LMFDB - Embedded newform 2205.4.a.x.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31662$ 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)

$f(q)$	$=$	$q+2.31662 q^{2} -2.63325 q^{4} +5.00000 q^{5} -24.6332 q^{8} +O(q^{10})$	$q+2.31662 q^{2} -2.63325 q^{4} +5.00000 q^{5} -24.6332 q^{8} +11.5831 q^{10} -46.2665 q^{11} +61.3325 q^{13} -36.0000 q^{16} +101.332 q^{17} -3.66750 q^{19} -13.1662 q^{20} -107.182 q^{22} -84.8655 q^{23} +25.0000 q^{25} +142.084 q^{26} -30.1980 q^{29} -188.997 q^{31} +113.668 q^{32} +234.749 q^{34} +18.0685 q^{37} -8.49623 q^{38} -123.166 q^{40} +481.662 q^{41} -97.7995 q^{43} +121.831 q^{44} -196.602 q^{46} +117.665 q^{47} +57.9156 q^{50} -161.504 q^{52} -667.995 q^{53} -231.332 q^{55} -69.9574 q^{58} +57.3350 q^{59} -738.997 q^{61} -437.836 q^{62} +551.325 q^{64} +306.662 q^{65} +552.396 q^{67} -266.834 q^{68} +740.264 q^{71} -233.325 q^{73} +41.8580 q^{74} +9.65745 q^{76} -1075.19 q^{79} -180.000 q^{80} +1115.83 q^{82} +683.325 q^{83} +506.662 q^{85} -226.565 q^{86} +1139.69 q^{88} -1380.32 q^{89} +223.472 q^{92} +272.586 q^{94} -18.3375 q^{95} -218.008 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8	$ 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8} - 10 q^{10} - 66 q^{11} - 10 q^{13} - 72 q^{16} + 70 q^{17} - 140 q^{19} + 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} + 450 q^{26} + 258 q^{29} + 20 q^{31} + 360 q^{32} + 370 q^{34} + 328 q^{37} + 580 q^{38} - 180 q^{40} + 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} - 30 q^{47} - 50 q^{50} - 920 q^{52} - 540 q^{53} - 330 q^{55} - 1314 q^{58} + 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} - 600 q^{68} + 1056 q^{71} + 860 q^{73} - 1296 q^{74} - 1440 q^{76} + 158 q^{79} - 360 q^{80} + 1900 q^{82} + 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 910 q^{94} - 700 q^{95} - 1630 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8 - 10 * q^10 - 66 * q^11 - 10 * q^13 - 72 * q^16 + 70 * q^17 - 140 * q^19 + 40 * q^20 - 22 * q^22 + 16 * q^23 + 50 * q^25 + 450 * q^26 + 258 * q^29 + 20 * q^31 + 360 * q^32 + 370 * q^34 + 328 * q^37 + 580 * q^38 - 180 * q^40 + 300 * q^41 - 116 * q^43 - 88 * q^44 - 632 * q^46 - 30 * q^47 - 50 * q^50 - 920 * q^52 - 540 * q^53 - 330 * q^55 - 1314 * q^58 + 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 - 50 * q^65 + 468 * q^67 - 600 * q^68 + 1056 * q^71 + 860 * q^73 - 1296 * q^74 - 1440 * q^76 + 158 * q^79 - 360 * q^80 + 1900 * q^82 + 40 * q^83 + 350 * q^85 - 148 * q^86 + 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 910 * q^94 - 700 * q^95 - 1630 * 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 $

$2$	2.31662	0.819051	0.409525	−	0.912299i	$-0.365694\pi$
$2$	2.31662	0.819051	0.409525	+	0.912299i	$0.365694\pi$
$3$	0	0
$3$	0	0
$4$	−2.63325	−0.329156
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−24.6332	−1.08865
$9$	0	0
$10$	11.5831	0.366291
$11$	−46.2665	−1.26817	−0.634085	−	0.773263i	$-0.718623\pi$
$11$	−46.2665	−1.26817	−0.634085	+	0.773263i	$0.718623\pi$
$12$	0	0
$13$	61.3325	1.30851	0.654253	−	0.756276i	$-0.272983\pi$
$13$	61.3325	1.30851	0.654253	+	0.756276i	$0.272983\pi$
$14$	0	0
$15$	0	0
$16$	−36.0000	−0.562500
$17$	101.332	1.44569	0.722845	−	0.691010i	$-0.242834\pi$
$17$	101.332	1.44569	0.722845	+	0.691010i	$0.242834\pi$
$18$	0	0
$19$	−3.66750	−0.0442833	−0.0221417	−	0.999755i	$-0.507048\pi$
$19$	−3.66750	−0.0442833	−0.0221417	+	0.999755i	$0.507048\pi$
$20$	−13.1662	−0.147203
$21$	0	0
$22$	−107.182	−1.03870
$23$	−84.8655	−0.769377	−0.384689	−	0.923046i	$-0.625691\pi$
$23$	−84.8655	−0.769377	−0.384689	+	0.923046i	$0.625691\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	142.084	1.07173
$27$	0	0
$28$	0	0
$29$	−30.1980	−0.193366	−0.0966832	−	0.995315i	$-0.530823\pi$
$29$	−30.1980	−0.193366	−0.0966832	+	0.995315i	$0.530823\pi$
$30$	0	0
$31$	−188.997	−1.09500	−0.547499	−	0.836806i	$-0.684420\pi$
$31$	−188.997	−1.09500	−0.547499	+	0.836806i	$0.684420\pi$
$32$	113.668	0.627930
$33$	0	0
$34$	234.749	1.18409
$35$	0	0
$36$	0	0
$37$	18.0685	0.0802823	0.0401411	−	0.999194i	$-0.487219\pi$
$37$	18.0685	0.0802823	0.0401411	+	0.999194i	$0.487219\pi$
$38$	−8.49623	−0.0362703
$39$	0	0
$40$	−123.166	−0.486857
$41$	481.662	1.83471	0.917354	−	0.398072i	$-0.130321\pi$
$41$	481.662	1.83471	0.917354	+	0.398072i	$0.130321\pi$
$42$	0	0
$43$	−97.7995	−0.346844	−0.173422	−	0.984848i	$-0.555482\pi$
$43$	−97.7995	−0.346844	−0.173422	+	0.984848i	$0.555482\pi$
$44$	121.831	0.417426
$45$	0	0
$46$	−196.602	−0.630159
$47$	117.665	0.365175	0.182587	−	0.983190i	$-0.441553\pi$
$47$	117.665	0.365175	0.182587	+	0.983190i	$0.441553\pi$
$48$	0	0
$49$	0	0
$50$	57.9156	0.163810
$51$	0	0
$52$	−161.504	−0.430703
$53$	−667.995	−1.73125	−0.865624	−	0.500694i	$-0.833078\pi$
$53$	−667.995	−1.73125	−0.865624	+	0.500694i	$0.833078\pi$
$54$	0	0
$55$	−231.332	−0.567143
$56$	0	0
$57$	0	0
$58$	−69.9574	−0.158377
$59$	57.3350	0.126515	0.0632575	−	0.997997i	$-0.479851\pi$
$59$	57.3350	0.126515	0.0632575	+	0.997997i	$0.479851\pi$
