LMFDB - Embedded newform 2205.4.a.bj.1.3

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:	$3$
Coefficient field:	3.3.4892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 4 $ x^3 - 11*x - 4
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			1.3
Root			$3.48535$ 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)

$f(q)$	$=$	$q+2.48535 q^{2} -1.82304 q^{4} +5.00000 q^{5} -24.4137 q^{8} +O(q^{10})$	$q+2.48535 q^{2} -1.82304 q^{4} +5.00000 q^{5} -24.4137 q^{8} +12.4267 q^{10} +26.0598 q^{11} -3.20425 q^{13} -46.0922 q^{16} -18.0325 q^{17} +34.2399 q^{19} -9.11519 q^{20} +64.7676 q^{22} +99.1824 q^{23} +25.0000 q^{25} -7.96368 q^{26} -212.096 q^{29} -293.170 q^{31} +80.7542 q^{32} -44.8170 q^{34} +20.7039 q^{37} +85.0981 q^{38} -122.068 q^{40} -207.842 q^{41} +5.96397 q^{43} -47.5079 q^{44} +246.503 q^{46} +401.899 q^{47} +62.1337 q^{50} +5.84147 q^{52} +7.31541 q^{53} +130.299 q^{55} -527.132 q^{58} +40.4089 q^{59} +358.484 q^{61} -728.631 q^{62} +569.440 q^{64} -16.0212 q^{65} +587.463 q^{67} +32.8739 q^{68} -803.405 q^{71} +656.051 q^{73} +51.4565 q^{74} -62.4206 q^{76} -1275.56 q^{79} -230.461 q^{80} -516.560 q^{82} +244.647 q^{83} -90.1624 q^{85} +14.8225 q^{86} -636.215 q^{88} -1411.08 q^{89} -180.813 q^{92} +998.859 q^{94} +171.199 q^{95} +952.212 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8	$ 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8} - 15 q^{10} + q^{11} + 79 q^{13} - 79 q^{16} - 72 q^{17} + 29 q^{19} + 5 q^{20} + 143 q^{22} + 63 q^{23} + 75 q^{25} - 339 q^{26} - 220 q^{29} - 136 q^{31} + 155 q^{32} + 220 q^{34} + 43 q^{37} - 21 q^{38} - 45 q^{40} - 599 q^{41} + 170 q^{43} - 135 q^{44} + 265 q^{46} - 3 q^{47} - 75 q^{50} + 701 q^{52} - 331 q^{53} + 5 q^{55} - 472 q^{58} - 1520 q^{59} + 1160 q^{61} - 748 q^{62} + 17 q^{64} + 395 q^{65} + 806 q^{67} - 684 q^{68} + 406 q^{71} + 1192 q^{73} - 959 q^{74} + 591 q^{76} - 2590 q^{79} - 395 q^{80} + 1191 q^{82} - 508 q^{83} - 360 q^{85} + 742 q^{86} - 749 q^{88} + 42 q^{89} + 211 q^{92} + 1167 q^{94} + 145 q^{95} + 1020 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8 - 15 * q^10 + q^11 + 79 * q^13 - 79 * q^16 - 72 * q^17 + 29 * q^19 + 5 * q^20 + 143 * q^22 + 63 * q^23 + 75 * q^25 - 339 * q^26 - 220 * q^29 - 136 * q^31 + 155 * q^32 + 220 * q^34 + 43 * q^37 - 21 * q^38 - 45 * q^40 - 599 * q^41 + 170 * q^43 - 135 * q^44 + 265 * q^46 - 3 * q^47 - 75 * q^50 + 701 * q^52 - 331 * q^53 + 5 * q^55 - 472 * q^58 - 1520 * q^59 + 1160 * q^61 - 748 * q^62 + 17 * q^64 + 395 * q^65 + 806 * q^67 - 684 * q^68 + 406 * q^71 + 1192 * q^73 - 959 * q^74 + 591 * q^76 - 2590 * q^79 - 395 * q^80 + 1191 * q^82 - 508 * q^83 - 360 * q^85 + 742 * q^86 - 749 * q^88 + 42 * q^89 + 211 * q^92 + 1167 * q^94 + 145 * q^95 + 1020 * 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 $

$2$	2.48535	0.878704	0.439352	−	0.898315i	$-0.355208\pi$
$2$	2.48535	0.878704	0.439352	+	0.898315i	$0.355208\pi$
$3$	0	0
$3$	0	0
$4$	−1.82304	−0.227880
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−24.4137	−1.07894
$9$	0	0
$10$	12.4267	0.392968
$11$	26.0598	0.714301	0.357151	−	0.934047i	$-0.383748\pi$
$11$	26.0598	0.714301	0.357151	+	0.934047i	$0.383748\pi$
$12$	0	0
$13$	−3.20425	−0.0683614	−0.0341807	−	0.999416i	$-0.510882\pi$
$13$	−3.20425	−0.0683614	−0.0341807	+	0.999416i	$0.510882\pi$
$14$	0	0
$15$	0	0
$16$	−46.0922	−0.720191
$17$	−18.0325	−0.257266	−0.128633	−	0.991692i	$-0.541059\pi$
$17$	−18.0325	−0.257266	−0.128633	+	0.991692i	$0.541059\pi$
$18$	0	0
$19$	34.2399	0.413430	0.206715	−	0.978401i	$-0.433723\pi$
$19$	34.2399	0.413430	0.206715	+	0.978401i	$0.433723\pi$
$20$	−9.11519	−0.101911
$21$	0	0
$22$	64.7676	0.627659
$23$	99.1824	0.899173	0.449586	−	0.893237i	$-0.351571\pi$
$23$	99.1824	0.899173	0.449586	+	0.893237i	$0.351571\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−7.96368	−0.0600694
$27$	0	0
$28$	0	0
$29$	−212.096	−1.35811	−0.679055	−	0.734088i	$-0.737610\pi$
$29$	−212.096	−1.35811	−0.679055	+	0.734088i	$0.737610\pi$
$30$	0	0
$31$	−293.170	−1.69855	−0.849274	−	0.527953i	$-0.822960\pi$
$31$	−293.170	−1.69855	−0.849274	+	0.527953i	$0.822960\pi$
