LMFDB - Embedded newform 2205.4.a.z.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.53113$ 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)

$f(q)$	$=$	$q-4.53113 q^{2} +12.5311 q^{4} +5.00000 q^{5} -20.5311 q^{8} +O(q^{10})$	$q-4.53113 q^{2} +12.5311 q^{4} +5.00000 q^{5} -20.5311 q^{8} -22.6556 q^{10} +19.0623 q^{11} +2.93774 q^{13} -7.21984 q^{16} -6.49806 q^{17} +5.43580 q^{19} +62.6556 q^{20} -86.3735 q^{22} -49.3774 q^{23} +25.0000 q^{25} -13.3113 q^{26} +291.494 q^{29} -244.307 q^{31} +196.963 q^{32} +29.4436 q^{34} -193.121 q^{37} -24.6303 q^{38} -102.656 q^{40} +315.113 q^{41} -300.996 q^{43} +238.872 q^{44} +223.735 q^{46} +86.5058 q^{47} -113.278 q^{50} +36.8132 q^{52} -509.677 q^{53} +95.3113 q^{55} -1320.80 q^{58} -83.3852 q^{59} +5.25291 q^{61} +1106.99 q^{62} -834.706 q^{64} +14.6887 q^{65} +205.992 q^{67} -81.4281 q^{68} -1004.31 q^{71} +1007.29 q^{73} +875.055 q^{74} +68.1168 q^{76} -863.237 q^{79} -36.0992 q^{80} -1427.82 q^{82} +1334.72 q^{83} -32.4903 q^{85} +1363.85 q^{86} -391.370 q^{88} +326.249 q^{89} -618.755 q^{92} -391.969 q^{94} +27.1790 q^{95} -1526.77 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8}+O(q^{10}) $ 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8	$ 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8} - 5 q^{10} + 22 q^{11} + 22 q^{13} - 87 q^{16} + 116 q^{17} - 102 q^{19} + 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} + 54 q^{26} + 196 q^{29} - 150 q^{31} + 15 q^{32} + 462 q^{34} - 96 q^{37} - 404 q^{38} - 165 q^{40} - 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} - 25 q^{50} + 122 q^{52} - 326 q^{53} + 110 q^{55} - 1658 q^{58} - 844 q^{59} + 204 q^{61} + 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} - 412 q^{76} - 888 q^{79} - 435 q^{80} - 3162 q^{82} + 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} + 588 q^{89} - 1560 q^{92} + 1280 q^{94} - 510 q^{95} - 522 q^{97}+O(q^{100}) $ 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8 - 5 * q^10 + 22 * q^11 + 22 * q^13 - 87 * q^16 + 116 * q^17 - 102 * q^19 + 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 + 54 * q^26 + 196 * q^29 - 150 * q^31 + 15 * q^32 + 462 * q^34 - 96 * q^37 - 404 * q^38 - 165 * q^40 - 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 - 25 * q^50 + 122 * q^52 - 326 * q^53 + 110 * q^55 - 1658 * q^58 - 844 * q^59 + 204 * q^61 + 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 - 412 * q^76 - 888 * q^79 - 435 * q^80 - 3162 * q^82 + 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 + 588 * q^89 - 1560 * q^92 + 1280 * q^94 - 510 * q^95 - 522 * 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 $

$2$	−4.53113	−1.60200	−0.800998	−	0.598667i	$-0.795697\pi$
$2$	−4.53113	−1.60200	−0.800998	+	0.598667i	$0.795697\pi$
$3$	0	0
$3$	0	0
$4$	12.5311	1.56639
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−20.5311	−0.907356
$9$	0	0
$10$	−22.6556	−0.716434
$11$	19.0623	0.522499	0.261249	−	0.965271i	$-0.415865\pi$
$11$	19.0623	0.522499	0.261249	+	0.965271i	$0.415865\pi$
$12$	0	0
$13$	2.93774	0.0626756	0.0313378	−	0.999509i	$-0.490023\pi$
$13$	2.93774	0.0626756	0.0313378	+	0.999509i	$0.490023\pi$
$14$	0	0
$15$	0	0
$16$	−7.21984	−0.112810
$17$	−6.49806	−0.0927066	−0.0463533	−	0.998925i	$-0.514760\pi$
$17$	−6.49806	−0.0927066	−0.0463533	+	0.998925i	$0.514760\pi$
$18$	0	0
$19$	5.43580	0.0656347	0.0328173	−	0.999461i	$-0.489552\pi$
$19$	5.43580	0.0656347	0.0328173	+	0.999461i	$0.489552\pi$
$20$	62.6556	0.700511
$21$	0	0
$22$	−86.3735	−0.837041
$23$	−49.3774	−0.447648	−0.223824	−	0.974630i	$-0.571854\pi$
$23$	−49.3774	−0.447648	−0.223824	+	0.974630i	$0.571854\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−13.3113	−0.100406
$27$	0	0
$28$	0	0
$29$	291.494	1.86652	0.933261	−	0.359200i	$-0.116950\pi$
$29$	291.494	1.86652	0.933261	+	0.359200i	$0.116950\pi$
$30$	0	0
$31$	−244.307	−1.41545	−0.707724	−	0.706489i	$-0.750278\pi$
$31$	−244.307	−1.41545	−0.707724	+	0.706489i	$0.750278\pi$
$32$	196.963	1.08808
$33$	0	0
$34$	29.4436	0.148516
$35$	0	0
$36$	0	0
$37$	−193.121	−0.858077	−0.429038	−	0.903286i	$-0.641147\pi$
$37$	−193.121	−0.858077	−0.429038	+	0.903286i	$0.641147\pi$
$38$	−24.6303	−0.105147
$39$	0	0
$40$	−102.656	−0.405782
$41$	315.113	1.20030	0.600151	−	0.799887i	$-0.295107\pi$
$41$	315.113	1.20030	0.600151	+	0.799887i	$0.295107\pi$
$42$	0	0
$43$	−300.996	−1.06748	−0.533738	−	0.845650i	$-0.679213\pi$
$43$	−300.996	−1.06748	−0.533738	+	0.845650i	$0.679213\pi$
$44$	238.872	0.818437
$45$	0	0
$46$	223.735	0.717130
$47$	86.5058	0.268472	0.134236	−	0.990949i	$-0.457142\pi$
$47$	86.5058	0.268472	0.134236	+	0.990949i	$0.457142\pi$
$48$	0	0
$49$	0	0
$50$	−113.278	−0.320399
$51$	0	0
$52$	36.8132	0.0981745
$53$	−509.677	−1.32093	−0.660467	−	0.750855i	$-0.729642\pi$
$53$	−509.677	−1.32093	−0.660467	+	0.750855i	$0.729642\pi$
$54$	0	0
$55$	95.3113	0.233669
$56$	0	0
$57$	0	0
$58$	−1320.80	−2.99016
$59$	−83.3852	−0.183997	−0.0919985	−	0.995759i	$-0.529326\pi$
