418 q^{19} -39.3180 q^{20} -11.3144 q^{22} -22.2207 q^{23} +25.0000 q^{25} -13.4660 q^{26} -101.285 q^{29} -249.956 q^{31} +69.3034 q^{32} -29.4470 q^{34} +7.55765 q^{37} +56.2904 q^{38} -29.2933 q^{40} +142.280 q^{41} -237.530 q^{43} +240.910 q^{44} -8.20644 q^{46} -331.129 q^{47} +9.23287 q^{50} +286.725 q^{52} +487.337 q^{53} -153.180 q^{55} -37.4059 q^{58} -717.355 q^{59} -354.592 q^{61} -92.3126 q^{62} -460.367 q^{64} -182.311 q^{65} +57.5883 q^{67} +626.997 q^{68} +696.174 q^{71} +261.035 q^{73} +2.79115 q^{74} -1198.56 q^{76} +271.344 q^{79} +303.726 q^{80} +52.5460 q^{82} +681.441 q^{83} -398.670 q^{85} -87.7234 q^{86} +179.487 q^{88} +160.668 q^{89} +174.735 q^{92} -122.291 q^{94} +762.092 q^{95} +167.841 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8	$ 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31} + 770 q^{32} + 812 q^{34} - 140 q^{37} - 20 q^{38} + 330 q^{40} - 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} - 628 q^{47} + 50 q^{50} + 1520 q^{52} + 676 q^{53} + 80 q^{55} - 2012 q^{58} - 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} + 2940 q^{68} + 224 q^{71} + 2640 q^{73} - 928 q^{74} - 1340 q^{76} + 1636 q^{79} + 1490 q^{80} - 1756 q^{82} - 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} + 1904 q^{89} + 1952 q^{92} - 3332 q^{94} + 1540 q^{95} + 516 q^{97}+O(q^{100}) $ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * 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 $

$2$	0.369315	0.130573	0.0652863	−	0.997867i	$-0.479204\pi$
$2$	0.369315	0.130573	0.0652863	+	0.997867i	$0.479204\pi$
$3$	0	0
$3$	0	0
$4$	−7.86361	−0.982951
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−5.85867	−0.258919
$9$	0	0
$10$	1.84657	0.0583938
$11$	−30.6361	−0.839739	−0.419870	−	0.907584i	$-0.637924\pi$
$11$	−30.6361	−0.839739	−0.419870	+	0.907584i	$0.637924\pi$
$12$	0	0
$13$	−36.4622	−0.777908	−0.388954	−	0.921257i	$-0.627164\pi$
$13$	−36.4622	−0.777908	−0.388954	+	0.921257i	$0.627164\pi$
$14$	0	0
$15$	0	0
$16$	60.7452	0.949143
$17$	−79.7341	−1.13755	−0.568775	−	0.822493i	$-0.692583\pi$
$17$	−79.7341	−1.13755	−0.568775	+	0.822493i	$0.692583\pi$
$18$	0	0
$19$	152.418	1.84038	0.920189	−	0.391474i	$-0.128035\pi$
$19$	152.418	1.84038	0.920189	+	0.391474i	$0.128035\pi$
$20$	−39.3180	−0.439589
$21$	0	0
$22$	−11.3144	−0.109647
$23$	−22.2207	−0.201450	−0.100725	−	0.994914i	$-0.532116\pi$
$23$	−22.2207	−0.201450	−0.100725	+	0.994914i	$0.532116\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−13.4660	−0.101573
$27$	0	0
$28$	0	0
$29$	−101.285	−0.648555	−0.324278	−	0.945962i	$-0.605121\pi$
$29$	−101.285	−0.648555	−0.324278	+	0.945962i	$0.605121\pi$
$30$	0	0
$31$	−249.956	−1.44818	−0.724089	−	0.689707i	$-0.757739\pi$
$31$	−249.956	−1.44818	−0.724089	+	0.689707i	$0.757739\pi$
$32$	69.3034	0.382851
$33$	0	0
$34$	−29.4470	−0.148533
$35$	0	0
$36$	0	0
$37$	7.55765	0.0335803	0.0167901	−	0.999859i	$-0.494655\pi$
$37$	7.55765	0.0335803	0.0167901	+	0.999859i	$0.494655\pi$
$38$	56.2904	0.240303
$39$	0	0
$40$	−29.2933	−0.115792
$41$	142.280	0.541960	0.270980	−	0.962585i	$-0.412652\pi$
$41$	142.280	0.541960	0.270980	+	0.962585i	$0.412652\pi$
$42$	0	0
$43$	−237.530	−0.842396	−0.421198	−	0.906969i	$-0.638390\pi$
$43$	−237.530	−0.842396	−0.421198	+	0.906969i	$0.638390\pi$
$44$	240.910	0.825422
$45$	0	0
$46$	−8.20644	−0.0263038
$47$	−331.129	−1.02766	−0.513830	−	0.857892i	$-0.671774\pi$
$47$	−331.129	−1.02766	−0.513830	+	0.857892i	$0.671774\pi$
$48$	0	0
$49$	0	0
$50$	9.23287	0.0261145
$51$	0	0
$52$	286.725	0.764645
$53$	487.337	1.26303	0.631517	−	0.775362i	$-0.282432\pi$
$53$	487.337	1.26303	0.631517	+	0.775362i	$0.282432\pi$
$54$	0	0
$55$	−153.180	−0.375543
$56$	0	0
$57$	0	0
$58$	−37.4059	−0.0846835
$59$	−717.355	−1.58291	−0.791454	−	0.611228i	$-0.790676\pi$
$59$	−717.355	−1.58291	−0.791454	+	0.611228i	$0.790676\pi$
$60$	0	0
$61$	−354.592	−0.744276	−0.372138	−	0.928177i	$-0.621375\pi$
$61$	−354.592	−0.744276	−0.372138	+	0.928177i	$0.621375\pi$
$62$	−92.3126	−0.189092
$63$	0	0
$64$	−460.367	−0.899153
$65$	−182.311	−0.347891
$66$	0	0
$67$	57.5883	0.105008	0.0525040	−	0.998621i	$-0.483280\pi$
$67$	57.5883	0.105008	0.0525040	+	0.998621i	$0.483280\pi$
$68$	626.997	1.11816
$69$	0	0
$70$	0	0
$71$	696.174	1.16367	0.581836	−	0.813306i	$-0.302335\pi$
$71$	696.174	1.16367	0.581836	+	0.813306i	$0.302335\pi$
$72$	0	0
$73$	261.035	0.418518	0.209259	−	0.977860i	$-0.432895\pi$
$73$	261.035	0.418518	0.209259	+	0.977860i	$0.432895\pi$
$74$	2.79115	0.00438466
$75$	0	0
$76$	−1198.56	−1.80900
$77$	0	0
$78$	0	0
$79$	271.344	0.386438	0.193219	−	0.981156i	$-0.438107\pi$
$79$	271.344	0.386438	0.193219	+	0.981156i	$0.438107\pi$
$80$	303.726	0.424470
$81$	0	0
$82$	52.5460	0.0707651
$83$	681.441	0.901179	0.450590	−	0.892731i	$-0.351214\pi$