$32$	80.7542	0.446108
$33$	0	0
$34$	−44.8170	−0.226060
$35$	0	0
$36$	0	0
$37$	20.7039	0.0919920	0.0459960	−	0.998942i	$-0.485354\pi$
$37$	20.7039	0.0919920	0.0459960	+	0.998942i	$0.485354\pi$
$38$	85.0981	0.363282
$39$	0	0
$40$	−122.068	−0.482518
$41$	−207.842	−0.791695	−0.395847	−	0.918316i	$-0.629549\pi$
$41$	−207.842	−0.791695	−0.395847	+	0.918316i	$0.629549\pi$
$42$	0	0
$43$	5.96397	0.0211511	0.0105755	−	0.999944i	$-0.496634\pi$
$43$	5.96397	0.0211511	0.0105755	+	0.999944i	$0.496634\pi$
$44$	−47.5079	−0.162775
$45$	0	0
$46$	246.503	0.790106
$47$	401.899	1.24730	0.623649	−	0.781705i	$-0.285650\pi$
$47$	401.899	1.24730	0.623649	+	0.781705i	$0.285650\pi$
$48$	0	0
$49$	0	0
$50$	62.1337	0.175741
$51$	0	0
$52$	5.84147	0.0155782
$53$	7.31541	0.0189594	0.00947970	−	0.999955i	$-0.496982\pi$
$53$	7.31541	0.0189594	0.00947970	+	0.999955i	$0.496982\pi$
$54$	0	0
$55$	130.299	0.319445
$56$	0	0
$57$	0	0
$58$	−527.132	−1.19338
$59$	40.4089	0.0891660	0.0445830	−	0.999006i	$-0.485804\pi$
$59$	40.4089	0.0891660	0.0445830	+	0.999006i	$0.485804\pi$
$60$	0	0
$61$	358.484	0.752445	0.376223	−	0.926529i	$-0.377223\pi$
$61$	358.484	0.752445	0.376223	+	0.926529i	$0.377223\pi$
$62$	−728.631	−1.49252
$63$	0	0
$64$	569.440	1.11219
$65$	−16.0212	−0.0305722
$66$	0	0
$67$	587.463	1.07119	0.535597	−	0.844474i	$-0.320087\pi$
$67$	587.463	1.07119	0.535597	+	0.844474i	$0.320087\pi$
$68$	32.8739	0.0586256
$69$	0	0
$70$	0	0
$71$	−803.405	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$71$	−803.405	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$72$	0	0
$73$	656.051	1.05185	0.525924	−	0.850531i	$-0.323720\pi$
$73$	656.051	1.05185	0.525924	+	0.850531i	$0.323720\pi$
$74$	51.4565	0.0808337
$75$	0	0
$76$	−62.4206	−0.0942123
$77$	0	0
$78$	0	0
$79$	−1275.56	−1.81660	−0.908302	−	0.418314i	$-0.862621\pi$
$79$	−1275.56	−1.81660	−0.908302	+	0.418314i	$0.862621\pi$
$80$	−230.461	−0.322079
$81$	0	0
$82$	−516.560	−0.695665
$83$	244.647	0.323537	0.161768	−	0.986829i	$-0.448280\pi$
$83$	244.647	0.323537	0.161768	+	0.986829i	$0.448280\pi$
$84$	0	0
$85$	−90.1624	−0.115053
$86$	14.8225	0.0185855
$87$	0	0
$88$	−636.215	−0.770690
$89$	−1411.08	−1.68061	−0.840307	−	0.542112i	$-0.817625\pi$
$89$	−1411.08	−1.68061	−0.840307	+	0.542112i	$0.817625\pi$
$90$	0	0
$91$	0	0
$92$	−180.813	−0.204903
$93$	0	0
$94$	998.859	1.09600
$95$	171.199	0.184892
$96$	0	0
$97$	952.212	0.996727	0.498363	−	0.866968i	$-0.333935\pi$
$97$	952.212	0.996727	0.498363	+	0.866968i	$0.333935\pi$
$98$	0	0
$99$	0	0
$100$	−45.5759	−0.0455759
$101$	−1441.46	−1.42011	−0.710055	−	0.704146i	$-0.751330\pi$
$101$	−1441.46	−1.42011	−0.710055	+	0.704146i	$0.751330\pi$
$102$	0	0
$103$	−1139.99	−1.09055	−0.545273	−	0.838259i	$-0.683574\pi$
$103$	−1139.99	−1.09055	−0.545273	+	0.838259i	$0.683574\pi$
$104$	78.2275	0.0737581
$105$	0	0
$106$	18.1813	0.0166597
$107$	−693.404	−0.626485	−0.313242	−	0.949673i	$-0.601415\pi$
$107$	−693.404	−0.626485	−0.313242	+	0.949673i	$0.601415\pi$
$108$	0	0
$109$	−1140.12	−1.00187	−0.500933	−	0.865486i	$-0.667010\pi$
$109$	−1140.12	−1.00187	−0.500933	+	0.865486i	$0.667010\pi$
$110$	323.838	0.280698
$111$	0	0
$112$	0	0
$113$	−1154.64	−0.961237	−0.480618	−	0.876930i	$-0.659588\pi$
$113$	−1154.64	−0.961237	−0.480618	+	0.876930i	$0.659588\pi$
$114$	0	0
$115$	495.912	0.402122
$116$	386.658	0.309486
$117$	0	0
$118$	100.430	0.0783505
$119$	0	0
$120$	0	0
$121$	−651.889	−0.489774
$122$	890.958	0.661176
$123$	0	0
$124$	534.461	0.387065
$125$	125.000	0.0894427
$126$	0	0
$127$	−1453.32	−1.01544	−0.507722	−	0.861521i	$-0.669512\pi$
$127$	−1453.32	−1.01544	−0.507722	+	0.861521i	$0.669512\pi$
$128$	769.225	0.531176
$129$	0	0
$130$	−39.8184	−0.0268639
$131$	−1539.58	−1.02682	−0.513410	−	0.858143i	$-0.671618\pi$
$131$	−1539.58	−1.02682	−0.513410	+	0.858143i	$0.671618\pi$
$132$	0	0
$133$	0	0
$134$	1460.05	0.941262
$135$	0	0
$136$	440.239	0.277575
$137$	−2915.54	−1.81819	−0.909093	−	0.416594i	$-0.863224\pi$
$137$	−2915.54	−1.81819	−0.909093	+	0.416594i	$0.863224\pi$
$138$	0	0
$139$	2338.76	1.42713	0.713565	−	0.700589i	$-0.247080\pi$
$139$	2338.76	1.42713	0.713565	+	0.700589i	$0.247080\pi$