$59$	−83.3852	−0.183997	−0.0919985	+	0.995759i	$0.529326\pi$
$60$	0	0
$61$	5.25291	0.0110257	0.00551283	−	0.999985i	$-0.498245\pi$
$61$	5.25291	0.0110257	0.00551283	+	0.999985i	$0.498245\pi$
$62$	1106.99	2.26754
$63$	0	0
$64$	−834.706	−1.63029
$65$	14.6887	0.0280294
$66$	0	0
$67$	205.992	0.375611	0.187806	−	0.982206i	$-0.439862\pi$
$67$	205.992	0.375611	0.187806	+	0.982206i	$0.439862\pi$
$68$	−81.4281	−0.145215
$69$	0	0
$70$	0	0
$71$	−1004.31	−1.67872	−0.839362	−	0.543573i	$-0.817071\pi$
$71$	−1004.31	−1.67872	−0.839362	+	0.543573i	$0.817071\pi$
$72$	0	0
$73$	1007.29	1.61499	0.807494	−	0.589876i	$-0.200823\pi$
$73$	1007.29	1.61499	0.807494	+	0.589876i	$0.200823\pi$
$74$	875.055	1.37464
$75$	0	0
$76$	68.1168	0.102810
$77$	0	0
$78$	0	0
$79$	−863.237	−1.22939	−0.614695	−	0.788765i	$-0.710721\pi$
$79$	−863.237	−1.22939	−0.614695	+	0.788765i	$0.710721\pi$
$80$	−36.0992	−0.0504502
$81$	0	0
$82$	−1427.82	−1.92288
$83$	1334.72	1.76512	0.882560	−	0.470200i	$-0.155818\pi$
$83$	1334.72	1.76512	0.882560	+	0.470200i	$0.155818\pi$
$84$	0	0
$85$	−32.4903	−0.0414596
$86$	1363.85	1.71009
$87$	0	0
$88$	−391.370	−0.474093
$89$	326.249	0.388565	0.194283	−	0.980946i	$-0.437762\pi$
$89$	326.249	0.388565	0.194283	+	0.980946i	$0.437762\pi$
$90$	0	0
$91$	0	0
$92$	−618.755	−0.701192
$93$	0	0
$94$	−391.969	−0.430091
$95$	27.1790	0.0293527
$96$	0	0
$97$	−1526.77	−1.59815	−0.799075	−	0.601232i	$-0.794677\pi$
$97$	−1526.77	−1.59815	−0.799075	+	0.601232i	$0.794677\pi$
$98$	0	0
$99$	0	0
$100$	313.278	0.313278
$101$	96.8716	0.0954365	0.0477182	−	0.998861i	$-0.484805\pi$
$101$	96.8716	0.0954365	0.0477182	+	0.998861i	$0.484805\pi$
$102$	0	0
$103$	−1321.99	−1.26466	−0.632329	−	0.774700i	$-0.717901\pi$
$103$	−1321.99	−1.26466	−0.632329	+	0.774700i	$0.717901\pi$
$104$	−60.3152	−0.0568691
$105$	0	0
$106$	2309.41	2.11613
$107$	−1745.71	−1.57724	−0.788619	−	0.614883i	$-0.789203\pi$
$107$	−1745.71	−1.57724	−0.788619	+	0.614883i	$0.789203\pi$
$108$	0	0
$109$	476.856	0.419032	0.209516	−	0.977805i	$-0.432811\pi$
$109$	476.856	0.419032	0.209516	+	0.977805i	$0.432811\pi$
$110$	−431.868	−0.374336
$111$	0	0
$112$	0	0
$113$	−1641.65	−1.36666	−0.683332	−	0.730108i	$-0.739470\pi$
$113$	−1641.65	−1.36666	−0.683332	+	0.730108i	$0.739470\pi$
$114$	0	0
$115$	−246.887	−0.200194
$116$	3652.75	2.92370
$117$	0	0
$118$	377.829	0.294763
$119$	0	0
$120$	0	0
$121$	−967.630	−0.726995
$122$	−23.8016	−0.0176631
$123$	0	0
$124$	−3061.45	−2.21715
$125$	125.000	0.0894427
$126$	0	0
$127$	−844.016	−0.589719	−0.294859	−	0.955541i	$-0.595273\pi$
$127$	−844.016	−0.589719	−0.294859	+	0.955541i	$0.595273\pi$
$128$	2206.46	1.52363
$129$	0	0
$130$	−66.5564	−0.0449030
$131$	−2796.20	−1.86493	−0.932463	−	0.361265i	$-0.882345\pi$
$131$	−2796.20	−1.86493	−0.932463	+	0.361265i	$0.882345\pi$
$132$	0	0
$133$	0	0
$134$	−933.377	−0.601728
$135$	0	0
$136$	133.413	0.0841179
$137$	2057.13	1.28287	0.641433	−	0.767179i	$-0.278340\pi$
$137$	2057.13	1.28287	0.641433	+	0.767179i	$0.278340\pi$
$138$	0	0
$139$	1745.12	1.06489	0.532444	−	0.846465i	$-0.321274\pi$
$139$	1745.12	1.06489	0.532444	+	0.846465i	$0.321274\pi$
$140$	0	0
$141$	0	0
$142$	4550.65	2.68931
$143$	56.0000	0.0327479
$144$	0	0
$145$	1457.47	0.834734
$146$	−4564.15	−2.58720
$147$	0	0
$148$	−2420.02	−1.34408
$149$	1173.57	0.645254	0.322627	−	0.946526i	$-0.395434\pi$
$149$	1173.57	0.645254	0.322627	+	0.946526i	$0.395434\pi$
$150$	0	0
$151$	1540.07	0.829994	0.414997	−	0.909823i	$-0.363783\pi$
$151$	1540.07	0.829994	0.414997	+	0.909823i	$0.363783\pi$
$152$	−111.603	−0.0595540
$153$	0	0
$154$	0	0
$155$	−1221.54	−0.633008
$156$	0	0
$157$	2544.53	1.29348	0.646738	−	0.762712i	$-0.276133\pi$
$157$	2544.53	1.29348	0.646738	+	0.762712i	$0.276133\pi$
$158$	3911.44	1.96948
$159$	0	0
$160$	984.815	0.486603
$161$	0	0
$162$	0	0
$163$	−594.708	−0.285774	−0.142887	−	0.989739i	$-0.545639\pi$
$163$	−594.708	−0.285774	−0.142887	+	0.989739i	$0.545639\pi$
$164$	3948.72	1.88014
$165$	0	0
$166$	−6047.81	−2.82772
$167$	928.498	0.430236	0.215118	−	0.976588i	$-0.430986\pi$
$167$	928.498	0.430236	0.215118	+	0.976588i	$0.430986\pi$
$168$	0	0
$169$	−2188.37	−0.996072
$170$	147.218	0.0664182
$171$	0	0
$172$	−3771.82	−1.67209
$173$	−315.642	−0.138716	−0.0693578	−	0.997592i	$-0.522095\pi$
$173$	−315.642	−0.138716	−0.0693578	+	0.997592i	$0.522095\pi$
$174$	0	0
$175$	0	0
$176$	−137.626	−0.0589431
$177$	0	0
$178$	−1478.28	−0.622480
$179$	1445.49	0.603581	0.301791	−	0.953374i	$-0.402416\pi$
$179$	1445.49	0.603581	0.301791	+	0.953374i	$0.402416\pi$
$180$	0	0
$181$	1843.81	0.757180	0.378590	−	0.925564i	$-0.376409\pi$
$181$	1843.81	0.757180	0.378590	+	0.925564i	$0.376409\pi$
$182$	0	0
$183$	0	0
$184$	1013.77	0.406176
$185$	−965.603	−0.383744
$186$	0	0
$187$	−123.868	−0.0484391
$188$	1084.02	0.420532
$189$	0	0
$190$	−123.152	−0.0470229
$191$	244.074	0.0924637	0.0462318	−	0.998931i	$-0.485279\pi$
$191$	244.074	0.0924637	0.0462318	+	0.998931i	$0.485279\pi$
$192$	0	0
$193$	−1733.03	−0.646355	−0.323178	−	0.946338i	$-0.604751\pi$