$60$	0	0
$61$	−738.997	−1.55113	−0.775565	−	0.631268i	$-0.782535\pi$
$61$	−738.997	−1.55113	−0.775565	+	0.631268i	$0.782535\pi$
$62$	−437.836	−0.896859
$63$	0	0
$64$	551.325	1.07681
$65$	306.662	0.585182
$66$	0	0
$67$	552.396	1.00725	0.503626	−	0.863922i	$-0.331999\pi$
$67$	552.396	1.00725	0.503626	+	0.863922i	$0.331999\pi$
$68$	−266.834	−0.475858
$69$	0	0
$70$	0	0
$71$	740.264	1.23737	0.618684	−	0.785640i	$-0.287666\pi$
$71$	740.264	1.23737	0.618684	+	0.785640i	$0.287666\pi$
$72$	0	0
$73$	−233.325	−0.374091	−0.187045	−	0.982351i	$-0.559891\pi$
$73$	−233.325	−0.374091	−0.187045	+	0.982351i	$0.559891\pi$
$74$	41.8580	0.0657553
$75$	0	0
$76$	9.65745	0.0145761
$77$	0	0
$78$	0	0
$79$	−1075.19	−1.53124	−0.765619	−	0.643294i	$-0.777567\pi$
$79$	−1075.19	−1.53124	−0.765619	+	0.643294i	$0.777567\pi$
$80$	−180.000	−0.251558
$81$	0	0
$82$	1115.83	1.50272
$83$	683.325	0.903671	0.451835	−	0.892101i	$-0.350769\pi$
$83$	683.325	0.903671	0.451835	+	0.892101i	$0.350769\pi$
$84$	0	0
$85$	506.662	0.646532
$86$	−226.565	−0.284083
$87$	0	0
$88$	1139.69	1.38059
$89$	−1380.32	−1.64397	−0.821985	−	0.569509i	$-0.807133\pi$
$89$	−1380.32	−1.64397	−0.821985	+	0.569509i	$0.807133\pi$
$90$	0	0
$91$	0	0
$92$	223.472	0.253245
$93$	0	0
$94$	272.586	0.299096
$95$	−18.3375	−0.0198041
$96$	0	0
$97$	−218.008	−0.228199	−0.114100	−	0.993469i	$-0.536398\pi$
$97$	−218.008	−0.228199	−0.114100	+	0.993469i	$0.536398\pi$
$98$	0	0
$99$	0	0
$100$	−65.8312	−0.0658312
$101$	−1474.33	−1.45249	−0.726243	−	0.687438i	$-0.758735\pi$
$101$	−1474.33	−1.45249	−0.726243	+	0.687438i	$0.758735\pi$
$102$	0	0
$103$	810.990	0.775818	0.387909	−	0.921698i	$-0.373198\pi$
$103$	810.990	0.775818	0.387909	+	0.921698i	$0.373198\pi$
$104$	−1510.82	−1.42450
$105$	0	0
$106$	−1547.49	−1.41798
$107$	−440.660	−0.398133	−0.199066	−	0.979986i	$-0.563791\pi$
$107$	−440.660	−0.398133	−0.199066	+	0.979986i	$0.563791\pi$
$108$	0	0
$109$	−1906.19	−1.67504	−0.837522	−	0.546404i	$-0.815996\pi$
$109$	−1906.19	−1.67504	−0.837522	+	0.546404i	$0.815996\pi$
$110$	−535.911	−0.464519
$111$	0	0
$112$	0	0
$113$	−962.470	−0.801252	−0.400626	−	0.916242i	$-0.631207\pi$
$113$	−962.470	−0.801252	−0.400626	+	0.916242i	$0.631207\pi$
$114$	0	0
$115$	−424.327	−0.344076
$116$	79.5188	0.0636478
$117$	0	0
$118$	132.824	0.103622
$119$	0	0
$120$	0	0
$121$	809.589	0.608256
$122$	−1711.98	−1.27045
$123$	0	0
$124$	497.678	0.360426
$125$	125.000	0.0894427
$126$	0	0
$127$	−1621.74	−1.13312	−0.566558	−	0.824022i	$-0.691725\pi$
$127$	−1621.74	−1.13312	−0.566558	+	0.824022i	$0.691725\pi$
$128$	367.873	0.254029
$129$	0	0
$130$	710.422	0.479293
$131$	−1380.32	−0.920602	−0.460301	−	0.887763i	$-0.652259\pi$
$131$	−1380.32	−0.920602	−0.460301	+	0.887763i	$0.652259\pi$
$132$	0	0
$133$	0	0
$134$	1279.69	0.824991
$135$	0	0
$136$	−2496.15	−1.57385
$137$	1949.66	1.21584	0.607921	−	0.793997i	$-0.292004\pi$
$137$	1949.66	1.21584	0.607921	+	0.793997i	$0.292004\pi$
$138$	0	0
$139$	−2800.00	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$139$	−2800.00	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$140$	0	0
$141$	0	0
$142$	1714.91	1.01347
$143$	−2837.64	−1.65941
$144$	0	0
$145$	−150.990	−0.0864761
$146$	−540.526	−0.306399
$147$	0	0
$148$	−47.5789	−0.0264254
$149$	1434.12	0.788506	0.394253	−	0.919002i	$-0.371003\pi$
$149$	1434.12	0.788506	0.394253	+	0.919002i	$0.371003\pi$
$150$	0	0
$151$	−1985.58	−1.07009	−0.535047	−	0.844822i	$-0.679706\pi$
$151$	−1985.58	−1.07009	−0.535047	+	0.844822i	$0.679706\pi$
$152$	90.3425	0.0482089
$153$	0	0
$154$	0	0
$155$	−944.987	−0.489698
$156$	0	0
$157$	−40.6600	−0.0206689	−0.0103345	−	0.999947i	$-0.503290\pi$
$157$	−40.6600	−0.0206689	−0.0103345	+	0.999947i	$0.503290\pi$
$158$	−2490.80	−1.25416
$159$	0	0
$160$	568.338	0.280819
$161$	0	0
$162$	0	0
$163$	−3953.98	−1.90000	−0.950000	−	0.312250i	$-0.898917\pi$
$163$	−3953.98	−1.90000	−0.950000	+	0.312250i	$0.898917\pi$
$164$	−1268.34	−0.603906
$165$	0	0
$166$	1583.01	0.740152
$167$	3380.30	1.56632	0.783161	−	0.621819i	$-0.213606\pi$
$167$	3380.30	1.56632	0.783161	+	0.621819i	$0.213606\pi$
$168$	0	0
$169$	1564.68	0.712187
$170$	1173.75	0.529543
$171$	0	0
$172$	257.530	0.114166
$173$	3206.66	1.40924	0.704619	−	0.709586i	$-0.251118\pi$
$173$	3206.66	1.40924	0.704619	+	0.709586i	$0.251118\pi$
$174$	0	0
$175$	0	0
$176$	1665.59	0.713346
$177$	0	0
$178$	−3197.68	−1.34649
$179$	−1442.65	−0.602395	−0.301198	−	0.953562i	$-0.597386\pi$
$179$	−1442.65	−0.602395	−0.301198	+	0.953562i	$0.597386\pi$
$180$	0	0
$181$	908.680	0.373158	0.186579	−	0.982440i	$-0.440260\pi$
$181$	908.680	0.373158	0.186579	+	0.982440i	$0.440260\pi$
$182$	0	0
$183$	0	0
$184$	2090.51	0.837580
$185$	90.3425	0.0359033
$186$	0	0
$187$	−4688.30	−1.83338
$188$	−309.841	−0.120199
$189$	0	0
$190$	−42.4812	−0.0162206
$191$	−2474.64	−0.937479	−0.468739	−	0.883336i	$-0.655292\pi$
$191$	−2474.64	−0.937479	−0.468739	+	0.883336i	$0.655292\pi$
$192$	0	0
$193$	3533.52	1.31787	0.658934	−	0.752201i	$-0.271008\pi$