$83$	681.441	0.901179	0.450590	+	0.892731i	$0.351214\pi$
$84$	0	0
$85$	−398.670	−0.508728
$86$	−87.7234	−0.109994
$87$	0	0
$88$	179.487	0.217424
$89$	160.668	0.191357	0.0956784	−	0.995412i	$-0.469498\pi$
$89$	160.668	0.191357	0.0956784	+	0.995412i	$0.469498\pi$
$90$	0	0
$91$	0	0
$92$	174.735	0.198015
$93$	0	0
$94$	−122.291	−0.134184
$95$	762.092	0.823042
$96$	0	0
$97$	167.841	0.175687	0.0878436	−	0.996134i	$-0.472002\pi$
$97$	167.841	0.175687	0.0878436	+	0.996134i	$0.472002\pi$
$98$	0	0
$99$	0	0
$100$	−196.590	−0.196590
$101$	−413.394	−0.407270	−0.203635	−	0.979047i	$-0.565276\pi$
$101$	−413.394	−0.407270	−0.203635	+	0.979047i	$0.565276\pi$
$102$	0	0
$103$	1451.11	1.38817	0.694086	−	0.719892i	$-0.255809\pi$
$103$	1451.11	1.38817	0.694086	+	0.719892i	$0.255809\pi$
$104$	213.620	0.201415
$105$	0	0
$106$	179.981	0.164918
$107$	1780.63	1.60878	0.804391	−	0.594100i	$-0.202492\pi$
$107$	1780.63	1.60878	0.804391	+	0.594100i	$0.202492\pi$
$108$	0	0
$109$	37.8920	0.0332972	0.0166486	−	0.999861i	$-0.494700\pi$
$109$	37.8920	0.0332972	0.0166486	+	0.999861i	$0.494700\pi$
$110$	−56.5718	−0.0490356
$111$	0	0
$112$	0	0
$113$	−457.320	−0.380717	−0.190359	−	0.981715i	$-0.560965\pi$
$113$	−457.320	−0.380717	−0.190359	+	0.981715i	$0.560965\pi$
$114$	0	0
$115$	−111.104	−0.0900910
$116$	796.463	0.637498
$117$	0	0
$118$	−264.930	−0.206684
$119$	0	0
$120$	0	0
$121$	−392.430	−0.294838
$122$	−130.956	−0.0971820
$123$	0	0
$124$	1965.56	1.42349
$125$	125.000	0.0894427
$126$	0	0
$127$	−2545.41	−1.77849	−0.889246	−	0.457429i	$-0.848771\pi$
$127$	−2545.41	−1.77849	−0.889246	+	0.457429i	$0.848771\pi$
$128$	−724.447	−0.500256
$129$	0	0
$130$	−67.3302	−0.0454250
$131$	970.873	0.647523	0.323762	−	0.946139i	$-0.395052\pi$
$131$	970.873	0.647523	0.323762	+	0.946139i	$0.395052\pi$
$132$	0	0
$133$	0	0
$134$	21.2682	0.0137112
$135$	0	0
$136$	467.135	0.294533
$137$	−183.635	−0.114518	−0.0572592	−	0.998359i	$-0.518236\pi$
$137$	−183.635	−0.114518	−0.0572592	+	0.998359i	$0.518236\pi$
$138$	0	0
$139$	−1078.90	−0.658356	−0.329178	−	0.944268i	$-0.606772\pi$
$139$	−1078.90	−0.658356	−0.329178	+	0.944268i	$0.606772\pi$
$140$	0	0
$141$	0	0
$142$	257.108	0.151944
$143$	1117.06	0.653240
$144$	0	0
$145$	−506.424	−0.290043
$146$	96.4040	0.0546469
$147$	0	0
$148$	−59.4304	−0.0330077
$149$	−1211.24	−0.665962	−0.332981	−	0.942934i	$-0.608054\pi$
$149$	−1211.24	−0.665962	−0.332981	+	0.942934i	$0.608054\pi$
$150$	0	0
$151$	−602.244	−0.324569	−0.162285	−	0.986744i	$-0.551886\pi$
$151$	−602.244	−0.324569	−0.162285	+	0.986744i	$0.551886\pi$
$152$	−892.969	−0.476509
$153$	0	0
$154$	0	0
$155$	−1249.78	−0.647644
$156$	0	0
$157$	1602.07	0.814389	0.407195	−	0.913341i	$-0.366507\pi$
$157$	1602.07	0.814389	0.407195	+	0.913341i	$0.366507\pi$
$158$	100.212	0.0504582
$159$	0	0
$160$	346.517	0.171216
$161$	0	0
$162$	0	0
$163$	−3091.61	−1.48561	−0.742803	−	0.669510i	$-0.766504\pi$
$163$	−3091.61	−1.48561	−0.742803	+	0.669510i	$0.766504\pi$
$164$	−1118.83	−0.532720
$165$	0	0
$166$	251.666	0.117669
$167$	3251.19	1.50649	0.753247	−	0.657738i	$-0.228487\pi$
$167$	3251.19	1.50649	0.753247	+	0.657738i	$0.228487\pi$
$168$	0	0
$169$	−867.505	−0.394859
$170$	−147.235	−0.0664259
$171$	0	0
$172$	1867.84	0.828034
$173$	−2254.55	−0.990809	−0.495405	−	0.868662i	$-0.664980\pi$
$173$	−2254.55	−0.990809	−0.495405	+	0.868662i	$0.664980\pi$
$174$	0	0
$175$	0	0
$176$	−1860.99	−0.797033
$177$	0	0
$178$	59.3370	0.0249859
$179$	2685.46	1.12135	0.560673	−	0.828038i	$-0.310543\pi$
$179$	2685.46	1.12135	0.560673	+	0.828038i	$0.310543\pi$
$180$	0	0
$181$	1293.25	0.531087	0.265543	−	0.964099i	$-0.414449\pi$
$181$	1293.25	0.531087	0.265543	+	0.964099i	$0.414449\pi$
$182$	0	0
$183$	0	0
$184$	130.184	0.0521591
$185$	37.7882	0.0150175
$186$	0	0
$187$	2442.74	0.955245
$188$	2603.86	1.01014
$189$	0	0
$190$	281.452	0.107467
$191$	139.878	0.0529905	0.0264953	−	0.999649i	$-0.491565\pi$
$191$	139.878	0.0529905	0.0264953	+	0.999649i	$0.491565\pi$
$192$	0	0
$193$	2384.81	0.889442	0.444721	−	0.895669i	$-0.353303\pi$
$193$	2384.81	0.889442	0.444721	+	0.895669i	$0.353303\pi$
$194$	61.9861	0.0229399
$195$	0	0
$196$	0	0
$197$	1008.67	0.364797	0.182399	−	0.983225i	$-0.441614\pi$
$197$	1008.67	0.364797	0.182399	+	0.983225i	$0.441614\pi$
$198$	0	0
$199$	995.036	0.354453	0.177227	−	0.984170i	$-0.443287\pi$
$199$	995.036	0.354453	0.177227	+	0.984170i	$0.443287\pi$
$200$	−146.467	−0.0517838
$201$	0	0
$202$	−152.672	−0.0531782
$203$	0	0
$204$	0	0
$205$	711.399	0.242372
$206$	535.915	0.181257
$207$	0	0
$208$	−2214.90	−0.738346
$209$	−4669.51	−1.54544
$210$	0	0
$211$	2307.30	0.752802	0.376401	−	0.926457i	$-0.377161\pi$
$211$	2307.30	0.752802	0.376401	+	0.926457i	$0.377161\pi$
$212$	−3832.22	−1.24150
$213$	0	0