$193$	−1733.03	−0.646355	−0.323178	+	0.946338i	$0.604751\pi$
$194$	6918.01	2.56023
$195$	0	0
$196$	0	0
$197$	−358.230	−0.129557	−0.0647787	−	0.997900i	$-0.520634\pi$
$197$	−358.230	−0.129557	−0.0647787	+	0.997900i	$0.520634\pi$
$198$	0	0
$199$	3203.63	1.14120	0.570601	−	0.821227i	$-0.306710\pi$
$199$	3203.63	1.14120	0.570601	+	0.821227i	$0.306710\pi$
$200$	−513.278	−0.181471
$201$	0	0
$202$	−438.938	−0.152889
$203$	0	0
$204$	0	0
$205$	1575.56	0.536791
$206$	5990.12	2.02598
$207$	0	0
$208$	−21.2100	−0.00707044
$209$	103.619	0.0342940
$210$	0	0
$211$	4943.16	1.61280	0.806401	−	0.591369i	$-0.201412\pi$
$211$	4943.16	1.61280	0.806401	+	0.591369i	$0.201412\pi$
$212$	−6386.83	−2.06910
$213$	0	0
$214$	7910.05	2.52673
$215$	−1504.98	−0.477390
$216$	0	0
$217$	0	0
$218$	−2160.70	−0.671288
$219$	0	0
$220$	1194.36	0.366016
$221$	−19.0896	−0.00581044
$222$	0	0
$223$	−3160.15	−0.948965	−0.474482	−	0.880265i	$-0.657365\pi$
$223$	−3160.15	−0.948965	−0.474482	+	0.880265i	$0.657365\pi$
$224$	0	0
$225$	0	0
$226$	7438.51	2.18939
$227$	−3651.11	−1.06755	−0.533773	−	0.845628i	$-0.679226\pi$
$227$	−3651.11	−1.06755	−0.533773	+	0.845628i	$0.679226\pi$
$228$	0	0
$229$	4083.70	1.17842	0.589210	−	0.807980i	$-0.299439\pi$
$229$	4083.70	1.17842	0.589210	+	0.807980i	$0.299439\pi$
$230$	1118.68	0.320710
$231$	0	0
$232$	−5984.70	−1.69360
$233$	3682.51	1.03540	0.517702	−	0.855561i	$-0.326788\pi$
$233$	3682.51	1.03540	0.517702	+	0.855561i	$0.326788\pi$
$234$	0	0
$235$	432.529	0.120064
$236$	−1044.91	−0.288211
$237$	0	0
$238$	0	0
$239$	−2658.78	−0.719591	−0.359796	−	0.933031i	$-0.617154\pi$
$239$	−2658.78	−0.719591	−0.359796	+	0.933031i	$0.617154\pi$
$240$	0	0
$241$	4820.39	1.28842	0.644209	−	0.764850i	$-0.277187\pi$
$241$	4820.39	1.28842	0.644209	+	0.764850i	$0.277187\pi$
$242$	4384.46	1.16464
$243$	0	0
$244$	65.8249	0.0172705
$245$	0	0
$246$	0	0
$247$	15.9690	0.00411369
$248$	5015.91	1.28432
$249$	0	0
$250$	−566.391	−0.143287
$251$	1672.27	0.420530	0.210265	−	0.977644i	$-0.432567\pi$
$251$	1672.27	0.420530	0.210265	+	0.977644i	$0.432567\pi$
$252$	0	0
$253$	−941.245	−0.233896
$254$	3824.34	0.944727
$255$	0	0
$256$	−3320.09	−0.810569
$257$	−3697.74	−0.897506	−0.448753	−	0.893656i	$-0.648132\pi$
$257$	−3697.74	−0.897506	−0.448753	+	0.893656i	$0.648132\pi$
$258$	0	0
$259$	0	0
$260$	184.066	0.0439050
$261$	0	0
$262$	12670.0	2.98760
$263$	−7319.00	−1.71600	−0.858002	−	0.513646i	$-0.828294\pi$
$263$	−7319.00	−1.71600	−0.858002	+	0.513646i	$0.828294\pi$
$264$	0	0
$265$	−2548.39	−0.590740
$266$	0	0
$267$	0	0
$268$	2581.32	0.588354
$269$	−815.097	−0.184749	−0.0923743	−	0.995724i	$-0.529446\pi$
$269$	−815.097	−0.184749	−0.0923743	+	0.995724i	$0.529446\pi$
$270$	0	0
$271$	5106.02	1.14453	0.572267	−	0.820068i	$-0.306064\pi$
$271$	5106.02	1.14453	0.572267	+	0.820068i	$0.306064\pi$
$272$	46.9150	0.0104582
$273$	0	0
$274$	−9321.13	−2.05515
$275$	476.556	0.104500
$276$	0	0
$277$	1398.72	0.303398	0.151699	−	0.988427i	$-0.451526\pi$
$277$	1398.72	0.303398	0.151699	+	0.988427i	$0.451526\pi$
$278$	−7907.39	−1.70595
$279$	0	0
$280$	0	0
$281$	7102.38	1.50780	0.753901	−	0.656988i	$-0.228170\pi$
$281$	7102.38	1.50780	0.753901	+	0.656988i	$0.228170\pi$
$282$	0	0
$283$	−4465.18	−0.937907	−0.468953	−	0.883223i	$-0.655369\pi$
$283$	−4465.18	−0.937907	−0.468953	+	0.883223i	$0.655369\pi$
$284$	−12585.1	−2.62954
$285$	0	0
$286$	−253.743	−0.0524621
$287$	0	0
$288$	0	0
$289$	−4870.78	−0.991405
$290$	−6603.99	−1.33724
$291$	0	0
$292$	12622.5	2.52970
$293$	7590.61	1.51348	0.756738	−	0.653718i	$-0.226792\pi$
$293$	7590.61	1.51348	0.756738	+	0.653718i	$0.226792\pi$
$294$	0	0
$295$	−416.926	−0.0822860
$296$	3964.98	0.778581
$297$	0	0
$298$	−5317.61	−1.03369
$299$	−145.058	−0.0280566
$300$	0	0
$301$	0	0
$302$	−6978.26	−1.32965
$303$	0	0
$304$	−39.2456	−0.00740425
$305$	26.2645	0.00493083
$306$	0	0
$307$	−9480.12	−1.76241	−0.881203	−	0.472737i	$-0.843266\pi$
$307$	−9480.12	−1.76241	−0.881203	+	0.472737i	$0.843266\pi$
$308$	0	0
$309$	0	0
$310$	5534.94	1.01408
$311$	7078.01	1.29054	0.645268	−	0.763956i	$-0.276746\pi$
$311$	7078.01	1.29054	0.645268	+	0.763956i	$0.276746\pi$
$312$	0	0
$313$	−5593.84	−1.01017	−0.505084	−	0.863070i	$-0.668539\pi$
$313$	−5593.84	−1.01017	−0.505084	+	0.863070i	$0.668539\pi$
$314$	−11529.6	−2.07214
$315$	0	0
$316$	−10817.3	−1.92571
$317$	−3567.81	−0.632139	−0.316070	−	0.948736i	$-0.602363\pi$
$317$	−3567.81	−0.632139	−0.316070	+	0.948736i	$0.602363\pi$
$318$	0	0
$319$	5556.54	0.975255
$320$	−4173.53	−0.729086
$321$	0	0
$322$	0	0
$323$	−35.3222	−0.00608477
$324$	0	0
$325$	73.4436	0.0125351
$326$	2694.70	0.457809
$327$	0	0
$328$	−6469.62	−1.08910
$329$	0	0
$330$	0	0
$331$	−4389.67	−0.728936	−0.364468	−	0.931216i	$-0.618749\pi$
$331$	−4389.67	−0.728936	−0.364468	+	0.931216i	$0.618749\pi$
$332$	16725.6	2.76487
$333$	0	0
$334$	−4207.14	−0.689236
$335$	1029.96	0.167978
$336$	0	0
$337$	2348.83	0.379671	0.189835	−	0.981816i	$-0.439205\pi$
$337$	2348.83	0.379671	0.189835	+	0.981816i	$0.439205\pi$