$193$	3533.52	1.31787	0.658934	+	0.752201i	$0.271008\pi$
$194$	−505.042	−0.186907
$195$	0	0
$196$	0	0
$197$	1952.57	0.706165	0.353083	−	0.935592i	$-0.385133\pi$
$197$	1952.57	0.706165	0.353083	+	0.935592i	$0.385133\pi$
$198$	0	0
$199$	−4064.67	−1.44792	−0.723962	−	0.689840i	$-0.757681\pi$
$199$	−4064.67	−1.44792	−0.723962	+	0.689840i	$0.757681\pi$
$200$	−615.831	−0.217729
$201$	0	0
$202$	−3415.46	−1.18966
$203$	0	0
$204$	0	0
$205$	2408.31	0.820507
$206$	1878.76	0.635434
$207$	0	0
$208$	−2207.97	−0.736034
$209$	169.683	0.0561588
$210$	0	0
$211$	−4325.34	−1.41123	−0.705613	−	0.708598i	$-0.749328\pi$
$211$	−4325.34	−1.41123	−0.705613	+	0.708598i	$0.749328\pi$
$212$	1759.00	0.569851
$213$	0	0
$214$	−1020.84	−0.326091
$215$	−488.997	−0.155113
$216$	0	0
$217$	0	0
$218$	−4415.92	−1.37194
$219$	0	0
$220$	609.156	0.186679
$221$	6214.97	1.89169
$222$	0	0
$223$	982.970	0.295177	0.147589	−	0.989049i	$-0.452849\pi$
$223$	982.970	0.295177	0.147589	+	0.989049i	$0.452849\pi$
$224$	0	0
$225$	0	0
$226$	−2229.68	−0.656266
$227$	−1660.96	−0.485648	−0.242824	−	0.970070i	$-0.578074\pi$
$227$	−1660.96	−0.485648	−0.242824	+	0.970070i	$0.578074\pi$
$228$	0	0
$229$	574.327	0.165732	0.0828660	−	0.996561i	$-0.473593\pi$
$229$	574.327	0.165732	0.0828660	+	0.996561i	$0.473593\pi$
$230$	−983.008	−0.281816
$231$	0	0
$232$	743.875	0.210508
$233$	2316.48	0.651320	0.325660	−	0.945487i	$-0.394414\pi$
$233$	2316.48	0.651320	0.325660	+	0.945487i	$0.394414\pi$
$234$	0	0
$235$	588.325	0.163311
$236$	−150.977	−0.0416432
$237$	0	0
$238$	0	0
$239$	3659.31	0.990382	0.495191	−	0.868784i	$-0.335098\pi$
$239$	3659.31	0.990382	0.495191	+	0.868784i	$0.335098\pi$
$240$	0	0
$241$	−2446.33	−0.653868	−0.326934	−	0.945047i	$-0.606015\pi$
$241$	−2446.33	−0.653868	−0.326934	+	0.945047i	$0.606015\pi$
$242$	1875.51	0.498193
$243$	0	0
$244$	1945.96	0.510564
$245$	0	0
$246$	0	0
$247$	−224.937	−0.0579450
$248$	4655.62	1.19207
$249$	0	0
$250$	289.578	0.0732581
$251$	−2909.29	−0.731605	−0.365802	−	0.930693i	$-0.619205\pi$
$251$	−2909.29	−0.731605	−0.365802	+	0.930693i	$0.619205\pi$
$252$	0	0
$253$	3926.43	0.975702
$254$	−3756.95	−0.928080
$255$	0	0
$256$	−3558.38	−0.868744
$257$	168.680	0.0409415	0.0204708	−	0.999790i	$-0.493483\pi$
$257$	168.680	0.0409415	0.0204708	+	0.999790i	$0.493483\pi$
$258$	0	0
$259$	0	0
$260$	−807.519	−0.192616
$261$	0	0
$262$	−3197.68	−0.754020
$263$	3244.47	0.760695	0.380347	−	0.924844i	$-0.375804\pi$
$263$	3244.47	0.760695	0.380347	+	0.924844i	$0.375804\pi$
$264$	0	0
$265$	−3339.97	−0.774238
$266$	0	0
$267$	0	0
$268$	−1454.60	−0.331543
$269$	−2848.65	−0.645671	−0.322836	−	0.946455i	$-0.604636\pi$
$269$	−2848.65	−0.645671	−0.322836	+	0.946455i	$0.604636\pi$
$270$	0	0
$271$	2850.98	0.639057	0.319529	−	0.947577i	$-0.396475\pi$
$271$	2850.98	0.639057	0.319529	+	0.947577i	$0.396475\pi$
$272$	−3647.97	−0.813201
$273$	0	0
$274$	4516.62	0.995837
$275$	−1156.66	−0.253634
$276$	0	0
$277$	2298.63	0.498597	0.249298	−	0.968427i	$-0.419800\pi$
$277$	2298.63	0.498597	0.249298	+	0.968427i	$0.419800\pi$
$278$	−6486.55	−1.39942
$279$	0	0
$280$	0	0
$281$	−6109.20	−1.29695	−0.648477	−	0.761234i	$-0.724594\pi$
$281$	−6109.20	−1.29695	−0.648477	+	0.761234i	$0.724594\pi$
$282$	0	0
$283$	−5854.95	−1.22983	−0.614913	−	0.788595i	$-0.710809\pi$
$283$	−5854.95	−1.22983	−0.614913	+	0.788595i	$0.710809\pi$
$284$	−1949.30	−0.407288
$285$	0	0
$286$	−6573.75	−1.35914
$287$	0	0
$288$	0	0
$289$	5355.27	1.09002
$290$	−349.787	−0.0708283
$291$	0	0
$292$	614.403	0.123134
$293$	−5135.34	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$293$	−5135.34	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$294$	0	0
$295$	286.675	0.0565792
$296$	−445.086	−0.0873990
$297$	0	0
$298$	3322.31	0.645827
$299$	−5205.01	−1.00673
$300$	0	0
$301$	0	0
$302$	−4599.85	−0.876462
$303$	0	0
$304$	132.030	0.0249094
$305$	−3694.99	−0.693686
$306$	0	0
$307$	−2102.97	−0.390954	−0.195477	−	0.980708i	$-0.562625\pi$
$307$	−2102.97	−0.390954	−0.195477	+	0.980708i	$0.562625\pi$
$308$	0	0
$309$	0	0
$310$	−2189.18	−0.401088
$311$	−5764.30	−1.05101	−0.525504	−	0.850791i	$-0.676123\pi$
$311$	−5764.30	−1.05101	−0.525504	+	0.850791i	$0.676123\pi$
$312$	0	0
$313$	1360.01	0.245599	0.122799	−	0.992432i	$-0.460813\pi$
$313$	1360.01	0.245599	0.122799	+	0.992432i	$0.460813\pi$
$314$	−94.1939	−0.0169289
$315$	0	0
$316$	2831.23	0.504017
$317$	−5138.95	−0.910511	−0.455256	−	0.890361i	$-0.650452\pi$
$317$	−5138.95	−0.910511	−0.455256	+	0.890361i	$0.650452\pi$
$318$	0	0
$319$	1397.16	0.245222
$320$	2756.62	0.481563
$321$	0	0
$322$	0	0
$323$	−371.637	−0.0640200
$324$	0	0
$325$	1533.31	0.261701
$326$	−9159.90	−1.55620
$327$	0	0
$328$	−11864.9	−1.99735
$329$	0	0
$330$	0	0
$331$	1825.70	0.303171	0.151585	−	0.988444i	$-0.451562\pi$
$331$	1825.70	0.303171	0.151585	+	0.988444i	$0.451562\pi$
$332$	−1799.37	−0.297449
$333$	0	0
$334$	7830.90	1.28290
$335$	2761.98	0.450457
$336$	0	0
$337$	153.985	0.0248905	0.0124452	−	0.999923i	$-0.496038\pi$
$337$	153.985	0.0248905	0.0124452	+	0.999923i	$0.496038\pi$