$140$	0	0
$141$	0	0
$142$	−1996.74	−1.18002
$143$	−83.5019	−0.0488307
$144$	0	0
$145$	−1060.48	−0.607365
$146$	1630.52	0.924263
$147$	0	0
$148$	−37.7440	−0.0209631
$149$	492.176	0.270608	0.135304	−	0.990804i	$-0.456799\pi$
$149$	492.176	0.270608	0.135304	+	0.990804i	$0.456799\pi$
$150$	0	0
$151$	1081.95	0.583098	0.291549	−	0.956556i	$-0.405829\pi$
$151$	1081.95	0.583098	0.291549	+	0.956556i	$0.405829\pi$
$152$	−835.922	−0.446067
$153$	0	0
$154$	0	0
$155$	−1465.85	−0.759613
$156$	0	0
$157$	2891.31	1.46975	0.734877	−	0.678200i	$-0.237240\pi$
$157$	2891.31	1.46975	0.734877	+	0.678200i	$0.237240\pi$
$158$	−3170.21	−1.59626
$159$	0	0
$160$	403.771	0.199506
$161$	0	0
$162$	0	0
$163$	2525.05	1.21336	0.606680	−	0.794946i	$-0.292501\pi$
$163$	2525.05	1.21336	0.606680	+	0.794946i	$0.292501\pi$
$164$	378.904	0.180411
$165$	0	0
$166$	608.034	0.284293
$167$	−2891.98	−1.34005	−0.670025	−	0.742338i	$-0.733717\pi$
$167$	−2891.98	−1.34005	−0.670025	+	0.742338i	$0.733717\pi$
$168$	0	0
$169$	−2186.73	−0.995327
$170$	−224.085	−0.101097
$171$	0	0
$172$	−10.8725	−0.00481990
$173$	−2904.94	−1.27664	−0.638320	−	0.769771i	$-0.720370\pi$
$173$	−2904.94	−1.27664	−0.638320	+	0.769771i	$0.720370\pi$
$174$	0	0
$175$	0	0
$176$	−1201.15	−0.514433
$177$	0	0
$178$	−3507.04	−1.47676
$179$	−23.1041	−0.00964737	−0.00482369	−	0.999988i	$-0.501535\pi$
$179$	−23.1041	−0.00964737	−0.00482369	+	0.999988i	$0.501535\pi$
$180$	0	0
$181$	3024.01	1.24184	0.620919	−	0.783875i	$-0.286760\pi$
$181$	3024.01	1.24184	0.620919	+	0.783875i	$0.286760\pi$
$182$	0	0
$183$	0	0
$184$	−2421.41	−0.970156
$185$	103.520	0.0411401
$186$	0	0
$187$	−469.922	−0.183765
$188$	−732.677	−0.284234
$189$	0	0
$190$	425.490	0.162465
$191$	−3896.90	−1.47628	−0.738140	−	0.674648i	$-0.764296\pi$
$191$	−3896.90	−1.47628	−0.738140	+	0.674648i	$0.764296\pi$
$192$	0	0
$193$	−4609.28	−1.71909	−0.859543	−	0.511064i	$-0.829252\pi$
$193$	−4609.28	−1.71909	−0.859543	+	0.511064i	$0.829252\pi$
$194$	2366.58	0.875828
$195$	0	0
$196$	0	0
$197$	−3119.11	−1.12806	−0.564029	−	0.825755i	$-0.690749\pi$
$197$	−3119.11	−1.12806	−0.564029	+	0.825755i	$0.690749\pi$
$198$	0	0
$199$	4148.93	1.47794	0.738970	−	0.673738i	$-0.235313\pi$
$199$	4148.93	1.47794	0.738970	+	0.673738i	$0.235313\pi$
$200$	−610.342	−0.215789
$201$	0	0
$202$	−3582.54	−1.24786
$203$	0	0
$204$	0	0
$205$	−1039.21	−0.354057
$206$	−2833.26	−0.958266
$207$	0	0
$208$	147.691	0.0492333
$209$	892.283	0.295314
$210$	0	0
$211$	−3835.74	−1.25148	−0.625742	−	0.780030i	$-0.715204\pi$
$211$	−3835.74	−1.25148	−0.625742	+	0.780030i	$0.715204\pi$
$212$	−13.3363	−0.00432047
$213$	0	0
$214$	−1723.35	−0.550495
$215$	29.8198	0.00945905
$216$	0	0
$217$	0	0
$218$	−2833.59	−0.880344
$219$	0	0
$220$	−237.540	−0.0727951
$221$	57.7805	0.0175871
$222$	0	0
$223$	750.330	0.225318	0.112659	−	0.993634i	$-0.464063\pi$
$223$	750.330	0.225318	0.112659	+	0.993634i	$0.464063\pi$
$224$	0	0
$225$	0	0
$226$	−2869.69	−0.844643
$227$	−628.839	−0.183866	−0.0919329	−	0.995765i	$-0.529305\pi$
$227$	−628.839	−0.183866	−0.0919329	+	0.995765i	$0.529305\pi$
$228$	0	0
$229$	2447.33	0.706219	0.353109	−	0.935582i	$-0.385124\pi$
$229$	2447.33	0.706219	0.353109	+	0.935582i	$0.385124\pi$
$230$	1232.52	0.353346
$231$	0	0
$232$	5178.04	1.46532
$233$	−3161.16	−0.888818	−0.444409	−	0.895824i	$-0.646586\pi$
$233$	−3161.16	−0.888818	−0.444409	+	0.895824i	$0.646586\pi$
$234$	0	0
$235$	2009.49	0.557808
$236$	−73.6670	−0.0203191
$237$	0	0
$238$	0	0
$239$	6018.30	1.62884	0.814418	−	0.580279i	$-0.197056\pi$
$239$	6018.30	1.62884	0.814418	+	0.580279i	$0.197056\pi$
$240$	0	0
$241$	−5718.43	−1.52845	−0.764225	−	0.644950i	$-0.776878\pi$
$241$	−5718.43	−1.52845	−0.764225	+	0.644950i	$0.776878\pi$
$242$	−1620.17	−0.430366
$243$	0	0
$244$	−653.530	−0.171467
$245$	0	0
$246$	0	0
$247$	−109.713	−0.0282627
$248$	7157.37	1.83263
$249$	0	0
$250$	310.669	0.0785937
$251$	1236.06	0.310834	0.155417	−	0.987849i	$-0.450328\pi$
$251$	1236.06	0.310834	0.155417	+	0.987849i	$0.450328\pi$
$252$	0	0
$253$	2584.67	0.642280
$254$	−3612.01	−0.892275
$255$	0	0
$256$	−2643.73	−0.645442