$338$	9915.79	1.59570
$339$	0	0
$340$	−407.140	−0.0649420
$341$	−4657.05	−0.739570
$342$	0	0
$343$	0	0
$344$	6179.79	0.968581
$345$	0	0
$346$	1430.21	0.222222
$347$	−558.436	−0.0863931	−0.0431965	−	0.999067i	$-0.513754\pi$
$347$	−558.436	−0.0863931	−0.0431965	+	0.999067i	$0.513754\pi$
$348$	0	0
$349$	−3233.89	−0.496006	−0.248003	−	0.968759i	$-0.579774\pi$
$349$	−3233.89	−0.496006	−0.248003	+	0.968759i	$0.579774\pi$
$350$	0	0
$351$	0	0
$352$	3754.56	0.568519
$353$	−7516.35	−1.13330	−0.566650	−	0.823959i	$-0.691761\pi$
$353$	−7516.35	−1.13330	−0.566650	+	0.823959i	$0.691761\pi$
$354$	0	0
$355$	−5021.54	−0.750748
$356$	4088.27	0.608646
$357$	0	0
$358$	−6549.70	−0.966935
$359$	6577.76	0.967021	0.483511	−	0.875338i	$-0.339361\pi$
$359$	6577.76	0.967021	0.483511	+	0.875338i	$0.339361\pi$
$360$	0	0
$361$	−6829.45	−0.995692
$362$	−8354.56	−1.21300
$363$	0	0
$364$	0	0
$365$	5036.44	0.722245
$366$	0	0
$367$	−8307.17	−1.18155	−0.590777	−	0.806835i	$-0.701179\pi$
$367$	−8307.17	−1.18155	−0.590777	+	0.806835i	$0.701179\pi$
$368$	356.497	0.0504992
$369$	0	0
$370$	4375.27	0.614756
$371$	0	0
$372$	0	0
$373$	−4551.09	−0.631760	−0.315880	−	0.948799i	$-0.602300\pi$
$373$	−4551.09	−0.631760	−0.315880	+	0.948799i	$0.602300\pi$
$374$	561.261	0.0775992
$375$	0	0
$376$	−1776.06	−0.243599
$377$	856.335	0.116985
$378$	0	0
$379$	−1788.29	−0.242370	−0.121185	−	0.992630i	$-0.538669\pi$
$379$	−1788.29	−0.242370	−0.121185	+	0.992630i	$0.538669\pi$
$380$	340.584	0.0459778
$381$	0	0
$382$	−1105.93	−0.148126
$383$	−1358.47	−0.181240	−0.0906199	−	0.995886i	$-0.528885\pi$
$383$	−1358.47	−0.181240	−0.0906199	+	0.995886i	$0.528885\pi$
$384$	0	0
$385$	0	0
$386$	7852.60	1.03546
$387$	0	0
$388$	−19132.2	−2.50333
$389$	−9722.54	−1.26723	−0.633615	−	0.773649i	$-0.718430\pi$
$389$	−9722.54	−1.26723	−0.633615	+	0.773649i	$0.718430\pi$
$390$	0	0
$391$	320.858	0.0414999
$392$	0	0
$393$	0	0
$394$	1623.19	0.207551
$395$	−4316.19	−0.549800
$396$	0	0
$397$	4788.04	0.605302	0.302651	−	0.953101i	$-0.402128\pi$
$397$	4788.04	0.605302	0.302651	+	0.953101i	$0.402128\pi$
$398$	−14516.1	−1.82820
$399$	0	0
$400$	−180.496	−0.0225620
$401$	−9681.41	−1.20565	−0.602826	−	0.797873i	$-0.705959\pi$
$401$	−9681.41	−1.20565	−0.602826	+	0.797873i	$0.705959\pi$
$402$	0	0
$403$	−717.712	−0.0887141
$404$	1213.91	0.149491
$405$	0	0
$406$	0	0
$407$	−3681.32	−0.448344
$408$	0	0
$409$	−11113.1	−1.34353	−0.671767	−	0.740763i	$-0.734464\pi$
$409$	−11113.1	−1.34353	−0.671767	+	0.740763i	$0.734464\pi$
$410$	−7139.09	−0.859937
$411$	0	0
$412$	−16566.1	−1.98095
$413$	0	0
$414$	0	0
$415$	6673.62	0.789386
$416$	578.627	0.0681959
$417$	0	0
$418$	−469.510	−0.0549389
$419$	−1230.09	−0.143421	−0.0717107	−	0.997425i	$-0.522846\pi$
$419$	−1230.09	−0.143421	−0.0717107	+	0.997425i	$0.522846\pi$
$420$	0	0
$421$	−12356.5	−1.43044	−0.715222	−	0.698897i	$-0.753674\pi$
$421$	−12356.5	−1.43044	−0.715222	+	0.698897i	$0.753674\pi$
$422$	−22398.1	−2.58370
$423$	0	0
$424$	10464.2	1.19856
$425$	−162.452	−0.0185413
$426$	0	0
$427$	0	0
$428$	−21875.7	−2.47057
$429$	0	0
$430$	6819.26	0.764777
$431$	7375.27	0.824256	0.412128	−	0.911126i	$-0.364786\pi$
$431$	7375.27	0.824256	0.412128	+	0.911126i	$0.364786\pi$
$432$	0	0
$433$	690.067	0.0765877	0.0382939	−	0.999267i	$-0.487808\pi$
$433$	690.067	0.0765877	0.0382939	+	0.999267i	$0.487808\pi$
$434$	0	0
$435$	0	0
$436$	5975.55	0.656369
$437$	−268.406	−0.0293812
$438$	0	0
$439$	−8408.79	−0.914191	−0.457095	−	0.889418i	$-0.651110\pi$
$439$	−8408.79	−0.914191	−0.457095	+	0.889418i	$0.651110\pi$
$440$	−1956.85	−0.212021
$441$	0	0
$442$	86.4976	0.00930830
$443$	−6568.55	−0.704473	−0.352236	−	0.935911i	$-0.614579\pi$
$443$	−6568.55	−0.704473	−0.352236	+	0.935911i	$0.614579\pi$
$444$	0	0
$445$	1631.25	0.173772
$446$	14319.0	1.52024
$447$	0	0
$448$	0	0
$449$	−2954.55	−0.310543	−0.155271	−	0.987872i	$-0.549625\pi$
$449$	−2954.55	−0.310543	−0.155271	+	0.987872i	$0.549625\pi$
$450$	0	0
$451$	6006.76	0.627156
$452$	−20571.7	−2.14073
$453$	0	0
$454$	16543.7	1.71020
$455$	0	0
$456$	0	0
$457$	−8144.84	−0.833697	−0.416849	−	0.908976i	$-0.636865\pi$
$457$	−8144.84	−0.833697	−0.416849	+	0.908976i	$0.636865\pi$
$458$	−18503.8	−1.88782
$459$	0	0
$460$	−3093.77	−0.313583
$461$	2495.26	0.252095	0.126048	−	0.992024i	$-0.459771\pi$
$461$	2495.26	0.252095	0.126048	+	0.992024i	$0.459771\pi$
$462$	0	0
$463$	−5755.66	−0.577728	−0.288864	−	0.957370i	$-0.593278\pi$
$463$	−5755.66	−0.577728	−0.288864	+	0.957370i	$0.593278\pi$
$464$	−2104.54	−0.210562
$465$	0	0
$466$	−16685.9	−1.65871
$467$	−4143.73	−0.410598	−0.205299	−	0.978699i	$-0.565817\pi$
$467$	−4143.73	−0.410598	−0.205299	+	0.978699i	$0.565817\pi$
$468$	0	0
$469$	0	0
$470$	−1959.84	−0.192342
$471$	0	0
$472$	1711.99	0.166951
$473$	−5737.67	−0.557755
$474$	0	0
$475$	135.895	0.0131269
$476$	0	0
$477$	0	0
$478$	12047.3	1.15278
$479$	−6765.96	−0.645396	−0.322698	−	0.946502i	$-0.604590\pi$
$479$	−6765.96	−0.645396	−0.322698	+	0.946502i	$0.604590\pi$
$480$	0	0
$481$	−567.339	−0.0537805