$338$	3624.76	0.583317
$339$	0	0
$340$	−1334.17	−0.212810
$341$	8744.25	1.38864
$342$	0	0
$343$	0	0
$344$	2409.12	0.377590
$345$	0	0
$346$	7428.63	1.15424
$347$	−4359.39	−0.674421	−0.337211	−	0.941429i	$-0.609483\pi$
$347$	−4359.39	−0.674421	−0.337211	+	0.941429i	$0.609483\pi$
$348$	0	0
$349$	−1689.00	−0.259054	−0.129527	−	0.991576i	$-0.541346\pi$
$349$	−1689.00	−0.259054	−0.129527	+	0.991576i	$0.541346\pi$
$350$	0	0
$351$	0	0
$352$	−5259.00	−0.796322
$353$	−3921.36	−0.591254	−0.295627	−	0.955303i	$-0.595529\pi$
$353$	−3921.36	−0.591254	−0.295627	+	0.955303i	$0.595529\pi$
$354$	0	0
$355$	3701.32	0.553368
$356$	3634.72	0.541123
$357$	0	0
$358$	−3342.08	−0.493392
$359$	2867.86	0.421616	0.210808	−	0.977528i	$-0.432391\pi$
$359$	2867.86	0.421616	0.210808	+	0.977528i	$0.432391\pi$
$360$	0	0
$361$	−6845.55	−0.998039
$362$	2105.07	0.305636
$363$	0	0
$364$	0	0
$365$	−1166.62	−0.167298
$366$	0	0
$367$	−11503.0	−1.63611	−0.818054	−	0.575142i	$-0.804947\pi$
$367$	−11503.0	−1.63611	−0.818054	+	0.575142i	$0.804947\pi$
$368$	3055.16	0.432775
$369$	0	0
$370$	209.290	0.0294066
$371$	0	0
$372$	0	0
$373$	5086.43	0.706073	0.353037	−	0.935610i	$-0.385149\pi$
$373$	5086.43	0.706073	0.353037	+	0.935610i	$0.385149\pi$
$374$	−10861.0	−1.50163
$375$	0	0
$376$	−2898.47	−0.397546
$377$	−1852.12	−0.253021
$378$	0	0
$379$	954.827	0.129409	0.0647047	−	0.997904i	$-0.479389\pi$
$379$	954.827	0.129409	0.0647047	+	0.997904i	$0.479389\pi$
$380$	48.2873	0.00651864
$381$	0	0
$382$	−5732.81	−0.767843
$383$	3083.91	0.411437	0.205719	−	0.978611i	$-0.434047\pi$
$383$	3083.91	0.411437	0.205719	+	0.978611i	$0.434047\pi$
$384$	0	0
$385$	0	0
$386$	8185.84	1.07940
$387$	0	0
$388$	574.068	0.0751131
$389$	6331.15	0.825198	0.412599	−	0.910913i	$-0.364621\pi$
$389$	6331.15	0.825198	0.412599	+	0.910913i	$0.364621\pi$
$390$	0	0
$391$	−8599.63	−1.11228
$392$	0	0
$393$	0	0
$394$	4523.36	0.578385
$395$	−5375.93	−0.684791
$396$	0	0
$397$	12133.2	1.53388	0.766939	−	0.641720i	$-0.221779\pi$
$397$	12133.2	1.53388	0.766939	+	0.641720i	$0.221779\pi$
$398$	−9416.32	−1.18592
$399$	0	0
$400$	−900.000	−0.112500
$401$	270.669	0.0337072	0.0168536	−	0.999858i	$-0.494635\pi$
$401$	270.669	0.0337072	0.0168536	+	0.999858i	$0.494635\pi$
$402$	0	0
$403$	−11591.7	−1.43281
$404$	3882.27	0.478095
$405$	0	0
$406$	0	0
$407$	−835.967	−0.101812
$408$	0	0
$409$	−4019.92	−0.485996	−0.242998	−	0.970027i	$-0.578131\pi$
$409$	−4019.92	−0.485996	−0.242998	+	0.970027i	$0.578131\pi$
$410$	5579.16	0.672036
$411$	0	0
$412$	−2135.54	−0.255365
$413$	0	0
$414$	0	0
$415$	3416.62	0.404134
$416$	6971.51	0.821650
$417$	0	0
$418$	393.091	0.0459969
$419$	2437.28	0.284175	0.142087	−	0.989854i	$-0.454619\pi$
$419$	2437.28	0.284175	0.142087	+	0.989854i	$0.454619\pi$
$420$	0	0
$421$	−4751.36	−0.550041	−0.275020	−	0.961438i	$-0.588685\pi$
$421$	−4751.36	−0.550041	−0.275020	+	0.961438i	$0.588685\pi$
$422$	−10020.2	−1.15586
$423$	0	0
$424$	16454.9	1.88472
$425$	2533.31	0.289138
$426$	0	0
$427$	0	0
$428$	1160.37	0.131048
$429$	0	0
$430$	−1132.82	−0.127046
$431$	−7925.19	−0.885714	−0.442857	−	0.896592i	$-0.646035\pi$
$431$	−7925.19	−0.885714	−0.442857	+	0.896592i	$0.646035\pi$
$432$	0	0
$433$	11487.3	1.27492	0.637462	−	0.770481i	$-0.279984\pi$
$433$	11487.3	1.27492	0.637462	+	0.770481i	$0.279984\pi$
$434$	0	0
$435$	0	0
$436$	5019.47	0.551351
$437$	311.245	0.0340706
$438$	0	0
$439$	9147.92	0.994548	0.497274	−	0.867594i	$-0.334334\pi$
$439$	9147.92	0.994548	0.497274	+	0.867594i	$0.334334\pi$
$440$	5698.47	0.617418
$441$	0	0
$442$	14397.8	1.54939
$443$	−1864.35	−0.199950	−0.0999752	−	0.994990i	$-0.531876\pi$
$443$	−1864.35	−0.199950	−0.0999752	+	0.994990i	$0.531876\pi$
$444$	0	0
$445$	−6901.59	−0.735206
$446$	2277.17	0.241765
$447$	0	0
$448$	0	0
$449$	−4490.88	−0.472022	−0.236011	−	0.971750i	$-0.575840\pi$
$449$	−4490.88	−0.472022	−0.236011	+	0.971750i	$0.575840\pi$
$450$	0	0
$451$	−22284.8	−2.32672
$452$	2534.42	0.263737
$453$	0	0
$454$	−3847.83	−0.397770
$455$	0	0
$456$	0	0
$457$	−14343.8	−1.46822	−0.734109	−	0.679032i	$-0.762400\pi$
$457$	−14343.8	−1.46822	−0.734109	+	0.679032i	$0.762400\pi$
$458$	1330.50	0.135743
$459$	0	0
$460$	1117.36	0.113255
$461$	−14558.7	−1.47086	−0.735429	−	0.677602i	$-0.763019\pi$
$461$	−14558.7	−1.47086	−0.735429	+	0.677602i	$0.763019\pi$
$462$	0	0
$463$	−1809.56	−0.181636	−0.0908178	−	0.995868i	$-0.528948\pi$
$463$	−1809.56	−0.181636	−0.0908178	+	0.995868i	$0.528948\pi$
$464$	1087.13	0.108769
$465$	0	0
$466$	5366.41	0.533464
$467$	5981.65	0.592715	0.296357	−	0.955077i	$-0.404228\pi$
$467$	5981.65	0.592715	0.296357	+	0.955077i	$0.404228\pi$
$468$	0	0
$469$	0	0
$470$	1362.93	0.133760
$471$	0	0
$472$	−1412.35	−0.137730
$473$	4524.84	0.439857
$474$	0	0
$475$	−91.6876	−0.00885666
$476$	0	0
$477$	0	0
$478$	8477.25	0.811173
$479$	11527.5	1.09959	0.549796	−	0.835299i	$-0.314705\pi$
$479$	11527.5	1.09959	0.549796	+	0.835299i	$0.314705\pi$
$480$	0	0
$481$	1108.19	0.105050
$482$	−5667.23	−0.535551
$483$	0	0
$484$	−2131.85	−0.200211