$257$	−1098.93	−0.266730	−0.133365	−	0.991067i	$-0.542578\pi$
$257$	−1098.93	−0.266730	−0.133365	+	0.991067i	$0.542578\pi$
$258$	0	0
$259$	0	0
$260$	29.2073	0.00696678
$261$	0	0
$262$	−3826.39	−0.902270
$263$	6723.48	1.57638	0.788189	−	0.615433i	$-0.211019\pi$
$263$	6723.48	1.57638	0.788189	+	0.615433i	$0.211019\pi$
$264$	0	0
$265$	36.5770	0.00847891
$266$	0	0
$267$	0	0
$268$	−1070.97	−0.244103
$269$	5140.13	1.16505	0.582527	−	0.812812i	$-0.302064\pi$
$269$	5140.13	1.16505	0.582527	+	0.812812i	$0.302064\pi$
$270$	0	0
$271$	−2915.01	−0.653410	−0.326705	−	0.945126i	$-0.605938\pi$
$271$	−2915.01	−0.653410	−0.326705	+	0.945126i	$0.605938\pi$
$272$	831.157	0.185280
$273$	0	0
$274$	−7246.14	−1.59765
$275$	651.494	0.142860
$276$	0	0
$277$	1785.93	0.387388	0.193694	−	0.981062i	$-0.437953\pi$
$277$	1785.93	0.387388	0.193694	+	0.981062i	$0.437953\pi$
$278$	5812.63	1.25402
$279$	0	0
$280$	0	0
$281$	−1264.76	−0.268503	−0.134252	−	0.990947i	$-0.542863\pi$
$281$	−1264.76	−0.268503	−0.134252	+	0.990947i	$0.542863\pi$
$282$	0	0
$283$	−8366.40	−1.75735	−0.878677	−	0.477418i	$-0.841573\pi$
$283$	−8366.40	−1.75735	−0.878677	+	0.477418i	$0.841573\pi$
$284$	1464.64	0.306022
$285$	0	0
$286$	−207.532	−0.0429077
$287$	0	0
$288$	0	0
$289$	−4587.83	−0.933814
$290$	−2635.66	−0.533694
$291$	0	0
$292$	−1196.01	−0.239695
$293$	−1552.76	−0.309602	−0.154801	−	0.987946i	$-0.549474\pi$
$293$	−1552.76	−0.309602	−0.154801	+	0.987946i	$0.549474\pi$
$294$	0	0
$295$	202.045	0.0398762
$296$	−505.459	−0.0992540
$297$	0	0
$298$	1223.23	0.237784
$299$	−317.805	−0.0614687
$300$	0	0
$301$	0	0
$302$	2689.02	0.512370
$303$	0	0
$304$	−1578.19	−0.297749
$305$	1792.42	0.336504
$306$	0	0
$307$	6847.06	1.27291	0.636453	−	0.771315i	$-0.280401\pi$
$307$	6847.06	1.27291	0.636453	+	0.771315i	$0.280401\pi$
$308$	0	0
$309$	0	0
$310$	−3643.16	−0.667475
$311$	2350.26	0.428524	0.214262	−	0.976776i	$-0.431265\pi$
$311$	2350.26	0.428524	0.214262	+	0.976776i	$0.431265\pi$
$312$	0	0
$313$	4091.85	0.738929	0.369465	−	0.929245i	$-0.379541\pi$
$313$	4091.85	0.738929	0.369465	+	0.929245i	$0.379541\pi$
$314$	7185.91	1.29148
$315$	0	0
$316$	2325.40	0.413967
$317$	993.691	0.176061	0.0880304	−	0.996118i	$-0.471943\pi$
$317$	993.691	0.176061	0.0880304	+	0.996118i	$0.471943\pi$
$318$	0	0
$319$	−5527.16	−0.970099
$320$	2847.20	0.497385
$321$	0	0
$322$	0	0
$323$	−617.430	−0.106361
$324$	0	0
$325$	−80.1062	−0.0136723
$326$	6275.64	1.06618
$327$	0	0
$328$	5074.19	0.854193
$329$	0	0
$330$	0	0
$331$	3060.14	0.508159	0.254079	−	0.967183i	$-0.418228\pi$
$331$	3060.14	0.508159	0.254079	+	0.967183i	$0.418228\pi$
$332$	−446.001	−0.0737274
$333$	0	0
$334$	−7187.59	−1.17751
$335$	2937.31	0.479052
$336$	0	0
$337$	−2761.72	−0.446411	−0.223205	−	0.974771i	$-0.571652\pi$
$337$	−2761.72	−0.446411	−0.223205	+	0.974771i	$0.571652\pi$
$338$	−5434.80	−0.874597
$339$	0	0
$340$	164.369	0.0262182
$341$	−7639.95	−1.21327
$342$	0	0
$343$	0	0
$344$	−145.602	−0.0228208
$345$	0	0
$346$	−7219.80	−1.12179
$347$	−12530.1	−1.93847	−0.969235	−	0.246138i	$-0.920838\pi$
$347$	−12530.1	−1.93847	−0.969235	+	0.246138i	$0.920838\pi$
$348$	0	0
$349$	−3744.12	−0.574263	−0.287132	−	0.957891i	$-0.592702\pi$
$349$	−3744.12	−0.574263	−0.287132	+	0.957891i	$0.592702\pi$
$350$	0	0
$351$	0	0
$352$	2104.43	0.318655
$353$	−2244.03	−0.338351	−0.169175	−	0.985586i	$-0.554110\pi$
$353$	−2244.03	−0.338351	−0.169175	+	0.985586i	$0.554110\pi$
$354$	0	0
$355$	−4017.02	−0.600567
$356$	2572.46	0.382978
$357$	0	0
$358$	−57.4217	−0.00847718
$359$	941.177	0.138366	0.0691830	−	0.997604i	$-0.477961\pi$
$359$	941.177	0.138366	0.0691830	+	0.997604i	$0.477961\pi$
$360$	0	0
$361$	−5686.63	−0.829076
$362$	7515.71	1.09121
$363$	0	0
$364$	0	0
$365$	3280.25	0.470401
$366$	0	0
$367$	−484.224	−0.0688727	−0.0344363	−	0.999407i	$-0.510964\pi$
$367$	−484.224	−0.0688727	−0.0344363	+	0.999407i	$0.510964\pi$
$368$	−4571.54	−0.647576
$369$	0	0
$370$	257.282	0.0361499
$371$	0	0
$372$	0	0
$373$	−10422.2	−1.44677	−0.723383	−	0.690447i	$-0.757414\pi$
$373$	−10422.2	−1.44677	−0.723383	+	0.690447i	$0.757414\pi$
$374$	−1167.92	−0.161475
$375$	0	0
$376$	−9811.83	−1.34576