$482$	−21841.8	−2.06404
$483$	0	0
$484$	−12125.5	−1.13876
$485$	−7633.87	−0.714714
$486$	0	0
$487$	6360.42	0.591824	0.295912	−	0.955215i	$-0.404377\pi$
$487$	6360.42	0.591824	0.295912	+	0.955215i	$0.404377\pi$
$488$	−107.848	−0.0100042
$489$	0	0
$490$	0	0
$491$	7072.54	0.650060	0.325030	−	0.945704i	$-0.394626\pi$
$491$	7072.54	0.650060	0.325030	+	0.945704i	$0.394626\pi$
$492$	0	0
$493$	−1894.15	−0.173039
$494$	−72.3576	−0.00659012
$495$	0	0
$496$	1763.86	0.159677
$497$	0	0
$498$	0	0
$499$	−18473.9	−1.65732	−0.828661	−	0.559751i	$-0.810897\pi$
$499$	−18473.9	−1.65732	−0.828661	+	0.559751i	$0.810897\pi$
$500$	1566.39	0.140102
$501$	0	0
$502$	−7577.28	−0.673687
$503$	11379.2	1.00869	0.504347	−	0.863501i	$-0.331733\pi$
$503$	11379.2	1.00869	0.504347	+	0.863501i	$0.331733\pi$
$504$	0	0
$505$	484.358	0.0426805
$506$	4264.90	0.374700
$507$	0	0
$508$	−10576.5	−0.923730
$509$	6064.48	0.528101	0.264051	−	0.964509i	$-0.414941\pi$
$509$	6064.48	0.528101	0.264051	+	0.964509i	$0.414941\pi$
$510$	0	0
$511$	0	0
$512$	−2607.89	−0.225105
$513$	0	0
$514$	16755.0	1.43780
$515$	−6609.96	−0.565572
$516$	0	0
$517$	1649.00	0.140276
$518$	0	0
$519$	0	0
$520$	−301.576	−0.0254326
$521$	2682.88	0.225603	0.112801	−	0.993618i	$-0.464018\pi$
$521$	2682.88	0.225603	0.112801	+	0.993618i	$0.464018\pi$
$522$	0	0
$523$	4309.02	0.360268	0.180134	−	0.983642i	$-0.442347\pi$
$523$	4309.02	0.360268	0.180134	+	0.983642i	$0.442347\pi$
$524$	−35039.6	−2.92120
$525$	0	0
$526$	33163.3	2.74903
$527$	1587.52	0.131221
$528$	0	0
$529$	−9728.87	−0.799611
$530$	11547.1	0.946363
$531$	0	0
$532$	0	0
$533$	925.720	0.0752296
$534$	0	0
$535$	−8728.56	−0.705362
$536$	−4229.25	−0.340813
$537$	0	0
$538$	3693.31	0.295966
$539$	0	0
$540$	0	0
$541$	4081.47	0.324355	0.162178	−	0.986762i	$-0.448148\pi$
$541$	4081.47	0.324355	0.162178	+	0.986762i	$0.448148\pi$
$542$	−23136.0	−1.83354
$543$	0	0
$544$	−1279.88	−0.100872
$545$	2384.28	0.187397
$546$	0	0
$547$	8844.82	0.691366	0.345683	−	0.938351i	$-0.387647\pi$
$547$	8844.82	0.691366	0.345683	+	0.938351i	$0.387647\pi$
$548$	25778.2	2.00947
$549$	0	0
$550$	−2159.34	−0.167408
$551$	1584.51	0.122509
$552$	0	0
$553$	0	0
$554$	−6337.80	−0.486042
$555$	0	0
$556$	21868.4	1.66803
$557$	11144.7	0.847787	0.423894	−	0.905712i	$-0.360663\pi$
$557$	11144.7	0.847787	0.423894	+	0.905712i	$0.360663\pi$
$558$	0	0
$559$	−884.249	−0.0669047
$560$	0	0
$561$	0	0
$562$	−32181.8	−2.41549
$563$	−21857.5	−1.63621	−0.818104	−	0.575071i	$-0.804975\pi$
$563$	−21857.5	−1.63621	−0.818104	+	0.575071i	$0.804975\pi$
$564$	0	0
$565$	−8208.23	−0.611191
$566$	20232.3	1.50252
$567$	0	0
$568$	20619.6	1.52320
$569$	−23496.4	−1.73115	−0.865573	−	0.500783i	$-0.833046\pi$
$569$	−23496.4	−1.73115	−0.865573	+	0.500783i	$0.833046\pi$
$570$	0	0
$571$	11067.8	0.811164	0.405582	−	0.914059i	$-0.367069\pi$
$571$	11067.8	0.811164	0.405582	+	0.914059i	$0.367069\pi$
$572$	701.743	0.0512961
$573$	0	0
$574$	0	0
$575$	−1234.44	−0.0895296
$576$	0	0
$577$	−20482.9	−1.47784	−0.738922	−	0.673791i	$-0.764665\pi$
$577$	−20482.9	−1.47784	−0.738922	+	0.673791i	$0.764665\pi$
$578$	22070.1	1.58823
$579$	0	0
$580$	18263.8	1.30752
$581$	0	0
$582$	0	0
$583$	−9715.60	−0.690187
$584$	−20680.8	−1.46537
$585$	0	0
$586$	−34394.1	−2.42458
$587$	−23444.3	−1.64847	−0.824235	−	0.566248i	$-0.808394\pi$
$587$	−23444.3	−1.64847	−0.824235	+	0.566248i	$0.808394\pi$
$588$	0	0
$589$	−1328.01	−0.0929025
$590$	1889.14	0.131822
$591$	0	0
$592$	1394.30	0.0967996
$593$	4404.69	0.305024	0.152512	−	0.988302i	$-0.451264\pi$
$593$	4404.69	0.305024	0.152512	+	0.988302i	$0.451264\pi$
$594$	0	0
$595$	0	0
$596$	14706.2	1.01072
$597$	0	0
$598$	657.277	0.0449466
$599$	−3327.05	−0.226945	−0.113472	−	0.993541i	$-0.536197\pi$
$599$	−3327.05	−0.226945	−0.113472	+	0.993541i	$0.536197\pi$
$600$	0	0
$601$	14244.8	0.966818	0.483409	−	0.875395i	$-0.339398\pi$
$601$	14244.8	0.966818	0.483409	+	0.875395i	$0.339398\pi$
$602$	0	0
$603$	0	0
$604$	19298.8	1.30010
$605$	−4838.15	−0.325122
$606$	0	0
$607$	−11446.5	−0.765402	−0.382701	−	0.923872i	$-0.625006\pi$
$607$	−11446.5	−0.765402	−0.382701	+	0.923872i	$0.625006\pi$
$608$	1070.65	0.0714156
$609$	0	0
$610$	−119.008	−0.00789917
$611$	254.132	0.0168266
$612$	0	0
$613$	−19436.4	−1.28063	−0.640316	−	0.768111i	$-0.721197\pi$
$613$	−19436.4	−1.28063	−0.640316	+	0.768111i	$0.721197\pi$
$614$	42955.6	2.82337
$615$	0	0
$616$	0	0
$617$	20530.1	1.33956	0.669781	−	0.742558i	$-0.266388\pi$
$617$	20530.1	1.33956	0.669781	+	0.742558i	$0.266388\pi$
$618$	0	0
$619$	−5833.35	−0.378776	−0.189388	−	0.981902i	$-0.560650\pi$
$619$	−5833.35	−0.378776	−0.189388	+	0.981902i	$0.560650\pi$
$620$	−15307.2	−0.991538
$621$	0	0
$622$	−32071.4	−2.06744
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	25346.4	1.61829
$627$	0	0
$628$	31885.9	2.02609
$629$	1254.91	0.0795493
$630$	0	0
$631$	24776.6	1.56314	0.781568	−	0.623820i	$-0.214420\pi$
$631$	24776.6	1.56314	0.781568	+	0.623820i	$0.214420\pi$
$632$	17723.2	1.11549
$633$	0	0
$634$	16166.2	1.01268