$214$	657.612	0.210063
$215$	−1187.65	−0.376731
$216$	0	0
$217$	0	0
$218$	13.9941	0.00434770
$219$	0	0
$220$	1204.55	0.369140
$221$	2907.28	0.884910
$222$	0	0
$223$	1629.00	0.489173	0.244587	−	0.969627i	$-0.421348\pi$
$223$	1629.00	0.489173	0.244587	+	0.969627i	$0.421348\pi$
$224$	0	0
$225$	0	0
$226$	−168.895	−0.0497112
$227$	2410.92	0.704926	0.352463	−	0.935826i	$-0.385344\pi$
$227$	2410.92	0.704926	0.352463	+	0.935826i	$0.385344\pi$
$228$	0	0
$229$	3666.59	1.05806	0.529028	−	0.848604i	$-0.322557\pi$
$229$	3666.59	1.05806	0.529028	+	0.848604i	$0.322557\pi$
$230$	−41.0322	−0.0117634
$231$	0	0
$232$	593.393	0.167923
$233$	3018.33	0.848658	0.424329	−	0.905508i	$-0.360510\pi$
$233$	3018.33	0.848658	0.424329	+	0.905508i	$0.360510\pi$
$234$	0	0
$235$	−1655.64	−0.459584
$236$	5640.99	1.55592
$237$	0	0
$238$	0	0
$239$	546.873	0.148010	0.0740048	−	0.997258i	$-0.476422\pi$
$239$	546.873	0.148010	0.0740048	+	0.997258i	$0.476422\pi$
$240$	0	0
$241$	3135.27	0.838010	0.419005	−	0.907984i	$-0.362379\pi$
$241$	3135.27	0.838010	0.419005	+	0.907984i	$0.362379\pi$
$242$	−144.930	−0.0384978
$243$	0	0
$244$	2788.37	0.731587
$245$	0	0
$246$	0	0
$247$	−5557.52	−1.43165
$248$	1464.41	0.374960
$249$	0	0
$250$	46.1644	0.0116788
$251$	914.967	0.230088	0.115044	−	0.993360i	$-0.463299\pi$
$251$	914.967	0.230088	0.115044	+	0.993360i	$0.463299\pi$
$252$	0	0
$253$	680.756	0.169165
$254$	−940.058	−0.232222
$255$	0	0
$256$	3415.38	0.833834
$257$	6334.18	1.53741	0.768707	−	0.639602i	$-0.220901\pi$
$257$	6334.18	1.53741	0.768707	+	0.639602i	$0.220901\pi$
$258$	0	0
$259$	0	0
$260$	1433.62	0.341960
$261$	0	0
$262$	358.558	0.0845487
$263$	−4126.75	−0.967553	−0.483777	−	0.875191i	$-0.660735\pi$
$263$	−4126.75	−0.967553	−0.483777	+	0.875191i	$0.660735\pi$
$264$	0	0
$265$	2436.68	0.564846
$266$	0	0
$267$	0	0
$268$	−452.852	−0.103218
$269$	5887.07	1.33435	0.667176	−	0.744900i	$-0.267503\pi$
$269$	5887.07	1.33435	0.667176	+	0.744900i	$0.267503\pi$
$270$	0	0
$271$	7237.72	1.62236	0.811181	−	0.584795i	$-0.198825\pi$
$271$	7237.72	1.62236	0.811181	+	0.584795i	$0.198825\pi$
$272$	−4843.46	−1.07970
$273$	0	0
$274$	−67.8192	−0.0149530
$275$	−765.902	−0.167948
$276$	0	0
$277$	5640.17	1.22341	0.611705	−	0.791086i	$-0.290484\pi$
$277$	5640.17	1.22341	0.611705	+	0.791086i	$0.290484\pi$
$278$	−398.456	−0.0859632
$279$	0	0
$280$	0	0
$281$	2593.05	0.550492	0.275246	−	0.961374i	$-0.411241\pi$
$281$	2593.05	0.550492	0.275246	+	0.961374i	$0.411241\pi$
$282$	0	0
$283$	2533.26	0.532109	0.266054	−	0.963958i	$-0.414280\pi$
$283$	2533.26	0.532109	0.266054	+	0.963958i	$0.414280\pi$
$284$	−5474.44	−1.14383
$285$	0	0
$286$	412.547	0.0852952
$287$	0	0
$288$	0	0
$289$	1444.52	0.294021
$290$	−187.030	−0.0378716
$291$	0	0
$292$	−2052.67	−0.411382
$293$	589.215	0.117482	0.0587411	−	0.998273i	$-0.481291\pi$
$293$	589.215	0.117482	0.0587411	+	0.998273i	$0.481291\pi$
$294$	0	0
$295$	−3586.77	−0.707898
$296$	−44.2777	−0.00869456
$297$	0	0
$298$	−447.327	−0.0869563
$299$	810.217	0.156709
$300$	0	0
$301$	0	0
$302$	−222.418	−0.0423798
$303$	0	0
$304$	9258.68	1.74678
$305$	−1772.96	−0.332851
$306$	0	0
$307$	5273.75	0.980420	0.490210	−	0.871604i	$-0.336920\pi$
$307$	5273.75	0.980420	0.490210	+	0.871604i	$0.336920\pi$
$308$	0	0
$309$	0	0
$310$	−461.563	−0.0845646
$311$	3295.08	0.600795	0.300397	−	0.953814i	$-0.402881\pi$
$311$	3295.08	0.600795	0.300397	+	0.953814i	$0.402881\pi$
$312$	0	0
$313$	−1819.37	−0.328553	−0.164276	−	0.986414i	$-0.552529\pi$
$313$	−1819.37	−0.328553	−0.164276	+	0.986414i	$0.552529\pi$
$314$	591.668	0.106337
$315$	0	0
$316$	−2133.75	−0.379850
$317$	−1887.91	−0.334496	−0.167248	−	0.985915i	$-0.553488\pi$
$317$	−1887.91	−0.334496	−0.167248	+	0.985915i	$0.553488\pi$
$318$	0	0
$319$	3102.97	0.544617
$320$	−2301.83	−0.402114
$321$	0	0
$322$	0	0
$323$	−12152.9	−2.09352
$324$	0	0
$325$	−911.556	−0.155582
$326$	−1141.78	−0.193979
$327$	0	0
$328$	−833.569	−0.140324
$329$	0	0
$330$	0	0
$331$	−456.387	−0.0757863	−0.0378932	−	0.999282i	$-0.512065\pi$
$331$	−456.387	−0.0757863	−0.0378932	+	0.999282i	$0.512065\pi$
$332$	−5358.58	−0.885815
$333$	0	0
$334$	1200.71	0.196707
$335$	287.942	0.0469610
$336$	0	0
$337$	8174.42	1.32133	0.660666	−	0.750680i	$-0.270274\pi$
$337$	8174.42	1.32133	0.660666	+	0.750680i	$0.270274\pi$
$338$	−320.383	−0.0515577
$339$	0	0
$340$	3134.99	0.500055
$341$	7657.69	1.21609
$342$	0	0
$343$	0	0
$344$	1391.61	0.218112
$345$	0	0
$346$	−832.637	−0.129372
$347$	7935.70	1.22770	0.613849	−	0.789424i	$-0.289621\pi$
$347$	7935.70	1.22770	0.613849	+	0.789424i	$0.289621\pi$
$348$	0	0
$349$	8649.40	1.32662	0.663312	−	0.748343i	$-0.269150\pi$
$349$	8649.40	1.32662	0.663312	+	0.748343i	$0.269150\pi$
$350$	0	0
$351$	0	0
$352$	−2123.19	−0.321495
$353$	4931.06	0.743495	0.371747	−	0.928334i	$-0.378759\pi$