$485$	−1090.04	−0.102054
$486$	0	0
$487$	15791.5	1.46936	0.734682	−	0.678411i	$-0.237331\pi$
$487$	15791.5	1.46936	0.734682	+	0.678411i	$0.237331\pi$
$488$	18203.9	1.68863
$489$	0	0
$490$	0	0
$491$	−13064.9	−1.20083	−0.600417	−	0.799687i	$-0.704999\pi$
$491$	−13064.9	−1.20083	−0.600417	+	0.799687i	$0.704999\pi$
$492$	0	0
$493$	−3060.04	−0.279548
$494$	−521.095	−0.0474599
$495$	0	0
$496$	6803.91	0.615937
$497$	0	0
$498$	0	0
$499$	−20135.8	−1.80642	−0.903209	−	0.429201i	$-0.858795\pi$
$499$	−20135.8	−1.80642	−0.903209	+	0.429201i	$0.858795\pi$
$500$	−329.156	−0.0294406
$501$	0	0
$502$	−6739.73	−0.599221
$503$	751.675	0.0666313	0.0333156	−	0.999445i	$-0.489393\pi$
$503$	751.675	0.0666313	0.0333156	+	0.999445i	$0.489393\pi$
$504$	0	0
$505$	−7371.64	−0.649571
$506$	9096.06	0.799149
$507$	0	0
$508$	4270.44	0.372972
$509$	12334.5	1.07410	0.537049	−	0.843551i	$-0.319539\pi$
$509$	12334.5	1.07410	0.537049	+	0.843551i	$0.319539\pi$
$510$	0	0
$511$	0	0
$512$	−11186.4	−0.965574
$513$	0	0
$514$	390.768	0.0335332
$515$	4054.95	0.346956
$516$	0	0
$517$	−5443.95	−0.463104
$518$	0	0
$519$	0	0
$520$	−7554.09	−0.637056
$521$	1736.43	0.146016	0.0730082	−	0.997331i	$-0.476740\pi$
$521$	1736.43	0.146016	0.0730082	+	0.997331i	$0.476740\pi$
$522$	0	0
$523$	−1421.42	−0.118842	−0.0594210	−	0.998233i	$-0.518925\pi$
$523$	−1421.42	−0.118842	−0.0594210	+	0.998233i	$0.518925\pi$
$524$	3634.72	0.303022
$525$	0	0
$526$	7516.23	0.623048
$527$	−19151.6	−1.58303
$528$	0	0
$529$	−4964.85	−0.408059
$530$	−7737.47	−0.634140
$531$	0	0
$532$	0	0
$533$	29541.6	2.40073
$534$	0	0
$535$	−2203.30	−0.178050
$536$	−13607.3	−1.09654
$537$	0	0
$538$	−6599.26	−0.528837
$539$	0	0
$540$	0	0
$541$	5773.27	0.458802	0.229401	−	0.973332i	$-0.426323\pi$
$541$	5773.27	0.458802	0.229401	+	0.973332i	$0.426323\pi$
$542$	6604.64	0.523420
$543$	0	0
$544$	11518.2	0.907793
$545$	−9530.94	−0.749102
$546$	0	0
$547$	−3941.30	−0.308076	−0.154038	−	0.988065i	$-0.549228\pi$
$547$	−3941.30	−0.308076	−0.154038	+	0.988065i	$0.549228\pi$
$548$	−5133.93	−0.400202
$549$	0	0
$550$	−2679.55	−0.207739
$551$	110.751	0.00856291
$552$	0	0
$553$	0	0
$554$	5325.06	0.408376
$555$	0	0
$556$	7373.10	0.562390
$557$	6951.74	0.528823	0.264412	−	0.964410i	$-0.414822\pi$
$557$	6951.74	0.528823	0.264412	+	0.964410i	$0.414822\pi$
$558$	0	0
$559$	−5998.29	−0.453847
$560$	0	0
$561$	0	0
$562$	−14152.7	−1.06227
$563$	24284.6	1.81789	0.908946	−	0.416913i	$-0.136888\pi$
$563$	24284.6	1.81789	0.908946	+	0.416913i	$0.136888\pi$
$564$	0	0
$565$	−4812.35	−0.358331
$566$	−13563.7	−1.00729
$567$	0	0
$568$	−18235.1	−1.34706
$569$	21563.4	1.58873	0.794363	−	0.607443i	$-0.207805\pi$
$569$	21563.4	1.58873	0.794363	+	0.607443i	$0.207805\pi$
$570$	0	0
$571$	−3689.56	−0.270409	−0.135204	−	0.990818i	$-0.543169\pi$
$571$	−3689.56	−0.270409	−0.135204	+	0.990818i	$0.543169\pi$
$572$	7472.21	0.546204
$573$	0	0
$574$	0	0
$575$	−2121.64	−0.153875
$576$	0	0
$577$	22183.9	1.60057	0.800285	−	0.599620i	$-0.204682\pi$
$577$	22183.9	1.60057	0.800285	+	0.599620i	$0.204682\pi$
$578$	12406.2	0.892783
$579$	0	0
$580$	397.594	0.0284641
$581$	0	0
$582$	0	0
$583$	30905.8	2.19552
$584$	5747.55	0.407252
$585$	0	0
$586$	−11896.7	−0.838646
$587$	10605.3	0.745705	0.372852	−	0.927891i	$-0.378380\pi$
$587$	10605.3	0.745705	0.372852	+	0.927891i	$0.378380\pi$
$588$	0	0
$589$	693.149	0.0484902
$590$	664.119	0.0463412
$591$	0	0
$592$	−650.466	−0.0451588
$593$	6277.25	0.434698	0.217349	−	0.976094i	$-0.430259\pi$
$593$	6277.25	0.434698	0.217349	+	0.976094i	$0.430259\pi$
$594$	0	0
$595$	0	0
$596$	−3776.39	−0.259542
$597$	0	0
$598$	−12058.1	−0.824567
$599$	9970.73	0.680122	0.340061	−	0.940403i	$-0.389552\pi$
$599$	9970.73	0.680122	0.340061	+	0.940403i	$0.389552\pi$
$600$	0	0
$601$	24619.2	1.67094	0.835472	−	0.549533i	$-0.185194\pi$
$601$	24619.2	1.67094	0.835472	+	0.549533i	$0.185194\pi$
$602$	0	0
$603$	0	0
$604$	5228.53	0.352228
$605$	4047.94	0.272020
$606$	0	0
$607$	−11252.9	−0.752460	−0.376230	−	0.926526i	$-0.622780\pi$
$607$	−11252.9	−0.752460	−0.376230	+	0.926526i	$0.622780\pi$
$608$	−416.876	−0.0278068
$609$	0	0
$610$	−8559.90	−0.568164
$611$	7216.69	0.477833
$612$	0	0
$613$	−15293.2	−1.00765	−0.503824	−	0.863807i	$-0.668074\pi$
$613$	−15293.2	−1.00765	−0.503824	+	0.863807i	$0.668074\pi$
$614$	−4871.79	−0.320211
$615$	0	0
$616$	0	0
$617$	17589.4	1.14769	0.573843	−	0.818966i	$-0.305452\pi$
$617$	17589.4	1.14769	0.573843	+	0.818966i	$0.305452\pi$
$618$	0	0
$619$	23467.4	1.52380	0.761900	−	0.647694i	$-0.224267\pi$
$619$	23467.4	1.52380	0.761900	+	0.647694i	$0.224267\pi$
$620$	2488.39	0.161187
$621$	0	0
$622$	−13353.7	−0.860829
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	3150.64	0.201158
$627$	0	0
$628$	107.068	0.00680330
$629$	1830.93	0.116063
$630$	0	0
$631$	−6040.86	−0.381114	−0.190557	−	0.981676i	$-0.561029\pi$
$631$	−6040.86	−0.381114	−0.190557	+	0.981676i	$0.561029\pi$
$632$	26485.3	1.66698
$633$	0	0
$634$	−11905.0	−0.745755
$635$	−8108.68	−0.506745
$636$	0	0
$637$	0	0