$377$	679.607	0.0928423
$378$	0	0
$379$	−6388.05	−0.865784	−0.432892	−	0.901446i	$-0.642507\pi$
$379$	−6388.05	−0.865784	−0.432892	+	0.901446i	$0.642507\pi$
$380$	−312.103	−0.0421330
$381$	0	0
$382$	−9685.15	−1.29721
$383$	−4982.79	−0.664775	−0.332387	−	0.943143i	$-0.607854\pi$
$383$	−4982.79	−0.664775	−0.332387	+	0.943143i	$0.607854\pi$
$384$	0	0
$385$	0	0
$386$	−11455.7	−1.51057
$387$	0	0
$388$	−1735.92	−0.227134
$389$	6748.65	0.879615	0.439807	−	0.898092i	$-0.355047\pi$
$389$	6748.65	0.879615	0.439807	+	0.898092i	$0.355047\pi$
$390$	0	0
$391$	−1788.50	−0.231326
$392$	0	0
$393$	0	0
$394$	−7752.08	−0.991228
$395$	−6377.80	−0.812410
$396$	0	0
$397$	−3982.29	−0.503439	−0.251720	−	0.967800i	$-0.580996\pi$
$397$	−3982.29	−0.503439	−0.251720	+	0.967800i	$0.580996\pi$
$398$	10311.5	1.29867
$399$	0	0
$400$	−1152.31	−0.144038
$401$	−6568.27	−0.817965	−0.408982	−	0.912542i	$-0.634116\pi$
$401$	−6568.27	−0.817965	−0.408982	+	0.912542i	$0.634116\pi$
$402$	0	0
$403$	939.391	0.116115
$404$	2627.84	0.323614
$405$	0	0
$406$	0	0
$407$	539.539	0.0657100
$408$	0	0
$409$	4956.77	0.599258	0.299629	−	0.954056i	$-0.403137\pi$
$409$	4956.77	0.599258	0.299629	+	0.954056i	$0.403137\pi$
$410$	−2582.80	−0.311111
$411$	0	0
$412$	2078.24	0.248513
$413$	0	0
$414$	0	0
$415$	1223.24	0.144690
$416$	−258.756	−0.0304966
$417$	0	0
$418$	2217.64	0.259493
$419$	5778.60	0.673754	0.336877	−	0.941549i	$-0.390629\pi$
$419$	5778.60	0.673754	0.336877	+	0.941549i	$0.390629\pi$
$420$	0	0
$421$	7463.23	0.863980	0.431990	−	0.901878i	$-0.357812\pi$
$421$	7463.23	0.863980	0.431990	+	0.901878i	$0.357812\pi$
$422$	−9533.16	−1.09968
$423$	0	0
$424$	−178.596	−0.0204561
$425$	−450.812	−0.0514531
$426$	0	0
$427$	0	0
$428$	1264.10	0.142763
$429$	0	0
$430$	74.1127	0.00831170
$431$	−3041.58	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$431$	−3041.58	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$432$	0	0
$433$	−4047.48	−0.449214	−0.224607	−	0.974449i	$-0.572110\pi$
$433$	−4047.48	−0.449214	−0.224607	+	0.974449i	$0.572110\pi$
$434$	0	0
$435$	0	0
$436$	2078.48	0.228305
$437$	3396.00	0.371745
$438$	0	0
$439$	2732.91	0.297118	0.148559	−	0.988904i	$-0.452537\pi$
$439$	2732.91	0.297118	0.148559	+	0.988904i	$0.452537\pi$
$440$	−3181.07	−0.344663
$441$	0	0
$442$	143.605	0.0154538
$443$	−6399.04	−0.686293	−0.343146	−	0.939282i	$-0.611493\pi$
$443$	−6399.04	−0.686293	−0.343146	+	0.939282i	$0.611493\pi$
$444$	0	0
$445$	−7055.42	−0.751593
$446$	1864.83	0.197987
$447$	0	0
$448$	0	0
$449$	16489.7	1.73318	0.866588	−	0.499024i	$-0.166308\pi$
$449$	16489.7	1.73318	0.866588	+	0.499024i	$0.166308\pi$
$450$	0	0
$451$	−5416.31	−0.565508
$452$	2104.96	0.219046
$453$	0	0
$454$	−1562.89	−0.161564
$455$	0	0
$456$	0	0
$457$	14375.9	1.47151	0.735753	−	0.677250i	$-0.236829\pi$
$457$	14375.9	1.47151	0.735753	+	0.677250i	$0.236829\pi$
$458$	6082.47	0.620557
$459$	0	0
$460$	−904.067	−0.0916355
$461$	1000.39	0.101069	0.0505346	−	0.998722i	$-0.483907\pi$
$461$	1000.39	0.101069	0.0505346	+	0.998722i	$0.483907\pi$
$462$	0	0
$463$	−626.125	−0.0628477	−0.0314239	−	0.999506i	$-0.510004\pi$
$463$	−626.125	−0.0628477	−0.0314239	+	0.999506i	$0.510004\pi$
$464$	9775.96	0.978099
$465$	0	0
$466$	−7856.60	−0.781008
$467$	919.040	0.0910665	0.0455333	−	0.998963i	$-0.485501\pi$
$467$	919.040	0.0910665	0.0455333	+	0.998963i	$0.485501\pi$
$468$	0	0
$469$	0	0
$470$	4994.29	0.490148
$471$	0	0
$472$	−986.531	−0.0962050
$473$	155.420	0.0151082
$474$	0	0
$475$	855.997	0.0826860
$476$	0	0
$477$	0	0
$478$	14957.6	1.43126
$479$	−15663.7	−1.49414	−0.747068	−	0.664747i	$-0.768539\pi$
$479$	−15663.7	−1.49414	−0.747068	+	0.664747i	$0.768539\pi$
$480$	0	0
$481$	−66.3405	−0.00628870
$482$	−14212.3	−1.34305
$483$	0	0
$484$	1188.42	0.111610
$485$	4761.06	0.445750
$486$	0	0
$487$	14440.5	1.34365	0.671827	−	0.740708i	$-0.265510\pi$
$487$	14440.5	1.34365	0.671827	+	0.740708i	$0.265510\pi$
$488$	−8751.91	−0.811845
$489$	0	0
$490$	0	0
$491$	19320.8	1.77584	0.887918	−	0.460001i	$-0.152151\pi$
$491$	19320.8	1.77584	0.887918	+	0.460001i	$0.152151\pi$
$492$	0	0
$493$	3824.61	0.349395
$494$	−272.675	−0.0248345
$495$	0	0