$353$	4931.06	0.743495	0.371747	+	0.928334i	$0.378759\pi$
$354$	0	0
$355$	3480.87	0.520410
$356$	−1263.43	−0.188094
$357$	0	0
$358$	991.781	0.146417
$359$	−8243.64	−1.21193	−0.605965	−	0.795491i	$-0.707213\pi$
$359$	−8243.64	−1.21193	−0.605965	+	0.795491i	$0.707213\pi$
$360$	0	0
$361$	16372.4	2.38699
$362$	477.617	0.0693453
$363$	0	0
$364$	0	0
$365$	1305.17	0.187167
$366$	0	0
$367$	−8179.11	−1.16334	−0.581670	−	0.813425i	$-0.697601\pi$
$367$	−8179.11	−1.16334	−0.581670	+	0.813425i	$0.697601\pi$
$368$	−1349.80	−0.191204
$369$	0	0
$370$	13.9558	0.00196088
$371$	0	0
$372$	0	0
$373$	−12795.5	−1.77620	−0.888102	−	0.459646i	$-0.847976\pi$
$373$	−12795.5	−1.77620	−0.888102	+	0.459646i	$0.847976\pi$
$374$	902.140	0.124729
$375$	0	0
$376$	1939.97	0.266081
$377$	3693.07	0.504516
$378$	0	0
$379$	−5735.40	−0.777329	−0.388664	−	0.921379i	$-0.627063\pi$
$379$	−5735.40	−0.777329	−0.388664	+	0.921379i	$0.627063\pi$
$380$	−5992.79	−0.809010
$381$	0	0
$382$	51.6589	0.00691910
$383$	−737.841	−0.0984384	−0.0492192	−	0.998788i	$-0.515673\pi$
$383$	−737.841	−0.0984384	−0.0492192	+	0.998788i	$0.515673\pi$
$384$	0	0
$385$	0	0
$386$	880.745	0.116137
$387$	0	0
$388$	−1319.83	−0.172692
$389$	13034.0	1.69884	0.849419	−	0.527718i	$-0.176952\pi$
$389$	13034.0	1.69884	0.849419	+	0.527718i	$0.176952\pi$
$390$	0	0
$391$	1771.75	0.229159
$392$	0	0
$393$	0	0
$394$	372.518	0.0476325
$395$	1356.72	0.172821
$396$	0	0
$397$	8313.64	1.05101	0.525503	−	0.850792i	$-0.323877\pi$
$397$	8313.64	1.05101	0.525503	+	0.850792i	$0.323877\pi$
$398$	367.481	0.0462819
$399$	0	0
$400$	1518.63	0.189829
$401$	14341.4	1.78598	0.892988	−	0.450080i	$-0.148604\pi$
$401$	14341.4	1.78598	0.892988	+	0.450080i	$0.148604\pi$
$402$	0	0
$403$	9113.97	1.12655
$404$	3250.77	0.400326
$405$	0	0
$406$	0	0
$407$	−231.537	−0.0281987
$408$	0	0
$409$	−6282.35	−0.759517	−0.379758	−	0.925086i	$-0.623993\pi$
$409$	−6282.35	−0.759517	−0.379758	+	0.925086i	$0.623993\pi$
$410$	262.730	0.0316471
$411$	0	0
$412$	−11410.9	−1.36450
$413$	0	0
$414$	0	0
$415$	3407.21	0.403020
$416$	−2526.96	−0.297823
$417$	0	0
$418$	−1724.52	−0.201792
$419$	−15226.1	−1.77528	−0.887640	−	0.460538i	$-0.847656\pi$
$419$	−15226.1	−1.77528	−0.887640	+	0.460538i	$0.847656\pi$
$420$	0	0
$421$	−2026.55	−0.234603	−0.117302	−	0.993096i	$-0.537424\pi$
$421$	−2026.55	−0.234603	−0.117302	+	0.993096i	$0.537424\pi$
$422$	852.121	0.0982953
$423$	0	0
$424$	−2855.14	−0.327024
$425$	−1993.35	−0.227510
$426$	0	0
$427$	0	0
$428$	−14002.2	−1.58135
$429$	0	0
$430$	−438.617	−0.0491907
$431$	−8983.56	−1.00400	−0.501999	−	0.864868i	$-0.667402\pi$
$431$	−8983.56	−1.00400	−0.501999	+	0.864868i	$0.667402\pi$
$432$	0	0
$433$	−6836.59	−0.758766	−0.379383	−	0.925240i	$-0.623864\pi$
$433$	−6836.59	−0.758766	−0.379383	+	0.925240i	$0.623864\pi$
$434$	0	0
$435$	0	0
$436$	−297.968	−0.0327295
$437$	−3386.85	−0.370743
$438$	0	0
$439$	7802.96	0.848325	0.424163	−	0.905586i	$-0.360568\pi$
$439$	7802.96	0.848325	0.424163	+	0.905586i	$0.360568\pi$
$440$	897.433	0.0972351
$441$	0	0
$442$	1073.70	0.115545
$443$	−5184.37	−0.556020	−0.278010	−	0.960578i	$-0.589675\pi$
$443$	−5184.37	−0.556020	−0.278010	+	0.960578i	$0.589675\pi$
$444$	0	0
$445$	803.339	0.0855773
$446$	601.612	0.0638726
$447$	0	0
$448$	0	0
$449$	772.951	0.0812424	0.0406212	−	0.999175i	$-0.487066\pi$
$449$	772.951	0.0812424	0.0406212	+	0.999175i	$0.487066\pi$
$450$	0	0
$451$	−4358.90	−0.455105
$452$	3596.18	0.374226
$453$	0	0
$454$	890.387	0.0920439
$455$	0	0
$456$	0	0
$457$	11532.7	1.18048	0.590239	−	0.807228i	$-0.299033\pi$
$457$	11532.7	1.18048	0.590239	+	0.807228i	$0.299033\pi$
$458$	1354.13	0.138153
$459$	0	0
$460$	873.675	0.0885550
$461$	−10400.7	−1.05078	−0.525391	−	0.850861i	$-0.676081\pi$
$461$	−10400.7	−1.05078	−0.525391	+	0.850861i	$0.676081\pi$
$462$	0	0
$463$	−13855.4	−1.39075	−0.695374	−	0.718648i	$-0.744761\pi$
$463$	−13855.4	−1.39075	−0.695374	+	0.718648i	$0.744761\pi$
$464$	−6152.56	−0.615572
$465$	0	0
$466$	1114.71	0.110811
$467$	14734.5	1.46002	0.730011	−	0.683435i	$-0.239515\pi$
$467$	14734.5	1.46002	0.730011	+	0.683435i	$0.239515\pi$
$468$	0	0
$469$	0	0
$470$	−611.453	−0.0600090
$471$	0	0
$472$	4202.74	0.409845
$473$	7277.00	0.707393
$474$	0	0
$475$	3810.46	0.368076
$476$	0	0
$477$	0	0
$478$	201.968	0.0193260
$479$	1471.23	0.140338	0.0701691	−	0.997535i	$-0.477646\pi$
$479$	1471.23	0.140338	0.0701691	+	0.997535i	$0.477646\pi$
$480$	0	0
$481$	−275.569	−0.0261224
$482$	1157.90	0.109421
$483$	0	0
$484$	3085.91	0.289812
$485$	839.204	0.0785697
$486$	0	0
$487$	5610.00	0.521998	0.260999	−	0.965339i	$-0.415948\pi$
$487$	5610.00	0.521998	0.260999	+	0.965339i	$0.415948\pi$
$488$	2077.44	0.192707
$489$	0	0
$490$	0	0
$491$	3193.98	0.293569	0.146784	−	0.989169i	$-0.453108\pi$