$638$	3236.68	0.200849
$639$	0	0
$640$	1839.37	0.113605
$641$	−25111.6	−1.54735	−0.773673	−	0.633586i	$-0.781582\pi$
$641$	−25111.6	−1.54735	−0.773673	+	0.633586i	$0.781582\pi$
$642$	0	0
$643$	−3095.03	−0.189823	−0.0949113	−	0.995486i	$-0.530257\pi$
$643$	−3095.03	−0.189823	−0.0949113	+	0.995486i	$0.530257\pi$
$644$	0	0
$645$	0	0
$646$	−860.944	−0.0524356
$647$	9178.63	0.557727	0.278863	−	0.960331i	$-0.410042\pi$
$647$	9178.63	0.557727	0.278863	+	0.960331i	$0.410042\pi$
$648$	0	0
$649$	−2652.69	−0.160443
$650$	3552.11	0.214346
$651$	0	0
$652$	10411.8	0.625397
$653$	14438.4	0.865265	0.432632	−	0.901570i	$-0.357585\pi$
$653$	14438.4	0.865265	0.432632	+	0.901570i	$0.357585\pi$
$654$	0	0
$655$	−6901.59	−0.411706
$656$	−17339.8	−1.03202
$657$	0	0
$658$	0	0
$659$	2900.64	0.171461	0.0857305	−	0.996318i	$-0.472678\pi$
$659$	2900.64	0.171461	0.0857305	+	0.996318i	$0.472678\pi$
$660$	0	0
$661$	−9976.52	−0.587053	−0.293526	−	0.955951i	$-0.594829\pi$
$661$	−9976.52	−0.587053	−0.293526	+	0.955951i	$0.594829\pi$
$662$	4229.46	0.248312
$663$	0	0
$664$	−16832.5	−0.983777
$665$	0	0
$666$	0	0
$667$	2562.77	0.148772
$668$	−8901.19	−0.515565
$669$	0	0
$670$	6398.47	0.368947
$671$	34190.8	1.96710
$672$	0	0
$673$	20760.8	1.18911	0.594554	−	0.804055i	$-0.297329\pi$
$673$	20760.8	1.18911	0.594554	+	0.804055i	$0.297329\pi$
$674$	356.725	0.0203866
$675$	0	0
$676$	−4120.18	−0.234421
$677$	3209.13	0.182181	0.0910907	−	0.995843i	$-0.470965\pi$
$677$	3209.13	0.182181	0.0910907	+	0.995843i	$0.470965\pi$
$678$	0	0
$679$	0	0
$680$	−12480.7	−0.703845
$681$	0	0
$682$	20257.2	1.13737
$683$	−4333.57	−0.242781	−0.121391	−	0.992605i	$-0.538735\pi$
$683$	−4333.57	−0.242781	−0.121391	+	0.992605i	$0.538735\pi$
$684$	0	0
$685$	9748.29	0.543741
$686$	0	0
$687$	0	0
$688$	3520.78	0.195100
$689$	−40969.8	−2.26535
$690$	0	0
$691$	14446.0	0.795297	0.397649	−	0.917538i	$-0.369826\pi$
$691$	14446.0	0.795297	0.397649	+	0.917538i	$0.369826\pi$
$692$	−8443.94	−0.463859
$693$	0	0
$694$	−10099.1	−0.552385
$695$	−14000.0	−0.764101
$696$	0	0
$697$	48808.1	2.65242
$698$	−3912.77	−0.212179
$699$	0	0
$700$	0	0
$701$	859.801	0.0463256	0.0231628	−	0.999732i	$-0.492626\pi$
$701$	859.801	0.0463256	0.0231628	+	0.999732i	$0.492626\pi$
$702$	0	0
$703$	−66.2663	−0.00355517
$704$	−25507.9	−1.36557
$705$	0	0
$706$	−9084.31	−0.484267
$707$	0	0
$708$	0	0
$709$	−7979.13	−0.422655	−0.211327	−	0.977415i	$-0.567779\pi$
$709$	−7979.13	−0.422655	−0.211327	+	0.977415i	$0.567779\pi$
$710$	8574.57	0.453236
$711$	0	0
$712$	34001.7	1.78970
$713$	16039.4	0.842467
$714$	0	0
$715$	−14188.2	−0.742110
$716$	3798.86	0.198282
$717$	0	0
$718$	6643.76	0.345325
$719$	−33703.3	−1.74815	−0.874076	−	0.485790i	$-0.838532\pi$
$719$	−33703.3	−1.74815	−0.874076	+	0.485790i	$0.838532\pi$
$720$	0	0
$721$	0	0
$722$	−15858.6	−0.817444
$723$	0	0
$724$	−2392.78	−0.122827
$725$	−754.950	−0.0386733
$726$	0	0
$727$	30277.0	1.54458	0.772290	−	0.635270i	$-0.219111\pi$
$727$	30277.0	1.54458	0.772290	+	0.635270i	$0.219111\pi$
$728$	0	0
$729$	0	0
$730$	−2702.63	−0.137026
$731$	−9910.27	−0.501429
$732$	0	0
$733$	−19363.9	−0.975749	−0.487874	−	0.872914i	$-0.662228\pi$
$733$	−19363.9	−0.975749	−0.487874	+	0.872914i	$0.662228\pi$
$734$	−26648.1	−1.34005
$735$	0	0
$736$	−9646.45	−0.483115
$737$	−25557.4	−1.27737
$738$	0	0
$739$	24952.4	1.24207	0.621035	−	0.783783i	$-0.286713\pi$
$739$	24952.4	1.24207	0.621035	+	0.783783i	$0.286713\pi$
$740$	−237.894	−0.0118178
$741$	0	0
$742$	0	0
$743$	8154.54	0.402640	0.201320	−	0.979526i	$-0.435477\pi$
$743$	8154.54	0.402640	0.201320	+	0.979526i	$0.435477\pi$
$744$	0	0
$745$	7170.58	0.352631
$746$	11783.3	0.578310
$747$	0	0
$748$	12345.5	0.603469
$749$	0	0
$750$	0	0
$751$	−4311.26	−0.209481	−0.104740	−	0.994500i	$-0.533401\pi$
$751$	−4311.26	−0.209481	−0.104740	+	0.994500i	$0.533401\pi$
$752$	−4235.94	−0.205411
$753$	0	0
$754$	−4290.66	−0.207237
$755$	−9927.91	−0.478561
$756$	0	0
$757$	3624.79	0.174036	0.0870179	−	0.996207i	$-0.472266\pi$
$757$	3624.79	0.174036	0.0870179	+	0.996207i	$0.472266\pi$
$758$	2211.98	0.105993
$759$	0	0
$760$	451.713	0.0215597
$761$	20576.4	0.980150	0.490075	−	0.871680i	$-0.336969\pi$
$761$	20576.4	0.980150	0.490075	+	0.871680i	$0.336969\pi$
$762$	0	0
$763$	0	0
$764$	6516.34	0.308577
$765$	0	0
$766$	7144.26	0.336988
$767$	3516.50	0.165546
$768$	0	0
$769$	3066.14	0.143781	0.0718907	−	0.997413i	$-0.477097\pi$
$769$	3066.14	0.143781	0.0718907	+	0.997413i	$0.477097\pi$
$770$	0	0
$771$	0	0
$772$	−9304.64	−0.433784
$773$	−19387.0	−0.902074	−0.451037	−	0.892505i	$-0.648946\pi$
$773$	−19387.0	−0.902074	−0.451037	+	0.892505i	$0.648946\pi$
$774$	0	0
$775$	−4724.94	−0.219000
$776$	5370.23	0.248428
$777$	0	0
$778$	14666.9	0.675879
$779$	−1766.50	−0.0812470
$780$	0	0
$781$	−34249.4	−1.56919
$782$	−19922.1	−0.911015
$783$	0	0
$784$	0	0
$785$	−203.300	−0.00924342
$786$	0	0
$787$	−43363.4	−1.96409	−0.982044	−	0.188651i	$-0.939588\pi$
$787$	−43363.4	−1.96409	−0.982044	+	0.188651i	$0.939588\pi$
$788$	−5141.59	−0.232439