$496$	13512.9	1.22328
$497$	0	0
$498$	0	0
$499$	−4533.19	−0.406680	−0.203340	−	0.979108i	$-0.565180\pi$
$499$	−4533.19	−0.406680	−0.203340	+	0.979108i	$0.565180\pi$
$500$	−227.880	−0.0203822
$501$	0	0
$502$	3072.04	0.273131
$503$	1296.32	0.114911	0.0574554	−	0.998348i	$-0.481701\pi$
$503$	1296.32	0.114911	0.0574554	+	0.998348i	$0.481701\pi$
$504$	0	0
$505$	−7207.32	−0.635092
$506$	6423.81	0.564374
$507$	0	0
$508$	2649.46	0.231399
$509$	−6533.61	−0.568953	−0.284477	−	0.958683i	$-0.591820\pi$
$509$	−6533.61	−0.568953	−0.284477	+	0.958683i	$0.591820\pi$
$510$	0	0
$511$	0	0
$512$	−12724.4	−1.09833
$513$	0	0
$514$	−2731.24	−0.234377
$515$	−5699.93	−0.487707
$516$	0	0
$517$	10473.4	0.890946
$518$	0	0
$519$	0	0
$520$	391.138	0.0329856
$521$	823.583	0.0692549	0.0346275	−	0.999400i	$-0.488976\pi$
$521$	823.583	0.0692549	0.0346275	+	0.999400i	$0.488976\pi$
$522$	0	0
$523$	17252.5	1.44244	0.721221	−	0.692705i	$-0.243581\pi$
$523$	17252.5	1.44244	0.721221	+	0.692705i	$0.243581\pi$
$524$	2806.71	0.233991
$525$	0	0
$526$	16710.2	1.38517
$527$	5286.59	0.436978
$528$	0	0
$529$	−2329.84	−0.191489
$530$	90.9067	0.00745045
$531$	0	0
$532$	0	0
$533$	665.978	0.0541214
$534$	0	0
$535$	−3467.02	−0.280173
$536$	−14342.1	−1.15576
$537$	0	0
$538$	12775.0	1.02374
$539$	0	0
$540$	0	0
$541$	−17144.9	−1.36250	−0.681252	−	0.732049i	$-0.738564\pi$
$541$	−17144.9	−1.36250	−0.681252	+	0.732049i	$0.738564\pi$
$542$	−7244.81	−0.574154
$543$	0	0
$544$	−1456.20	−0.114768
$545$	−5700.59	−0.448048
$546$	0	0
$547$	−14217.3	−1.11131	−0.555656	−	0.831412i	$-0.687533\pi$
$547$	−14217.3	−1.11131	−0.555656	+	0.831412i	$0.687533\pi$
$548$	5315.14	0.414328
$549$	0	0
$550$	1619.19	0.125532
$551$	−7262.13	−0.561483
$552$	0	0
$553$	0	0
$554$	4438.67	0.340399
$555$	0	0
$556$	−4263.65	−0.325214
$557$	19391.8	1.47515	0.737575	−	0.675266i	$-0.235971\pi$
$557$	19391.8	1.47515	0.737575	+	0.675266i	$0.235971\pi$
$558$	0	0
$559$	−19.1100	−0.00144592
$560$	0	0
$561$	0	0
$562$	−3143.38	−0.235935
$563$	−4162.41	−0.311589	−0.155794	−	0.987789i	$-0.549794\pi$
$563$	−4162.41	−0.311589	−0.155794	+	0.987789i	$0.549794\pi$
$564$	0	0
$565$	−5773.22	−0.429878
$566$	−20793.4	−1.54419
$567$	0	0
$568$	19614.1	1.44892
$569$	10696.1	0.788057	0.394028	−	0.919098i	$-0.371081\pi$
$569$	10696.1	0.788057	0.394028	+	0.919098i	$0.371081\pi$
$570$	0	0
$571$	12471.0	0.913998	0.456999	−	0.889467i	$-0.348924\pi$
$571$	12471.0	0.913998	0.456999	+	0.889467i	$0.348924\pi$
$572$	152.227	0.0111275
$573$	0	0
$574$	0	0
$575$	2479.56	0.179835
$576$	0	0
$577$	15702.0	1.13290	0.566451	−	0.824095i	$-0.308316\pi$
$577$	15702.0	1.13290	0.566451	+	0.824095i	$0.308316\pi$
$578$	−11402.4	−0.820546
$579$	0	0
$580$	1933.29	0.138406
$581$	0	0
$582$	0	0
$583$	190.638	0.0135427
$584$	−16016.6	−1.13488
$585$	0	0
$586$	−3859.16	−0.272049
$587$	22423.3	1.57668	0.788339	−	0.615241i	$-0.210941\pi$
$587$	22423.3	1.57668	0.788339	+	0.615241i	$0.210941\pi$
$588$	0	0
$589$	−10038.1	−0.702230
$590$	502.151	0.0350394
$591$	0	0
$592$	−954.289	−0.0662518
$593$	533.937	0.0369750	0.0184875	−	0.999829i	$-0.494115\pi$
$593$	533.937	0.0369750	0.0184875	+	0.999829i	$0.494115\pi$
$594$	0	0
$595$	0	0
$596$	−897.255	−0.0616661
$597$	0	0
$598$	−789.857	−0.0540128
$599$	5626.60	0.383801	0.191900	−	0.981414i	$-0.438535\pi$
$599$	5626.60	0.383801	0.191900	+	0.981414i	$0.438535\pi$
$600$	0	0
$601$	−8563.09	−0.581191	−0.290596	−	0.956846i	$-0.593853\pi$
$601$	−8563.09	−0.581191	−0.290596	+	0.956846i	$0.593853\pi$
$602$	0	0
$603$	0	0
$604$	−1972.43	−0.132876
$605$	−3259.45	−0.219034
$606$	0	0
$607$	20733.3	1.38639	0.693194	−	0.720751i	$-0.256203\pi$
$607$	20733.3	1.38639	0.693194	+	0.720751i	$0.256203\pi$
$608$	2765.01	0.184434
$609$	0	0
$610$	4454.79	0.295687
$611$	−1287.78	−0.0852670
$612$	0	0
$613$	12450.6	0.820349	0.410174	−	0.912007i	$-0.365468\pi$
$613$	12450.6	0.820349	0.410174	+	0.912007i	$0.365468\pi$
$614$	17017.3	1.11851
$615$	0	0
$616$	0	0
$617$	−6173.86	−0.402837	−0.201418	−	0.979505i	$-0.564555\pi$
$617$	−6173.86	−0.402837	−0.201418	+	0.979505i	$0.564555\pi$
$618$	0	0
$619$	26671.6	1.73186	0.865930	−	0.500165i	$-0.166727\pi$