$491$	3193.98	0.293569	0.146784	+	0.989169i	$0.453108\pi$
$492$	0	0
$493$	8075.85	0.737764
$494$	−2052.47	−0.186933
$495$	0	0
$496$	−15183.6	−1.37453
$497$	0	0
$498$	0	0
$499$	8724.32	0.782673	0.391337	−	0.920248i	$-0.372013\pi$
$499$	8724.32	0.782673	0.391337	+	0.920248i	$0.372013\pi$
$500$	−982.951	−0.0879178
$501$	0	0
$502$	337.911	0.0300432
$503$	14636.7	1.29745	0.648726	−	0.761022i	$-0.275302\pi$
$503$	14636.7	1.29745	0.648726	+	0.761022i	$0.275302\pi$
$504$	0	0
$505$	−2066.97	−0.182136
$506$	251.413	0.0220883
$507$	0	0
$508$	20016.1	1.74817
$509$	−8530.56	−0.742850	−0.371425	−	0.928463i	$-0.621131\pi$
$509$	−8530.56	−0.742850	−0.371425	+	0.928463i	$0.621131\pi$
$510$	0	0
$511$	0	0
$512$	7056.93	0.609131
$513$	0	0
$514$	2339.31	0.200744
$515$	7255.53	0.620809
$516$	0	0
$517$	10144.5	0.862967
$518$	0	0
$519$	0	0
$520$	1068.10	0.0900756
$521$	−13743.0	−1.15564	−0.577822	−	0.816163i	$-0.696097\pi$
$521$	−13743.0	−1.15564	−0.577822	+	0.816163i	$0.696097\pi$
$522$	0	0
$523$	−2236.17	−0.186961	−0.0934806	−	0.995621i	$-0.529799\pi$
$523$	−2236.17	−0.186961	−0.0934806	+	0.995621i	$0.529799\pi$
$524$	−7634.56	−0.636483
$525$	0	0
$526$	−1524.07	−0.126336
$527$	19930.0	1.64737
$528$	0	0
$529$	−11673.2	−0.959418
$530$	899.903	0.0737534
$531$	0	0
$532$	0	0
$533$	−5187.84	−0.421595
$534$	0	0
$535$	8903.14	0.719470
$536$	−337.391	−0.0271885
$537$	0	0
$538$	2174.18	0.174230
$539$	0	0
$540$	0	0
$541$	−19487.9	−1.54870	−0.774352	−	0.632755i	$-0.781924\pi$
$541$	−19487.9	−1.54870	−0.774352	+	0.632755i	$0.781924\pi$
$542$	2673.00	0.211836
$543$	0	0
$544$	−5525.84	−0.435512
$545$	189.460	0.0148910
$546$	0	0
$547$	−15949.3	−1.24670	−0.623349	−	0.781944i	$-0.714228\pi$
$547$	−15949.3	−1.24670	−0.623349	+	0.781944i	$0.714228\pi$
$548$	1444.04	0.112566
$549$	0	0
$550$	−282.859	−0.0219294
$551$	−15437.7	−1.19359
$552$	0	0
$553$	0	0
$554$	2083.00	0.159744
$555$	0	0
$556$	8484.08	0.647132
$557$	6349.74	0.483029	0.241514	−	0.970397i	$-0.422356\pi$
$557$	6349.74	0.483029	0.241514	+	0.970397i	$0.422356\pi$
$558$	0	0
$559$	8660.88	0.655306
$560$	0	0
$561$	0	0
$562$	957.651	0.0718791
$563$	14805.0	1.10827	0.554135	−	0.832427i	$-0.313049\pi$
$563$	14805.0	1.10827	0.554135	+	0.832427i	$0.313049\pi$
$564$	0	0
$565$	−2286.60	−0.170262
$566$	935.571	0.0694788
$567$	0	0
$568$	−4078.65	−0.301297
$569$	−24578.7	−1.81088	−0.905442	−	0.424469i	$-0.860461\pi$
$569$	−24578.7	−1.81088	−0.905442	+	0.424469i	$0.860461\pi$
$570$	0	0
$571$	−15609.9	−1.14405	−0.572027	−	0.820235i	$-0.693843\pi$
$571$	−15609.9	−1.14405	−0.572027	+	0.820235i	$0.693843\pi$
$572$	−8784.12	−0.642103
$573$	0	0
$574$	0	0
$575$	−555.518	−0.0402899
$576$	0	0
$577$	−15111.3	−1.09028	−0.545141	−	0.838344i	$-0.683524\pi$
$577$	−15111.3	−1.09028	−0.545141	+	0.838344i	$0.683524\pi$
$578$	533.484	0.0383910
$579$	0	0
$580$	3982.32	0.285098
$581$	0	0
$582$	0	0
$583$	−14930.1	−1.06062
$584$	−1529.31	−0.108362
$585$	0	0
$586$	217.606	0.0153399
$587$	14983.2	1.05353	0.526766	−	0.850010i	$-0.323404\pi$
$587$	14983.2	1.05353	0.526766	+	0.850010i	$0.323404\pi$
$588$	0	0
$589$	−38098.0	−2.66519
$590$	−1324.65	−0.0924320
$591$	0	0
$592$	459.090	0.0318725
$593$	6931.01	0.479971	0.239985	−	0.970777i	$-0.422857\pi$
$593$	6931.01	0.479971	0.239985	+	0.970777i	$0.422857\pi$
$594$	0	0
$595$	0	0
$596$	9524.68	0.654608
$597$	0	0
$598$	299.225	0.0204619
$599$	−12309.2	−0.839632	−0.419816	−	0.907609i	$-0.637905\pi$
$599$	−12309.2	−0.839632	−0.419816	+	0.907609i	$0.637905\pi$
$600$	0	0
$601$	−15293.6	−1.03800	−0.519001	−	0.854774i	$-0.673696\pi$
$601$	−15293.6	−1.03800	−0.519001	+	0.854774i	$0.673696\pi$
$602$	0	0
$603$	0	0
$604$	4735.81	0.319036
$605$	−1962.15	−0.131856
$606$	0	0
$607$	5338.80	0.356994	0.178497	−	0.983940i	$-0.442877\pi$
$607$	5338.80	0.356994	0.178497	+	0.983940i	$0.442877\pi$
$608$	10563.1	0.704590
$609$	0	0
$610$	−654.780	−0.0434611
$611$	12073.7	0.799426
$612$	0	0
$613$	−28848.6	−1.90079	−0.950396	−	0.311043i	$-0.899322\pi$
$613$	−28848.6	−1.90079	−0.950396	+	0.311043i	$0.899322\pi$
$614$	1947.68	0.128016
$615$	0	0
$616$	0	0
$617$	12346.6	0.805602	0.402801	−	0.915287i	$-0.368037\pi$
$617$	12346.6	0.805602	0.402801	+	0.915287i	$0.368037\pi$
$618$	0	0
$619$	6383.16	0.414477	0.207238	−	0.978290i	$-0.433552\pi$
$619$	6383.16	0.414477	0.207238	+	0.978290i	$0.433552\pi$
$620$	9827.79	0.636603
$621$	0	0
$622$	1216.92	0.0784473
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−671.921	−0.0429000
$627$	0	0
$628$	−12598.1	−0.800505
$629$	−602.602	−0.0381992
$630$	0	0
$631$	25708.6	1.62194	0.810969	−	0.585090i	$-0.198941\pi$
$631$	25708.6	1.62194	0.810969	+	0.585090i	$0.198941\pi$
$632$	−1589.72	−0.100056
$633$	0	0
$634$	−697.232	−0.0436760
$635$	−12727.0	−0.795366
$636$	0	0
$637$	0	0