$635$	−4220.08	−0.263730
$636$	0	0
$637$	0	0
$638$	−25177.4	−1.56235
$639$	0	0
$640$	11032.3	0.681390
$641$	−27219.4	−1.67723	−0.838613	−	0.544728i	$-0.816633\pi$
$641$	−27219.4	−1.67723	−0.838613	+	0.544728i	$0.816633\pi$
$642$	0	0
$643$	−7091.79	−0.434950	−0.217475	−	0.976066i	$-0.569782\pi$
$643$	−7091.79	−0.434950	−0.217475	+	0.976066i	$0.569782\pi$
$644$	0	0
$645$	0	0
$646$	160.049	0.00974777
$647$	27773.0	1.68758	0.843792	−	0.536670i	$-0.180318\pi$
$647$	27773.0	1.68758	0.843792	+	0.536670i	$0.180318\pi$
$648$	0	0
$649$	−1589.51	−0.0961382
$650$	−332.782	−0.0200812
$651$	0	0
$652$	−7452.37	−0.447634
$653$	−21380.4	−1.28129	−0.640643	−	0.767839i	$-0.721332\pi$
$653$	−21380.4	−1.28129	−0.640643	+	0.767839i	$0.721332\pi$
$654$	0	0
$655$	−13981.0	−0.834020
$656$	−2275.06	−0.135406
$657$	0	0
$658$	0	0
$659$	17232.3	1.01863	0.509315	−	0.860580i	$-0.329899\pi$
$659$	17232.3	1.01863	0.509315	+	0.860580i	$0.329899\pi$
$660$	0	0
$661$	−26577.7	−1.56392	−0.781962	−	0.623326i	$-0.785781\pi$
$661$	−26577.7	−1.56392	−0.781962	+	0.623326i	$0.785781\pi$
$662$	19890.1	1.16775
$663$	0	0
$664$	−27403.4	−1.60159
$665$	0	0
$666$	0	0
$667$	−14393.2	−0.835544
$668$	11635.1	0.673917
$669$	0	0
$670$	−4666.89	−0.269101
$671$	100.132	0.00576090
$672$	0	0
$673$	−31695.2	−1.81540	−0.907698	−	0.419624i	$-0.862162\pi$
$673$	−31695.2	−1.81540	−0.907698	+	0.419624i	$0.862162\pi$
$674$	−10642.9	−0.608231
$675$	0	0
$676$	−27422.7	−1.56024
$677$	20440.3	1.16039	0.580195	−	0.814477i	$-0.302976\pi$
$677$	20440.3	1.16039	0.580195	+	0.814477i	$0.302976\pi$
$678$	0	0
$679$	0	0
$680$	667.063	0.0376187
$681$	0	0
$682$	21101.7	1.18479
$683$	22896.9	1.28276	0.641381	−	0.767223i	$-0.278362\pi$
$683$	22896.9	1.28276	0.641381	+	0.767223i	$0.278362\pi$
$684$	0	0
$685$	10285.7	0.573715
$686$	0	0
$687$	0	0
$688$	2173.14	0.120422
$689$	−1497.30	−0.0827904
$690$	0	0
$691$	23764.0	1.30829	0.654143	−	0.756371i	$-0.273029\pi$
$691$	23764.0	1.30829	0.654143	+	0.756371i	$0.273029\pi$
$692$	−3955.35	−0.217283
$693$	0	0
$694$	2530.34	0.138401
$695$	8725.62	0.476233
$696$	0	0
$697$	−2047.62	−0.111276
$698$	14653.2	0.794600
$699$	0	0
$700$	0	0
$701$	−26259.5	−1.41485	−0.707423	−	0.706791i	$-0.750142\pi$
$701$	−26259.5	−1.41485	−0.707423	+	0.706791i	$0.750142\pi$
$702$	0	0
$703$	−1049.77	−0.0563196
$704$	−15911.4	−0.851822
$705$	0	0
$706$	34057.6	1.81554
$707$	0	0
$708$	0	0
$709$	12783.0	0.677116	0.338558	−	0.940945i	$-0.390061\pi$
$709$	12783.0	0.677116	0.338558	+	0.940945i	$0.390061\pi$
$710$	22753.2	1.20270
$711$	0	0
$712$	−6698.26	−0.352567
$713$	12063.3	0.633623
$714$	0	0
$715$	280.000	0.0146453
$716$	18113.6	0.945444
$717$	0	0
$718$	−29804.7	−1.54916
$719$	−27609.0	−1.43205	−0.716025	−	0.698075i	$-0.754040\pi$
$719$	−27609.0	−1.43205	−0.716025	+	0.698075i	$0.754040\pi$
$720$	0	0
$721$	0	0
$722$	30945.1	1.59509
$723$	0	0
$724$	23105.1	1.18604
$725$	7287.35	0.373304
$726$	0	0
$727$	−31306.2	−1.59709	−0.798544	−	0.601937i	$-0.794396\pi$
$727$	−31306.2	−1.59709	−0.798544	+	0.601937i	$0.794396\pi$
$728$	0	0
$729$	0	0
$730$	−22820.8	−1.15703
$731$	1955.89	0.0989621
$732$	0	0
$733$	15765.8	0.794441	0.397220	−	0.917723i	$-0.369975\pi$
$733$	15765.8	0.794441	0.397220	+	0.917723i	$0.369975\pi$
$734$	37640.8	1.89285
$735$	0	0
$736$	−9725.53	−0.487076
$737$	3926.68	0.196256
$738$	0	0
$739$	3966.51	0.197443	0.0987216	−	0.995115i	$-0.468525\pi$
$739$	3966.51	0.197443	0.0987216	+	0.995115i	$0.468525\pi$
$740$	−12100.1	−0.601093
$741$	0	0
$742$	0	0
$743$	−8224.50	−0.406094	−0.203047	−	0.979169i	$-0.565084\pi$
$743$	−8224.50	−0.406094	−0.203047	+	0.979169i	$0.565084\pi$
$744$	0	0
$745$	5867.86	0.288566
$746$	20621.6	1.01208
$747$	0	0
$748$	−1552.20	−0.0758745
$749$	0	0
$750$	0	0
$751$	18929.2	0.919754	0.459877	−	0.887983i	$-0.347894\pi$
$751$	18929.2	0.919754	0.459877	+	0.887983i	$0.347894\pi$
$752$	−624.558	−0.0302863
$753$	0	0
$754$	−3880.16	−0.187410
$755$	7700.35	0.371185
$756$	0	0
$757$	−34906.8	−1.67597	−0.837984	−	0.545695i	$-0.816266\pi$
$757$	−34906.8	−1.67597	−0.837984	+	0.545695i	$0.816266\pi$
$758$	8102.96	0.388276
$759$	0	0
$760$	−558.016	−0.0266334
$761$	−13683.4	−0.651803	−0.325902	−	0.945404i	$-0.605668\pi$
$761$	−13683.4	−0.651803	−0.325902	+	0.945404i	$0.605668\pi$
$762$	0	0
$763$	0	0
$764$	3058.52	0.144834
$765$	0	0
$766$	6155.42	0.290345
$767$	−244.964	−0.0115321
$768$	0	0
$769$	−41837.3	−1.96189	−0.980943	−	0.194294i	$-0.937758\pi$
$769$	−41837.3	−1.96189	−0.980943	+	0.194294i	$0.937758\pi$
$770$	0	0
$771$	0	0
$772$	−21716.9	−1.01245
$773$	19640.0	0.913843	0.456921	−	0.889507i	$-0.348952\pi$
$773$	19640.0	0.913843	0.456921	+	0.889507i	$0.348952\pi$
$774$	0	0
$775$	−6107.69	−0.283090
$776$	31346.4	1.45009
$777$	0	0
$778$	44054.1	2.03010
$779$	1712.89	0.0787814
$780$	0	0
$781$	−19144.4	−0.877131
$782$	−1453.85	−0.0664827
$783$	0	0
$784$	0	0
$785$	12722.7	0.578460
$786$	0	0
$787$	−24935.3	−1.12941	−0.564705	−	0.825293i	$-0.691010\pi$
$787$	−24935.3	−1.12941	−0.564705	+	0.825293i	$0.691010\pi$