$789$	0	0
$790$	−12454.0	−0.560878
$791$	0	0
$792$	0	0
$793$	−45324.6	−2.02966
$794$	28108.2	1.25632
$795$	0	0
$796$	10703.3	0.476593
$797$	17132.6	0.761439	0.380720	−	0.924691i	$-0.375676\pi$
$797$	17132.6	0.761439	0.380720	+	0.924691i	$0.375676\pi$
$798$	0	0
$799$	11923.3	0.527929
$800$	2841.69	0.125586
$801$	0	0
$802$	627.039	0.0276079
$803$	10795.1	0.474411
$804$	0	0
$805$	0	0
$806$	−26853.6	−1.17355
$807$	0	0
$808$	36317.5	1.58124
$809$	1080.49	0.0469566	0.0234783	−	0.999724i	$-0.492526\pi$
$809$	1080.49	0.0469566	0.0234783	+	0.999724i	$0.492526\pi$
$810$	0	0
$811$	−19593.9	−0.848378	−0.424189	−	0.905574i	$-0.639441\pi$
$811$	−19593.9	−0.848378	−0.424189	+	0.905574i	$0.639441\pi$
$812$	0	0
$813$	0	0
$814$	−1936.62	−0.0833889
$815$	−19769.9	−0.849706
$816$	0	0
$817$	358.680	0.0153594
$818$	−9312.66	−0.398056
$819$	0	0
$820$	−6341.69	−0.270075
$821$	−5123.80	−0.217810	−0.108905	−	0.994052i	$-0.534734\pi$
$821$	−5123.80	−0.217810	−0.108905	+	0.994052i	$0.534734\pi$
$822$	0	0
$823$	−13184.1	−0.558405	−0.279202	−	0.960232i	$-0.590070\pi$
$823$	−13184.1	−0.558405	−0.279202	+	0.960232i	$0.590070\pi$
$824$	−19977.3	−0.844591
$825$	0	0
$826$	0	0
$827$	−24658.7	−1.03684	−0.518421	−	0.855126i	$-0.673480\pi$
$827$	−24658.7	−1.03684	−0.518421	+	0.855126i	$0.673480\pi$
$828$	0	0
$829$	28562.3	1.19664	0.598318	−	0.801259i	$-0.295836\pi$
$829$	28562.3	1.19664	0.598318	+	0.801259i	$0.295836\pi$
$830$	7915.04	0.331006
$831$	0	0
$832$	33814.1	1.40901
$833$	0	0
$834$	0	0
$835$	16901.5	0.700481
$836$	−446.817	−0.0184850
$837$	0	0
$838$	5646.27	0.232753
$839$	31106.0	1.27997	0.639987	−	0.768386i	$-0.278940\pi$
$839$	31106.0	1.27997	0.639987	+	0.768386i	$0.278940\pi$
$840$	0	0
$841$	−23477.1	−0.962609
$842$	−11007.1	−0.450511
$843$	0	0
$844$	11389.7	0.464514
$845$	7823.38	0.318500
$846$	0	0
$847$	0	0
$848$	24047.8	0.973827
$849$	0	0
$850$	5868.73	0.236819
$851$	−1533.39	−0.0617674
$852$	0	0
$853$	20567.9	0.825596	0.412798	−	0.910823i	$-0.364552\pi$
$853$	20567.9	0.825596	0.412798	+	0.910823i	$0.364552\pi$
$854$	0	0
$855$	0	0
$856$	10854.9	0.433426
$857$	−6459.44	−0.257468	−0.128734	−	0.991679i	$-0.541091\pi$
$857$	−6459.44	−0.257468	−0.128734	+	0.991679i	$0.541091\pi$
$858$	0	0
$859$	−48214.4	−1.91508	−0.957541	−	0.288298i	$-0.906910\pi$
$859$	−48214.4	−1.91508	−0.957541	+	0.288298i	$0.906910\pi$
$860$	1287.65	0.0510565
$861$	0	0
$862$	−18359.7	−0.725445
$863$	31709.5	1.25076	0.625378	−	0.780322i	$-0.284945\pi$
$863$	31709.5	1.25076	0.625378	+	0.780322i	$0.284945\pi$
$864$	0	0
$865$	16033.3	0.630230
$866$	26611.7	1.04423
$867$	0	0
$868$	0	0
$869$	49745.1	1.94187
$870$	0	0
$871$	33879.8	1.31800
$872$	46955.6	1.82353
$873$	0	0
$874$	721.037	0.0279055
$875$	0	0
$876$	0	0
$877$	25654.8	0.987799	0.493900	−	0.869519i	$-0.335571\pi$
$877$	25654.8	0.987799	0.493900	+	0.869519i	$0.335571\pi$
$878$	21192.3	0.814585
$879$	0	0
$880$	8327.97	0.319018
$881$	11470.4	0.438647	0.219323	−	0.975652i	$-0.429615\pi$
$881$	11470.4	0.438647	0.219323	+	0.975652i	$0.429615\pi$
$882$	0	0
$883$	39124.0	1.49108	0.745542	−	0.666459i	$-0.232191\pi$
$883$	39124.0	1.49108	0.745542	+	0.666459i	$0.232191\pi$
$884$	−16365.6	−0.622663
$885$	0	0
$886$	−4319.01	−0.163770
$887$	−15585.8	−0.589987	−0.294994	−	0.955499i	$-0.595318\pi$
$887$	−15585.8	−0.589987	−0.294994	+	0.955499i	$0.595318\pi$
$888$	0	0
$889$	0	0
$890$	−15988.4	−0.602171
$891$	0	0
$892$	−2588.40	−0.0971594
$893$	−431.537	−0.0161711
$894$	0	0
$895$	−7213.25	−0.269399
$896$	0	0
$897$	0	0
$898$	−10403.7	−0.386610
$899$	5707.34	0.211736
$900$	0	0
$901$	−67689.6	−2.50285
$902$	−51625.6	−1.90570
$903$	0	0
$904$	23708.8	0.872280
$905$	4543.40	0.166881
$906$	0	0
$907$	27596.1	1.01027	0.505134	−	0.863041i	$-0.331443\pi$
$907$	27596.1	1.01027	0.505134	+	0.863041i	$0.331443\pi$
$908$	4373.73	0.159854
$909$	0	0
$910$	0	0
$911$	14396.2	0.523565	0.261782	−	0.965127i	$-0.415690\pi$
$911$	14396.2	0.523565	0.261782	+	0.965127i	$0.415690\pi$
$912$	0	0
$913$	−31615.1	−1.14601
$914$	−33229.2	−1.20254
$915$	0	0
$916$	−1512.35	−0.0545517
$917$	0	0
$918$	0	0
$919$	10279.6	0.368978	0.184489	−	0.982835i	$-0.440937\pi$
$919$	10279.6	0.368978	0.184489	+	0.982835i	$0.440937\pi$
$920$	10452.6	0.374577
$921$	0	0
$922$	−33727.0	−1.20471
$923$	45402.2	1.61910
$924$	0	0
$925$	451.713	0.0160565
$926$	−4192.06	−0.148769
$927$	0	0
$928$	−3432.53	−0.121421
$929$	6499.87	0.229552	0.114776	−	0.993391i	$-0.463385\pi$
$929$	6499.87	0.229552	0.114776	+	0.993391i	$0.463385\pi$
$930$	0	0
$931$	0	0
$932$	−6099.86	−0.214386
$933$	0	0
$934$	13857.2	0.485463
$935$	−23441.5	−0.819913
$936$	0	0
$937$	10269.8	0.358056	0.179028	−	0.983844i	$-0.442705\pi$
$937$	10269.8	0.358056	0.179028	+	0.983844i	$0.442705\pi$
$938$	0	0
$939$	0	0
$940$	−1549.21	−0.0537548
$941$	34396.2	1.19159	0.595794	−	0.803137i	$-0.296837\pi$
$941$	34396.2	1.19159	0.595794	+	0.803137i	$0.296837\pi$
$942$	0	0
$943$	−40876.5	−1.41158
$944$	−2064.06	−0.0711647
$945$	0	0
$946$	10482.4	0.360265