$638$	1145.97	0.0711120
$639$	0	0
$640$	−3622.24	−0.223721
$641$	18617.5	1.14719	0.573594	−	0.819139i	$-0.305549\pi$
$641$	18617.5	1.14719	0.573594	+	0.819139i	$0.305549\pi$
$642$	0	0
$643$	−21168.8	−1.29832	−0.649158	−	0.760654i	$-0.724879\pi$
$643$	−21168.8	−1.29832	−0.649158	+	0.760654i	$0.724879\pi$
$644$	0	0
$645$	0	0
$646$	−4488.26	−0.273357
$647$	5952.47	0.361693	0.180847	−	0.983511i	$-0.442116\pi$
$647$	5952.47	0.361693	0.180847	+	0.983511i	$0.442116\pi$
$648$	0	0
$649$	21976.9	1.32923
$650$	−336.651	−0.0203147
$651$	0	0
$652$	24311.2	1.46028
$653$	3040.63	0.182219	0.0911096	−	0.995841i	$-0.470959\pi$
$653$	3040.63	0.182219	0.0911096	+	0.995841i	$0.470959\pi$
$654$	0	0
$655$	4854.36	0.289581
$656$	8642.81	0.514398
$657$	0	0
$658$	0	0
$659$	3335.95	0.197193	0.0985966	−	0.995127i	$-0.468565\pi$
$659$	3335.95	0.197193	0.0985966	+	0.995127i	$0.468565\pi$
$660$	0	0
$661$	−1547.90	−0.0910838	−0.0455419	−	0.998962i	$-0.514501\pi$
$661$	−1547.90	−0.0910838	−0.0455419	+	0.998962i	$0.514501\pi$
$662$	−168.550	−0.00989561
$663$	0	0
$664$	−3992.34	−0.233332
$665$	0	0
$666$	0	0
$667$	2250.62	0.130651
$668$	−25566.0	−1.48081
$669$	0	0
$670$	106.341	0.00613181
$671$	10863.3	0.624998
$672$	0	0
$673$	19632.4	1.12448	0.562239	−	0.826975i	$-0.309940\pi$
$673$	19632.4	1.12448	0.562239	+	0.826975i	$0.309940\pi$
$674$	3018.94	0.172530
$675$	0	0
$676$	6821.72	0.388127
$677$	−30318.2	−1.72116	−0.860579	−	0.509317i	$-0.829898\pi$
$677$	−30318.2	−1.72116	−0.860579	+	0.509317i	$0.829898\pi$
$678$	0	0
$679$	0	0
$680$	2335.68	0.131719
$681$	0	0
$682$	2828.10	0.158788
$683$	−28970.8	−1.62304	−0.811520	−	0.584325i	$-0.801359\pi$
$683$	−28970.8	−1.62304	−0.811520	+	0.584325i	$0.801359\pi$
$684$	0	0
$685$	−918.176	−0.0512142
$686$	0	0
$687$	0	0
$688$	−14428.8	−0.799554
$689$	−17769.4	−0.982525
$690$	0	0
$691$	−3434.91	−0.189103	−0.0945514	−	0.995520i	$-0.530142\pi$
$691$	−3434.91	−0.189103	−0.0945514	+	0.995520i	$0.530142\pi$
$692$	17728.9	0.973917
$693$	0	0
$694$	2930.77	0.160304
$695$	−5394.52	−0.294426
$696$	0	0
$697$	−11344.5	−0.616507
$698$	3194.35	0.173221
$699$	0	0
$700$	0	0
$701$	−16304.4	−0.878471	−0.439235	−	0.898372i	$-0.644751\pi$
$701$	−16304.4	−0.878471	−0.439235	+	0.898372i	$0.644751\pi$
$702$	0	0
$703$	1151.92	0.0618004
$704$	14103.8	0.755054
$705$	0	0
$706$	1821.11	0.0970800
$707$	0	0
$708$	0	0
$709$	16094.4	0.852519	0.426260	−	0.904601i	$-0.359831\pi$
$709$	16094.4	0.852519	0.426260	+	0.904601i	$0.359831\pi$
$710$	1285.54	0.0679512
$711$	0	0
$712$	−941.299	−0.0495459
$713$	5554.21	0.291735
$714$	0	0
$715$	5585.30	0.292138
$716$	−21117.4	−1.10223
$717$	0	0
$718$	−3044.50	−0.158245
$719$	−8651.93	−0.448766	−0.224383	−	0.974501i	$-0.572037\pi$
$719$	−8651.93	−0.448766	−0.224383	+	0.974501i	$0.572037\pi$
$720$	0	0
$721$	0	0
$722$	6046.56	0.311676
$723$	0	0
$724$	−10169.6	−0.522032
$725$	−2532.12	−0.129711
$726$	0	0
$727$	−6999.43	−0.357076	−0.178538	−	0.983933i	$-0.557137\pi$
$727$	−6999.43	−0.357076	−0.178538	+	0.983933i	$0.557137\pi$
$728$	0	0
$729$	0	0
$730$	482.020	0.0244388
$731$	18939.3	0.958268
$732$	0	0
$733$	5167.74	0.260402	0.130201	−	0.991488i	$-0.458438\pi$
$733$	5167.74	0.260402	0.130201	+	0.991488i	$0.458438\pi$
$734$	−3020.67	−0.151900
$735$	0	0
$736$	−1539.97	−0.0771251
$737$	−1764.28	−0.0881793
$738$	0	0
$739$	−12319.1	−0.613214	−0.306607	−	0.951836i	$-0.599194\pi$
$739$	−12319.1	−0.613214	−0.306607	+	0.951836i	$0.599194\pi$
$740$	−297.152	−0.0147615
$741$	0	0
$742$	0	0
$743$	−16942.4	−0.836548	−0.418274	−	0.908321i	$-0.637365\pi$
$743$	−16942.4	−0.836548	−0.418274	+	0.908321i	$0.637365\pi$
$744$	0	0
$745$	−6056.18	−0.297827
$746$	−4725.56	−0.231923
$747$	0	0
$748$	−19208.8	−0.938959
$749$	0	0
$750$	0	0
$751$	32917.9	1.59946	0.799728	−	0.600363i	$-0.204977\pi$
$751$	32917.9	1.59946	0.799728	+	0.600363i	$0.204977\pi$
$752$	−20114.5	−0.975397
$753$	0	0
$754$	1363.90	0.0658760
$755$	−3011.22	−0.145152
$756$	0	0
$757$	−9433.45	−0.452926	−0.226463	−	0.974020i	$-0.572716\pi$
$757$	−9433.45	−0.452926	−0.226463	+	0.974020i	$0.572716\pi$
$758$	−2118.17	−0.101498
$759$	0	0
$760$	−4464.84	−0.213101
$761$	34551.0	1.64582	0.822912	−	0.568170i	$-0.192348\pi$
$761$	34551.0	1.64582	0.822912	+	0.568170i	$0.192348\pi$
$762$	0	0
$763$	0	0
$764$	−1099.94	−0.0520871
$765$	0	0
$766$	−272.495	−0.0128533
$767$	26156.4	1.23136
$768$	0	0
$769$	−348.011	−0.0163194	−0.00815970	−	0.999967i	$-0.502597\pi$
$769$	−348.011	−0.0163194	−0.00815970	+	0.999967i	$0.502597\pi$
$770$	0	0
$771$	0	0
$772$	−18753.2	−0.874277
$773$	30083.9	1.39979	0.699897	−	0.714243i	$-0.253229\pi$
$773$	30083.9	1.39979	0.699897	+	0.714243i	$0.253229\pi$
$774$	0	0
$775$	−6248.91	−0.289635
$776$	−983.323	−0.0454887
$777$	0	0
$778$	4813.64	0.221822
$779$	21686.1	0.997412
$780$	0	0