$619$	26671.6	1.73186	0.865930	+	0.500165i	$0.166727\pi$
$620$	2672.30	0.173101
$621$	0	0
$622$	5841.22	0.376546
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	10169.7	0.649300
$627$	0	0
$628$	−5270.96	−0.334927
$629$	−373.343	−0.0236664
$630$	0	0
$631$	25175.3	1.58829	0.794147	−	0.607726i	$-0.207918\pi$
$631$	25175.3	1.58829	0.794147	+	0.607726i	$0.207918\pi$
$632$	31141.1	1.96001
$633$	0	0
$634$	2469.67	0.154705
$635$	−7266.61	−0.454121
$636$	0	0
$637$	0	0
$638$	−13736.9	−0.852430
$639$	0	0
$640$	3846.12	0.237549
$641$	7902.14	0.486920	0.243460	−	0.969911i	$-0.421718\pi$
$641$	7902.14	0.486920	0.243460	+	0.969911i	$0.421718\pi$
$642$	0	0
$643$	18223.2	1.11766	0.558828	−	0.829284i	$-0.311251\pi$
$643$	18223.2	1.11766	0.558828	+	0.829284i	$0.311251\pi$
$644$	0	0
$645$	0	0
$646$	−1534.53	−0.0934601
$647$	−23979.5	−1.45708	−0.728539	−	0.685004i	$-0.759800\pi$
$647$	−23979.5	−1.45708	−0.728539	+	0.685004i	$0.759800\pi$
$648$	0	0
$649$	1053.05	0.0636914
$650$	−199.092	−0.0120139
$651$	0	0
$652$	−4603.27	−0.276500
$653$	−10918.4	−0.654316	−0.327158	−	0.944970i	$-0.606091\pi$
$653$	−10918.4	−0.654316	−0.327158	+	0.944970i	$0.606091\pi$
$654$	0	0
$655$	−7697.88	−0.459208
$656$	9579.90	0.570171
$657$	0	0
$658$	0	0
$659$	910.217	0.0538043	0.0269021	−	0.999638i	$-0.491436\pi$
$659$	910.217	0.0538043	0.0269021	+	0.999638i	$0.491436\pi$
$660$	0	0
$661$	10642.9	0.626263	0.313132	−	0.949710i	$-0.398622\pi$
$661$	10642.9	0.626263	0.313132	+	0.949710i	$0.398622\pi$
$662$	7605.52	0.446521
$663$	0	0
$664$	−5972.74	−0.349077
$665$	0	0
$666$	0	0
$667$	−21036.2	−1.22118
$668$	5272.20	0.305370
$669$	0	0
$670$	7300.25	0.420945
$671$	9342.00	0.537472
$672$	0	0
$673$	12314.8	0.705351	0.352676	−	0.935746i	$-0.385272\pi$
$673$	12314.8	0.705351	0.352676	+	0.935746i	$0.385272\pi$
$674$	−6863.83	−0.392263
$675$	0	0
$676$	3986.50	0.226815
$677$	9815.89	0.557246	0.278623	−	0.960401i	$-0.410122\pi$
$677$	9815.89	0.557246	0.278623	+	0.960401i	$0.410122\pi$
$678$	0	0
$679$	0	0
$680$	2201.20	0.124135
$681$	0	0
$682$	−18987.9	−1.06611
$683$	−4354.72	−0.243966	−0.121983	−	0.992532i	$-0.538925\pi$
$683$	−4354.72	−0.243966	−0.121983	+	0.992532i	$0.538925\pi$
$684$	0	0
$685$	−14577.7	−0.813117
$686$	0	0
$687$	0	0
$688$	−274.893	−0.0152328
$689$	−23.4404	−0.00129609
$690$	0	0
$691$	−13489.2	−0.742622	−0.371311	−	0.928509i	$-0.621092\pi$
$691$	−13489.2	−0.742622	−0.371311	+	0.928509i	$0.621092\pi$
$692$	5295.82	0.290920
$693$	0	0
$694$	−31141.6	−1.70334
$695$	11693.8	0.638232
$696$	0	0
$697$	3747.91	0.203676
$698$	−9305.44	−0.504607
$699$	0	0
$700$	0	0
$701$	17175.4	0.925399	0.462700	−	0.886515i	$-0.346881\pi$
$701$	17175.4	0.925399	0.462700	+	0.886515i	$0.346881\pi$
$702$	0	0
$703$	708.900	0.0380322
$704$	14839.5	0.794437
$705$	0	0
$706$	−5577.21	−0.297310
$707$	0	0
$708$	0	0
$709$	1813.47	0.0960596	0.0480298	−	0.998846i	$-0.484706\pi$
$709$	1813.47	0.0960596	0.0480298	+	0.998846i	$0.484706\pi$
$710$	−9983.71	−0.527721
$711$	0	0
$712$	34449.7	1.81328
$713$	−29077.4	−1.52729
$714$	0	0
$715$	−417.510	−0.0218377
$716$	42.1196	0.00219844
$717$	0	0
$718$	2339.15	0.121583
$719$	22701.9	1.17752	0.588760	−	0.808308i	$-0.299617\pi$
$719$	22701.9	1.17752	0.588760	+	0.808308i	$0.299617\pi$
$720$	0	0
$721$	0	0
$722$	−14133.3	−0.728512
$723$	0	0
$724$	−5512.88	−0.282990
$725$	−5302.39	−0.271622
$726$	0	0
$727$	−29667.1	−1.51347	−0.756734	−	0.653723i	$-0.773206\pi$
$727$	−29667.1	−1.51347	−0.756734	+	0.653723i	$0.773206\pi$
$728$	0	0
$729$	0	0
$730$	8152.58	0.413343
$731$	−107.545	−0.00544145
$732$	0	0
$733$	−10574.7	−0.532856	−0.266428	−	0.963855i	$-0.585844\pi$
$733$	−10574.7	−0.532856	−0.266428	+	0.963855i	$0.585844\pi$
$734$	−1203.46	−0.0605187
$735$	0	0
$736$	8009.39	0.401128
$737$	15309.1	0.765155
$738$	0	0
$739$	31529.5	1.56946	0.784730	−	0.619837i	$-0.212801\pi$
$739$	31529.5	1.56946	0.784730	+	0.619837i	$0.212801\pi$
$740$	−188.720	−0.00937498
$741$	0	0
$742$	0	0
$743$	−8886.27	−0.438769	−0.219385	−	0.975638i	$-0.570405\pi$
$743$	−8886.27	−0.438769	−0.219385	+	0.975638i	$0.570405\pi$
$744$	0	0
$745$	2460.88	0.121020
$746$	−25902.9	−1.27128