$947$	−27192.1	−0.933078	−0.466539	−	0.884501i	$-0.654499\pi$
$947$	−27192.1	−0.933078	−0.466539	+	0.884501i	$0.654499\pi$
$948$	0	0
$949$	−14310.4	−0.489500
$950$	−212.406	−0.00725406
$951$	0	0
$952$	0	0
$953$	−49965.2	−1.69836	−0.849178	−	0.528107i	$-0.822902\pi$
$953$	−49965.2	−1.69836	−0.849178	+	0.528107i	$0.822902\pi$
$954$	0	0
$955$	−12373.2	−0.419253
$956$	−9635.88	−0.325990
$957$	0	0
$958$	26704.9	0.900622
$959$	0	0
$960$	0	0
$961$	5929.05	0.199022
$962$	2567.25	0.0860411
$963$	0	0
$964$	6441.80	0.215225
$965$	17667.6	0.589368
$966$	0	0
$967$	−16755.5	−0.557208	−0.278604	−	0.960406i	$-0.589872\pi$
$967$	−16755.5	−0.557208	−0.278604	+	0.960406i	$0.589872\pi$
$968$	−19942.8	−0.662176
$969$	0	0
$970$	−2525.21	−0.0835872
$971$	−37617.4	−1.24325	−0.621626	−	0.783314i	$-0.713528\pi$
$971$	−37617.4	−1.24325	−0.621626	+	0.783314i	$0.713528\pi$
$972$	0	0
$973$	0	0
$974$	36583.0	1.20348
$975$	0	0
$976$	26603.9	0.872511
$977$	27690.9	0.906767	0.453384	−	0.891315i	$-0.350217\pi$
$977$	27690.9	0.906767	0.453384	+	0.891315i	$0.350217\pi$
$978$	0	0
$979$	63862.5	2.08483
$980$	0	0
$981$	0	0
$982$	−30266.4	−0.983544
$983$	−22754.2	−0.738299	−0.369149	−	0.929370i	$-0.620351\pi$
$983$	−22754.2	−0.738299	−0.369149	+	0.929370i	$0.620351\pi$
$984$	0	0
$985$	9762.83	0.315807
$986$	−7088.96	−0.228964
$987$	0	0
$988$	592.316	0.0190729
$989$	8299.80	0.266854
$990$	0	0
$991$	−55470.9	−1.77809	−0.889047	−	0.457816i	$-0.848632\pi$
$991$	−55470.9	−1.77809	−0.889047	+	0.457816i	$0.848632\pi$
$992$	−21482.9	−0.687583
$993$	0	0
$994$	0	0
$995$	−20323.4	−0.647531
$996$	0	0
$997$	15181.9	0.482264	0.241132	−	0.970492i	$-0.422481\pi$
$997$	15181.9	0.482264	0.241132	+	0.970492i	$0.422481\pi$
$998$	−46647.1	−1.47955
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.x.1.2		2
3.2	odd	2		245.4.a.i.1.1	✓	2
7.6	odd	2		2205.4.a.w.1.2		2
15.14	odd	2		1225.4.a.q.1.2		2
21.2	odd	6		245.4.e.k.116.2		4
21.5	even	6		245.4.e.j.116.2		4
21.11	odd	6		245.4.e.k.226.2		4
21.17	even	6		245.4.e.j.226.2		4
21.20	even	2		245.4.a.j.1.1	yes	2
105.104	even	2		1225.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.i.1.1	✓	2	3.2	odd	2
245.4.a.j.1.1	yes	2	21.20	even	2
245.4.e.j.116.2		4	21.5	even	6
245.4.e.j.226.2		4	21.17	even	6
245.4.e.k.116.2		4	21.2	odd	6
245.4.e.k.226.2		4	21.11	odd	6
1225.4.a.p.1.2		2	105.104	even	2
1225.4.a.q.1.2		2	15.14	odd	2
2205.4.a.w.1.2		2	7.6	odd	2
2205.4.a.x.1.2		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)

\(f(q)\)	\(=\)	\(q+2.31662 q^{2} -2.63325 q^{4} +5.00000 q^{5} -24.6332 q^{8} +O(q^{10})\)	\(q+2.31662 q^{2} -2.63325 q^{4} +5.00000 q^{5} -24.6332 q^{8} +11.5831 q^{10} -46.2665 q^{11} +61.3325 q^{13} -36.0000 q^{16} +101.332 q^{17} -3.66750 q^{19} -13.1662 q^{20} -107.182 q^{22} -84.8655 q^{23} +25.0000 q^{25} +142.084 q^{26} -30.1980 q^{29} -188.997 q^{31} +113.668 q^{32} +234.749 q^{34} +18.0685 q^{37} -8.49623 q^{38} -123.166 q^{40} +481.662 q^{41} -97.7995 q^{43} +121.831 q^{44} -196.602 q^{46} +117.665 q^{47} +57.9156 q^{50} -161.504 q^{52} -667.995 q^{53} -231.332 q^{55} -69.9574 q^{58} +57.3350 q^{59} -738.997 q^{61} -437.836 q^{62} +551.325 q^{64} +306.662 q^{65} +552.396 q^{67} -266.834 q^{68} +740.264 q^{71} -233.325 q^{73} +41.8580 q^{74} +9.65745 q^{76} -1075.19 q^{79} -180.000 q^{80} +1115.83 q^{82} +683.325 q^{83} +506.662 q^{85} -226.565 q^{86} +1139.69 q^{88} -1380.32 q^{89} +223.472 q^{92} +272.586 q^{94} -18.3375 q^{95} -218.008 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8	\( 2 q - 2 q^{2} + 8 q^{4} + 10 q^{5} - 36 q^{8} - 10 q^{10} - 66 q^{11} - 10 q^{13} - 72 q^{16} + 70 q^{17} - 140 q^{19} + 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} + 450 q^{26} + 258 q^{29} + 20 q^{31} + 360 q^{32} + 370 q^{34} + 328 q^{37} + 580 q^{38} - 180 q^{40} + 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} - 30 q^{47} - 50 q^{50} - 920 q^{52} - 540 q^{53} - 330 q^{55} - 1314 q^{58} + 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} - 600 q^{68} + 1056 q^{71} + 860 q^{73} - 1296 q^{74} - 1440 q^{76} + 158 q^{79} - 360 q^{80} + 1900 q^{82} + 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 910 q^{94} - 700 q^{95} - 1630 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 8 * q^4 + 10 * q^5 - 36 * q^8 - 10 * q^10 - 66 * q^11 - 10 * q^13 - 72 * q^16 + 70 * q^17 - 140 * q^19 + 40 * q^20 - 22 * q^22 + 16 * q^23 + 50 * q^25 + 450 * q^26 + 258 * q^29 + 20 * q^31 + 360 * q^32 + 370 * q^34 + 328 * q^37 + 580 * q^38 - 180 * q^40 + 300 * q^41 - 116 * q^43 - 88 * q^44 - 632 * q^46 - 30 * q^47 - 50 * q^50 - 920 * q^52 - 540 * q^53 - 330 * q^55 - 1314 * q^58 + 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 - 50 * q^65 + 468 * q^67 - 600 * q^68 + 1056 * q^71 + 860 * q^73 - 1296 * q^74 - 1440 * q^76 + 158 * q^79 - 360 * q^80 + 1900 * q^82 + 40 * q^83 + 350 * q^85 - 148 * q^86 + 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 910 * q^94 - 700 * q^95 - 1630 * q^97

Introduction

L-functions

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