$781$	−21328.1	−0.977181
$782$	654.333	0.0299219
$783$	0	0
$784$	0	0
$785$	8010.35	0.364206
$786$	0	0
$787$	31958.4	1.44751	0.723757	−	0.690055i	$-0.242413\pi$
$787$	31958.4	1.44751	0.723757	+	0.690055i	$0.242413\pi$
$788$	−7931.81	−0.358578
$789$	0	0
$790$	501.058	0.0225656
$791$	0	0
$792$	0	0
$793$	12929.2	0.578979
$794$	3070.35	0.137233
$795$	0	0
$796$	−7824.57	−0.348410
$797$	364.165	0.0161849	0.00809247	−	0.999967i	$-0.497424\pi$
$797$	364.165	0.0161849	0.00809247	+	0.999967i	$0.497424\pi$
$798$	0	0
$799$	26402.2	1.16902
$800$	1732.59	0.0765702
$801$	0	0
$802$	5296.50	0.233199
$803$	−7997.08	−0.351446
$804$	0	0
$805$	0	0
$806$	3365.92	0.147096
$807$	0	0
$808$	2421.94	0.105450
$809$	−30714.8	−1.33482	−0.667412	−	0.744688i	$-0.732598\pi$
$809$	−30714.8	−1.33482	−0.667412	+	0.744688i	$0.732598\pi$
$810$	0	0
$811$	18967.6	0.821260	0.410630	−	0.911802i	$-0.365309\pi$
$811$	18967.6	0.821260	0.410630	+	0.911802i	$0.365309\pi$
$812$	0	0
$813$	0	0
$814$	−85.5100	−0.00368197
$815$	−15458.1	−0.664383
$816$	0	0
$817$	−36204.0	−1.55033
$818$	−2320.17	−0.0991720
$819$	0	0
$820$	−5594.16	−0.238240
$821$	14817.0	0.629860	0.314930	−	0.949115i	$-0.398019\pi$
$821$	14817.0	0.629860	0.314930	+	0.949115i	$0.398019\pi$
$822$	0	0
$823$	−26588.1	−1.12613	−0.563063	−	0.826414i	$-0.690377\pi$
$823$	−26588.1	−1.12613	−0.563063	+	0.826414i	$0.690377\pi$
$824$	−8501.54	−0.359424
$825$	0	0
$826$	0	0
$827$	11287.0	0.474592	0.237296	−	0.971437i	$-0.423739\pi$
$827$	11287.0	0.474592	0.237296	+	0.971437i	$0.423739\pi$
$828$	0	0
$829$	14448.6	0.605332	0.302666	−	0.953097i	$-0.402123\pi$
$829$	14448.6	0.605332	0.302666	+	0.953097i	$0.402123\pi$
$830$	1258.33	0.0526233
$831$	0	0
$832$	16786.0	0.699459
$833$	0	0
$834$	0	0
$835$	16255.9	0.673724
$836$	36719.2	1.51909
$837$	0	0
$838$	−5623.21	−0.231803
$839$	−40032.0	−1.64727	−0.823635	−	0.567120i	$-0.808057\pi$
$839$	−40032.0	−1.64727	−0.823635	+	0.567120i	$0.808057\pi$
$840$	0	0
$841$	−14130.4	−0.579376
$842$	−748.434	−0.0306327
$843$	0	0
$844$	−18143.7	−0.739967
$845$	−4337.53	−0.176586
$846$	0	0
$847$	0	0
$848$	29603.3	1.19880
$849$	0	0
$850$	−736.175	−0.0297066
$851$	−167.936	−0.00676473
$852$	0	0
$853$	10864.4	0.436098	0.218049	−	0.975938i	$-0.430031\pi$
$853$	10864.4	0.436098	0.218049	+	0.975938i	$0.430031\pi$
$854$	0	0
$855$	0	0
$856$	−10432.1	−0.416544
$857$	−42013.7	−1.67463	−0.837316	−	0.546719i	$-0.815877\pi$
$857$	−42013.7	−1.67463	−0.837316	+	0.546719i	$0.815877\pi$
$858$	0	0
$859$	16135.6	0.640908	0.320454	−	0.947264i	$-0.396165\pi$
$859$	16135.6	0.640908	0.320454	+	0.947264i	$0.396165\pi$
$860$	9339.22	0.370308
$861$	0	0
$862$	−3317.76	−0.131094
$863$	46246.8	1.82417	0.912086	−	0.409998i	$-0.134471\pi$
$863$	46246.8	1.82417	0.912086	+	0.409998i	$0.134471\pi$
$864$	0	0
$865$	−11272.7	−0.443103
$866$	−2524.85	−0.0990739
$867$	0	0
$868$	0	0
$869$	−8312.93	−0.324507
$870$	0	0
$871$	−2099.80	−0.0816865
$872$	−221.996	−0.00862127
$873$	0	0
$874$	−1250.81	−0.0484089
$875$	0	0
$876$	0	0
$877$	−25094.1	−0.966212	−0.483106	−	0.875562i	$-0.660491\pi$
$877$	−25094.1	−0.966212	−0.483106	+	0.875562i	$0.660491\pi$
$878$	2881.75	0.110768
$879$	0	0
$880$	−9304.97	−0.356444
$881$	27546.6	1.05342	0.526712	−	0.850044i	$-0.323424\pi$
$881$	27546.6	1.05342	0.526712	+	0.850044i	$0.323424\pi$
$882$	0	0
$883$	5825.31	0.222013	0.111006	−	0.993820i	$-0.464593\pi$
$883$	5825.31	0.222013	0.111006	+	0.993820i	$0.464593\pi$
$884$	−22861.7	−0.869823
$885$	0	0
$886$	−1914.67	−0.0726010
$887$	−6214.29	−0.235237	−0.117619	−	0.993059i	$-0.537526\pi$
$887$	−6214.29	−0.235237	−0.117619	+	0.993059i	$0.537526\pi$
$888$	0	0
$889$	0	0
$890$	296.685	0.0111740
$891$	0	0
$892$	−12809.8	−0.480833
$893$	−50470.1	−1.89128
$894$	0	0
$895$	13427.3	0.501481
$896$	0	0
$897$	0	0
$898$	285.462	0.0106080
$899$	25316.8	0.939223
$900$	0	0
$901$	−38857.3	−1.43677
$902$	−1609.80	−0.0594242
$903$	0	0
$904$	2679.28	0.0985748
$905$	6466.26	0.237509
$906$	0	0
$907$	24415.9	0.893845	0.446923	−	0.894573i	$-0.352520\pi$
$907$	24415.9	0.893845	0.446923	+	0.894573i	$0.352520\pi$
$908$	−18958.5	−0.692907
$909$	0	0
$910$	0	0
$911$	3493.35	0.127047	0.0635236	−	0.997980i	$-0.479766\pi$
$911$	3493.35	0.127047	0.0635236	+	0.997980i	$0.479766\pi$
$912$	0	0
$913$	−20876.7	−0.756755
$914$	4259.21	0.154138
$915$	0	0
$916$	−28832.6	−1.04002
$917$	0	0
$918$	0	0
$919$	15656.7	0.561989	0.280995	−	0.959709i	$-0.409336\pi$
$919$	15656.7	0.561989	0.280995	+	0.959709i	$0.409336\pi$
$920$	650.919	0.0233263
$921$	0	0
$922$	−3841.14	−0.137203
$923$	−25384.1	−0.905230
$924$	0	0
$925$	188.941	0.00671605
$926$	−5117.02	−0.181594
$927$	0	0
$928$	−7019.38	−0.248300
$929$	30504.5	1.07731	0.538654	−	0.842527i	$-0.318933\pi$
$929$	30504.5	1.07731	0.538654	+	0.842527i	$0.318933\pi$