$788$	−4489.02	−0.202938
$789$	0	0
$790$	19557.2	0.880777
$791$	0	0
$792$	0	0
$793$	15.4317	0.000691041 0
$794$	−21695.2	−0.969692
$795$	0	0
$796$	40145.1	1.78757
$797$	−1168.33	−0.0519251	−0.0259625	−	0.999663i	$-0.508265\pi$
$797$	−1168.33	−0.0519251	−0.0259625	+	0.999663i	$0.508265\pi$
$798$	0	0
$799$	−562.120	−0.0248891
$800$	4924.08	0.217615
$801$	0	0
$802$	43867.7	1.93145
$803$	19201.2	0.843829
$804$	0	0
$805$	0	0
$806$	3252.05	0.142120
$807$	0	0
$808$	−1988.88	−0.0865949
$809$	35175.7	1.52869	0.764345	−	0.644807i	$-0.223062\pi$
$809$	35175.7	1.52869	0.764345	+	0.644807i	$0.223062\pi$
$810$	0	0
$811$	15256.5	0.660577	0.330288	−	0.943880i	$-0.392854\pi$
$811$	15256.5	0.660577	0.330288	+	0.943880i	$0.392854\pi$
$812$	0	0
$813$	0	0
$814$	16680.5	0.718245
$815$	−2973.54	−0.127802
$816$	0	0
$817$	−1636.16	−0.0700635
$818$	50354.7	2.15233
$819$	0	0
$820$	19743.6	0.840825
$821$	15971.9	0.678956	0.339478	−	0.940614i	$-0.389750\pi$
$821$	15971.9	0.678956	0.339478	+	0.940614i	$0.389750\pi$
$822$	0	0
$823$	2312.41	0.0979409	0.0489705	−	0.998800i	$-0.484406\pi$
$823$	2312.41	0.0979409	0.0489705	+	0.998800i	$0.484406\pi$
$824$	27142.0	1.14750
$825$	0	0
$826$	0	0
$827$	−10422.4	−0.438238	−0.219119	−	0.975698i	$-0.570318\pi$
$827$	−10422.4	−0.438238	−0.219119	+	0.975698i	$0.570318\pi$
$828$	0	0
$829$	13213.4	0.553584	0.276792	−	0.960930i	$-0.410729\pi$
$829$	13213.4	0.553584	0.276792	+	0.960930i	$0.410729\pi$
$830$	−30239.0	−1.26459
$831$	0	0
$832$	−2452.15	−0.102179
$833$	0	0
$834$	0	0
$835$	4642.49	0.192407
$836$	1298.46	0.0537179
$837$	0	0
$838$	5573.67	0.229761
$839$	−10119.6	−0.416409	−0.208205	−	0.978085i	$-0.566762\pi$
$839$	−10119.6	−0.416409	−0.208205	+	0.978085i	$0.566762\pi$
$840$	0	0
$841$	60579.9	2.48390
$842$	55988.7	2.29157
$843$	0	0
$844$	61943.4	2.52628
$845$	−10941.8	−0.445457
$846$	0	0
$847$	0	0
$848$	3679.79	0.149015
$849$	0	0
$850$	736.089	0.0297031
$851$	9535.80	0.384116
$852$	0	0
$853$	−35378.1	−1.42007	−0.710037	−	0.704165i	$-0.751322\pi$
$853$	−35378.1	−1.42007	−0.710037	+	0.704165i	$0.751322\pi$
$854$	0	0
$855$	0	0
$856$	35841.4	1.43112
$857$	6697.57	0.266960	0.133480	−	0.991052i	$-0.457385\pi$
$857$	6697.57	0.266960	0.133480	+	0.991052i	$0.457385\pi$
$858$	0	0
$859$	24298.4	0.965135	0.482568	−	0.875859i	$-0.339704\pi$
$859$	24298.4	0.965135	0.482568	+	0.875859i	$0.339704\pi$
$860$	−18859.1	−0.747779
$861$	0	0
$862$	−33418.3	−1.32045
$863$	−24942.9	−0.983853	−0.491926	−	0.870637i	$-0.663707\pi$
$863$	−24942.9	−0.983853	−0.491926	+	0.870637i	$0.663707\pi$
$864$	0	0
$865$	−1578.21	−0.0620355
$866$	−3126.78	−0.122693
$867$	0	0
$868$	0	0
$869$	−16455.3	−0.642355
$870$	0	0
$871$	605.152	0.0235417
$872$	−9790.39	−0.380212
$873$	0	0
$874$	1216.18	0.0470686
$875$	0	0
$876$	0	0
$877$	−16276.6	−0.626705	−0.313353	−	0.949637i	$-0.601452\pi$
$877$	−16276.6	−0.626705	−0.313353	+	0.949637i	$0.601452\pi$
$878$	38101.3	1.46453
$879$	0	0
$880$	−688.132	−0.0263602
$881$	26636.5	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$881$	26636.5	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$882$	0	0
$883$	−21788.3	−0.830392	−0.415196	−	0.909732i	$-0.636287\pi$
$883$	−21788.3	−0.830392	−0.415196	+	0.909732i	$0.636287\pi$
$884$	−239.215	−0.00910142
$885$	0	0
$886$	29763.0	1.12856
$887$	−26813.2	−1.01499	−0.507496	−	0.861654i	$-0.669429\pi$
$887$	−26813.2	−1.01499	−0.507496	+	0.861654i	$0.669429\pi$
$888$	0	0
$889$	0	0
$890$	−7391.38	−0.278382
$891$	0	0
$892$	−39600.2	−1.48645
$893$	470.229	0.0176211
$894$	0	0
$895$	7227.45	0.269930
$896$	0	0
$897$	0	0
$898$	13387.4	0.497488
$899$	−71214.2	−2.64196
$900$	0	0
$901$	3311.91	0.122459
$902$	−27217.4	−1.00470
$903$	0	0
$904$	33704.8	1.24005
$905$	9219.07	0.338621
$906$	0	0
$907$	15543.0	0.569014	0.284507	−	0.958674i	$-0.408170\pi$
$907$	15543.0	0.569014	0.284507	+	0.958674i	$0.408170\pi$
$908$	−45752.6	−1.67219
$909$	0	0
$910$	0	0
$911$	−48711.1	−1.77154	−0.885768	−	0.464128i	$-0.846368\pi$
$911$	−48711.1	−1.77154	−0.885768	+	0.464128i	$0.846368\pi$
$912$	0	0
$913$	25442.8	0.922273
$914$	36905.3	1.33558
$915$	0	0
$916$	51173.3	1.84587
$917$	0	0
$918$	0	0
$919$	1030.47	0.0369883	0.0184941	−	0.999829i	$-0.494113\pi$
$919$	1030.47	0.0369883	0.0184941	+	0.999829i	$0.494113\pi$
$920$	5068.87	0.181648
$921$	0	0
$922$	−11306.3	−0.403855
$923$	−2950.40	−0.105215
$924$	0	0
$925$	−4828.02	−0.171615
$926$	26079.6	0.925518
$927$	0	0
$928$	57413.6	2.03092
$929$	879.756	0.0310698	0.0155349	−	0.999879i	$-0.495055\pi$
$929$	879.756	0.0310698	0.0155349	+	0.999879i	$0.495055\pi$
$930$	0	0
$931$	0	0
$932$	46146.0	1.62185
$933$	0	0
$934$	18775.8	0.657776
$935$	−619.339	−0.0216626
$936$	0	0
$937$	18668.1	0.650864	0.325432	−	0.945565i	$-0.394490\pi$
$937$	18668.1	0.650864	0.325432	+	0.945565i	$0.394490\pi$
$938$	0	0
$939$	0	0
$940$	5420.08	0.188067
$941$	29613.4	1.02590	0.512948	−	0.858420i	$-0.328553\pi$
$941$	29613.4	1.02590	0.512948	+	0.858420i	$0.328553\pi$
$942$	0	0
$943$	−15559.5	−0.537313