$930$	0	0
$931$	0	0
$932$	−23735.0	−0.834189
$933$	0	0
$934$	5441.66	0.190639
$935$	12213.7	0.427199
$936$	0	0
$937$	50290.7	1.75339	0.876694	−	0.481049i	$-0.159744\pi$
$937$	50290.7	1.75339	0.876694	+	0.481049i	$0.159744\pi$
$938$	0	0
$939$	0	0
$940$	13019.3	0.451748
$941$	−18681.7	−0.647189	−0.323595	−	0.946196i	$-0.604891\pi$
$941$	−18681.7	−0.647189	−0.323595	+	0.946196i	$0.604891\pi$
$942$	0	0
$943$	−3161.56	−0.109178
$944$	−43575.8	−1.50241
$945$	0	0
$946$	2687.50	0.0923660
$947$	−16332.6	−0.560440	−0.280220	−	0.959936i	$-0.590407\pi$
$947$	−16332.6	−0.560440	−0.280220	+	0.959936i	$0.590407\pi$
$948$	0	0
$949$	−9517.91	−0.325568
$950$	1407.26	0.0480606
$951$	0	0
$952$	0	0
$953$	34233.6	1.16362	0.581812	−	0.813323i	$-0.302344\pi$
$953$	34233.6	1.16362	0.581812	+	0.813323i	$0.302344\pi$
$954$	0	0
$955$	699.388	0.0236981
$956$	−4300.40	−0.145486
$957$	0	0
$958$	543.345	0.0183243
$959$	0	0
$960$	0	0
$961$	32687.2	1.09722
$962$	−101.772	−0.00341086
$963$	0	0
$964$	−24654.5	−0.823723
$965$	11924.0	0.397770
$966$	0	0
$967$	53792.0	1.78887	0.894434	−	0.447200i	$-0.147579\pi$
$967$	53792.0	1.78887	0.894434	+	0.447200i	$0.147579\pi$
$968$	2299.11	0.0763392
$969$	0	0
$970$	309.931	0.0102590
$971$	−826.331	−0.0273102	−0.0136551	−	0.999907i	$-0.504347\pi$
$971$	−826.331	−0.0273102	−0.0136551	+	0.999907i	$0.504347\pi$
$972$	0	0
$973$	0	0
$974$	2071.85	0.0681586
$975$	0	0
$976$	−21539.8	−0.706425
$977$	−8703.52	−0.285005	−0.142503	−	0.989794i	$-0.545515\pi$
$977$	−8703.52	−0.285005	−0.142503	+	0.989794i	$0.545515\pi$
$978$	0	0
$979$	−4922.23	−0.160690
$980$	0	0
$981$	0	0
$982$	1179.58	0.0383320
$983$	−41000.9	−1.33034	−0.665170	−	0.746692i	$-0.731641\pi$
$983$	−41000.9	−1.33034	−0.665170	+	0.746692i	$0.731641\pi$
$984$	0	0
$985$	5043.37	0.163142
$986$	2982.53	0.0963317
$987$	0	0
$988$	43702.1	1.40724
$989$	5278.09	0.169700
$990$	0	0
$991$	−18341.0	−0.587912	−0.293956	−	0.955819i	$-0.594972\pi$
$991$	−18341.0	−0.587912	−0.293956	+	0.955819i	$0.594972\pi$
$992$	−17322.8	−0.554436
$993$	0	0
$994$	0	0
$995$	4975.18	0.158516
$996$	0	0
$997$	−57335.4	−1.82129	−0.910646	−	0.413187i	$-0.864416\pi$
$997$	−57335.4	−1.82129	−0.910646	+	0.413187i	$0.864416\pi$
$998$	3222.02	0.102196
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ca.1.4		6
3.2	odd	2		245.4.a.p.1.3	yes	6
7.6	odd	2		2205.4.a.bz.1.4		6
15.14	odd	2		1225.4.a.bi.1.4		6
21.2	odd	6		245.4.e.p.116.4		12
21.5	even	6		245.4.e.q.116.4		12
21.11	odd	6		245.4.e.p.226.4		12
21.17	even	6		245.4.e.q.226.4		12
21.20	even	2		245.4.a.o.1.3	✓	6
105.104	even	2		1225.4.a.bj.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.3	✓	6	21.20	even	2
245.4.a.p.1.3	yes	6	3.2	odd	2
245.4.e.p.116.4		12	21.2	odd	6
245.4.e.p.226.4		12	21.11	odd	6
245.4.e.q.116.4		12	21.5	even	6
245.4.e.q.226.4		12	21.17	even	6
1225.4.a.bi.1.4		6	15.14	odd	2
1225.4.a.bj.1.4		6	105.104	even	2
2205.4.a.bz.1.4		6	7.6	odd	2
2205.4.a.ca.1.4		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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+0.369315 q^{2} -7.86361 q^{4} +5.00000 q^{5} -5.85867 q^{8} +O(q^{10})\)	\(q+0.369315 q^{2} -7.86361 q^{4} +5.00000 q^{5} -5.85867 q^{8} +1.84657 q^{10} -30.6361 q^{11} -36.4622 q^{13} +60.7452 q^{16} -79.7341 q^{17} +152.418 q^{19} -39.3180 q^{20} -11.3144 q^{22} -22.2207 q^{23} +25.0000 q^{25} -13.4660 q^{26} -101.285 q^{29} -249.956 q^{31} +69.3034 q^{32} -29.4470 q^{34} +7.55765 q^{37} +56.2904 q^{38} -29.2933 q^{40} +142.280 q^{41} -237.530 q^{43} +240.910 q^{44} -8.20644 q^{46} -331.129 q^{47} +9.23287 q^{50} +286.725 q^{52} +487.337 q^{53} -153.180 q^{55} -37.4059 q^{58} -717.355 q^{59} -354.592 q^{61} -92.3126 q^{62} -460.367 q^{64} -182.311 q^{65} +57.5883 q^{67} +626.997 q^{68} +696.174 q^{71} +261.035 q^{73} +2.79115 q^{74} -1198.56 q^{76} +271.344 q^{79} +303.726 q^{80} +52.5460 q^{82} +681.441 q^{83} -398.670 q^{85} -87.7234 q^{86} +179.487 q^{88} +160.668 q^{89} +174.735 q^{92} -122.291 q^{94} +762.092 q^{95} +167.841 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8	\( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31} + 770 q^{32} + 812 q^{34} - 140 q^{37} - 20 q^{38} + 330 q^{40} - 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} - 628 q^{47} + 50 q^{50} + 1520 q^{52} + 676 q^{53} + 80 q^{55} - 2012 q^{58} - 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} + 2940 q^{68} + 224 q^{71} + 2640 q^{73} - 928 q^{74} - 1340 q^{76} + 1636 q^{79} + 1490 q^{80} - 1756 q^{82} - 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} + 1904 q^{89} + 1952 q^{92} - 3332 q^{94} + 1540 q^{95} + 516 q^{97}+O(q^{100}) \) 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Introduction

L-functions

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