$944$	602.028	0.0207567
$945$	0	0
$946$	25998.1	0.893521
$947$	−20738.9	−0.711640	−0.355820	−	0.934554i	$-0.615798\pi$
$947$	−20738.9	−0.711640	−0.355820	+	0.934554i	$0.615798\pi$
$948$	0	0
$949$	2959.15	0.101220
$950$	−615.758	−0.0210293
$951$	0	0
$952$	0	0
$953$	45776.5	1.55598	0.777988	−	0.628279i	$-0.216240\pi$
$953$	45776.5	1.55598	0.777988	+	0.628279i	$0.216240\pi$
$954$	0	0
$955$	1220.37	0.0413510
$956$	−33317.5	−1.12716
$957$	0	0
$958$	30657.4	1.03392
$959$	0	0
$960$	0	0
$961$	29895.1	1.00349
$962$	2570.68	0.0861561
$963$	0	0
$964$	60404.9	2.01817
$965$	−8665.17	−0.289059
$966$	0	0
$967$	34461.0	1.14601	0.573005	−	0.819552i	$-0.305778\pi$
$967$	34461.0	1.14601	0.573005	+	0.819552i	$0.305778\pi$
$968$	19866.5	0.659643
$969$	0	0
$970$	34590.1	1.14497
$971$	−22762.8	−0.752309	−0.376154	−	0.926557i	$-0.622754\pi$
$971$	−22762.8	−0.752309	−0.376154	+	0.926557i	$0.622754\pi$
$972$	0	0
$973$	0	0
$974$	−28819.9	−0.948099
$975$	0	0
$976$	−37.9251	−0.00124381
$977$	−4809.57	−0.157494	−0.0787470	−	0.996895i	$-0.525092\pi$
$977$	−4809.57	−0.157494	−0.0787470	+	0.996895i	$0.525092\pi$
$978$	0	0
$979$	6219.04	0.203025
$980$	0	0
$981$	0	0
$982$	−32046.6	−1.04139
$983$	−27591.6	−0.895256	−0.447628	−	0.894220i	$-0.647731\pi$
$983$	−27591.6	−0.895256	−0.447628	+	0.894220i	$0.647731\pi$
$984$	0	0
$985$	−1791.15	−0.0579399
$986$	8582.63	0.277207
$987$	0	0
$988$	200.109	0.00644365
$989$	14862.4	0.477854
$990$	0	0
$991$	−22263.4	−0.713643	−0.356822	−	0.934173i	$-0.616140\pi$
$991$	−22263.4	−0.713643	−0.356822	+	0.934173i	$0.616140\pi$
$992$	−48119.5	−1.54012
$993$	0	0
$994$	0	0
$995$	16018.2	0.510361
$996$	0	0
$997$	−30378.2	−0.964983	−0.482491	−	0.875901i	$-0.660268\pi$
$997$	−30378.2	−0.964983	−0.482491	+	0.875901i	$0.660268\pi$
$998$	83707.4	2.65502
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.z.1.1		2
3.2	odd	2		735.4.a.p.1.2		2
7.6	odd	2		315.4.a.i.1.1		2
21.20	even	2		105.4.a.f.1.2	✓	2
35.34	odd	2		1575.4.a.w.1.2		2
84.83	odd	2		1680.4.a.bg.1.2		2
105.62	odd	4		525.4.d.h.274.4		4
105.83	odd	4		525.4.d.h.274.1		4
105.104	even	2		525.4.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.2	✓	2	21.20	even	2
315.4.a.i.1.1		2	7.6	odd	2
525.4.a.k.1.1		2	105.104	even	2
525.4.d.h.274.1		4	105.83	odd	4
525.4.d.h.274.4		4	105.62	odd	4
735.4.a.p.1.2		2	3.2	odd	2
1575.4.a.w.1.2		2	35.34	odd	2
1680.4.a.bg.1.2		2	84.83	odd	2
2205.4.a.z.1.1		2	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)

\(f(q)\)	\(=\)	\(q-4.53113 q^{2} +12.5311 q^{4} +5.00000 q^{5} -20.5311 q^{8} +O(q^{10})\)	\(q-4.53113 q^{2} +12.5311 q^{4} +5.00000 q^{5} -20.5311 q^{8} -22.6556 q^{10} +19.0623 q^{11} +2.93774 q^{13} -7.21984 q^{16} -6.49806 q^{17} +5.43580 q^{19} +62.6556 q^{20} -86.3735 q^{22} -49.3774 q^{23} +25.0000 q^{25} -13.3113 q^{26} +291.494 q^{29} -244.307 q^{31} +196.963 q^{32} +29.4436 q^{34} -193.121 q^{37} -24.6303 q^{38} -102.656 q^{40} +315.113 q^{41} -300.996 q^{43} +238.872 q^{44} +223.735 q^{46} +86.5058 q^{47} -113.278 q^{50} +36.8132 q^{52} -509.677 q^{53} +95.3113 q^{55} -1320.80 q^{58} -83.3852 q^{59} +5.25291 q^{61} +1106.99 q^{62} -834.706 q^{64} +14.6887 q^{65} +205.992 q^{67} -81.4281 q^{68} -1004.31 q^{71} +1007.29 q^{73} +875.055 q^{74} +68.1168 q^{76} -863.237 q^{79} -36.0992 q^{80} -1427.82 q^{82} +1334.72 q^{83} -32.4903 q^{85} +1363.85 q^{86} -391.370 q^{88} +326.249 q^{89} -618.755 q^{92} -391.969 q^{94} +27.1790 q^{95} -1526.77 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8}+O(q^{10}) \) 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8	\( 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8} - 5 q^{10} + 22 q^{11} + 22 q^{13} - 87 q^{16} + 116 q^{17} - 102 q^{19} + 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} + 54 q^{26} + 196 q^{29} - 150 q^{31} + 15 q^{32} + 462 q^{34} - 96 q^{37} - 404 q^{38} - 165 q^{40} - 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} - 25 q^{50} + 122 q^{52} - 326 q^{53} + 110 q^{55} - 1658 q^{58} - 844 q^{59} + 204 q^{61} + 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} - 412 q^{76} - 888 q^{79} - 435 q^{80} - 3162 q^{82} + 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} + 588 q^{89} - 1560 q^{92} + 1280 q^{94} - 510 q^{95} - 522 q^{97}+O(q^{100}) \) 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8 - 5 * q^10 + 22 * q^11 + 22 * q^13 - 87 * q^16 + 116 * q^17 - 102 * q^19 + 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 + 54 * q^26 + 196 * q^29 - 150 * q^31 + 15 * q^32 + 462 * q^34 - 96 * q^37 - 404 * q^38 - 165 * q^40 - 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 - 25 * q^50 + 122 * q^52 - 326 * q^53 + 110 * q^55 - 1658 * q^58 - 844 * q^59 + 204 * q^61 + 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 - 412 * q^76 - 888 * q^79 - 435 * q^80 - 3162 * q^82 + 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 + 588 * q^89 - 1560 * q^92 + 1280 * q^94 - 510 * q^95 - 522 * q^97

Introduction

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