$747$	0	0
$748$	856.685	0.0418764
$749$	0	0
$750$	0	0
$751$	−20482.3	−0.995221	−0.497610	−	0.867401i	$-0.665789\pi$
$751$	−20482.3	−0.995221	−0.497610	+	0.867401i	$0.665789\pi$
$752$	−18524.4	−0.898292
$753$	0	0
$754$	1689.06	0.0815809
$755$	5409.75	0.260769
$756$	0	0
$757$	−9959.68	−0.478191	−0.239096	−	0.970996i	$-0.576851\pi$
$757$	−9959.68	−0.478191	−0.239096	+	0.970996i	$0.576851\pi$
$758$	−15876.5	−0.760768
$759$	0	0
$760$	−4179.61	−0.199487
$761$	−4556.69	−0.217057	−0.108528	−	0.994093i	$-0.534614\pi$
$761$	−4556.69	−0.217057	−0.108528	+	0.994093i	$0.534614\pi$
$762$	0	0
$763$	0	0
$764$	7104.19	0.336414
$765$	0	0
$766$	−12384.0	−0.584140
$767$	−129.480	−0.00609552
$768$	0	0
$769$	23304.9	1.09284	0.546421	−	0.837511i	$-0.315990\pi$
$769$	23304.9	1.09284	0.546421	+	0.837511i	$0.315990\pi$
$770$	0	0
$771$	0	0
$772$	8402.90	0.391745
$773$	39591.4	1.84218	0.921090	−	0.389349i	$-0.127300\pi$
$773$	39591.4	1.84218	0.921090	+	0.389349i	$0.127300\pi$
$774$	0	0
$775$	−7329.26	−0.339709
$776$	−23247.0	−1.07541
$777$	0	0
$778$	16772.8	0.772921
$779$	−7116.49	−0.327310
$780$	0	0
$781$	−20936.5	−0.959242
$782$	−4445.06	−0.203267
$783$	0	0
$784$	0	0
$785$	14456.5	0.657294
$786$	0	0
$787$	−22391.1	−1.01418	−0.507089	−	0.861894i	$-0.669278\pi$
$787$	−22391.1	−1.01418	−0.507089	+	0.861894i	$0.669278\pi$
$788$	5686.25	0.257061
$789$	0	0
$790$	−15851.1	−0.713868
$791$	0	0
$792$	0	0
$793$	−1148.67	−0.0514382
$794$	−9897.38	−0.442374
$795$	0	0
$796$	−7563.66	−0.336793
$797$	−16409.1	−0.729285	−0.364642	−	0.931148i	$-0.618809\pi$
$797$	−16409.1	−0.729285	−0.364642	+	0.931148i	$0.618809\pi$
$798$	0	0
$799$	−7247.23	−0.320887
$800$	2018.85	0.0892216
$801$	0	0
$802$	−16324.4	−0.718748
$803$	17096.5	0.751337
$804$	0	0
$805$	0	0
$806$	2334.71	0.102031
$807$	0	0
$808$	35191.5	1.53222
$809$	−43547.5	−1.89252	−0.946260	−	0.323406i	$-0.895172\pi$
$809$	−43547.5	−1.89252	−0.946260	+	0.323406i	$0.895172\pi$
$810$	0	0
$811$	−23780.0	−1.02963	−0.514814	−	0.857302i	$-0.672139\pi$
$811$	−23780.0	−1.02963	−0.514814	+	0.857302i	$0.672139\pi$
$812$	0	0
$813$	0	0
$814$	1340.94	0.0577396
$815$	12625.3	0.542631
$816$	0	0
$817$	204.206	0.00874449
$818$	12319.3	0.526571
$819$	0	0
$820$	1894.52	0.0806823
$821$	−31231.4	−1.32763	−0.663814	−	0.747898i	$-0.731063\pi$
$821$	−31231.4	−1.32763	−0.663814	+	0.747898i	$0.731063\pi$
$822$	0	0
$823$	3335.92	0.141292	0.0706458	−	0.997501i	$-0.477494\pi$
$823$	3335.92	0.141292	0.0706458	+	0.997501i	$0.477494\pi$
$824$	27831.3	1.17664
$825$	0	0
$826$	0	0
$827$	−32281.3	−1.35735	−0.678677	−	0.734437i	$-0.737446\pi$
$827$	−32281.3	−1.35735	−0.678677	+	0.734437i	$0.737446\pi$
$828$	0	0
$829$	29512.9	1.23646	0.618230	−	0.785998i	$-0.287850\pi$
$829$	29512.9	1.23646	0.618230	+	0.785998i	$0.287850\pi$
$830$	3040.17	0.127140
$831$	0	0
$832$	−1824.63	−0.0760308
$833$	0	0
$834$	0	0
$835$	−14459.9	−0.599289
$836$	−1626.67	−0.0672960
$837$	0	0
$838$	14361.8	0.592030
$839$	17025.0	0.700558	0.350279	−	0.936645i	$-0.386087\pi$
$839$	17025.0	0.700558	0.350279	+	0.936645i	$0.386087\pi$
$840$	0	0
$841$	20595.6	0.844462
$842$	18548.7	0.759183
$843$	0	0
$844$	6992.70	0.285188
$845$	−10933.7	−0.445124
$846$	0	0
$847$	0	0
$848$	−337.184	−0.0136544
$849$	0	0
$850$	−1120.42	−0.0452121
$851$	2053.46	0.0827166
$852$	0	0
$853$	19970.0	0.801593	0.400796	−	0.916167i	$-0.368733\pi$
$853$	19970.0	0.801593	0.400796	+	0.916167i	$0.368733\pi$
$854$	0	0
$855$	0	0
$856$	16928.5	0.675941
$857$	4689.93	0.186937	0.0934685	−	0.995622i	$-0.470205\pi$
$857$	4689.93	0.186937	0.0934685	+	0.995622i	$0.470205\pi$
$858$	0	0
$859$	−34402.5	−1.36647	−0.683235	−	0.730198i	$-0.739428\pi$
$859$	−34402.5	−1.36647	−0.683235	+	0.730198i	$0.739428\pi$
$860$	−54.3627	−0.00215553
$861$	0	0
$862$	−7559.39	−0.298694
$863$	−3317.85	−0.130870	−0.0654351	−	0.997857i	$-0.520844\pi$
$863$	−3317.85	−0.130870	−0.0654351	+	0.997857i	$0.520844\pi$
$864$	0	0
$865$	−14524.7	−0.570931
$866$	−10059.4	−0.394726
$867$	0	0
$868$	0	0
$869$	−33240.8	−1.29760
$870$	0	0
$871$	−1882.38	−0.0732283
$872$	27834.5	1.08096
$873$	0	0
$874$	8440.24	0.326654
$875$	0	0
$876$	0	0
$877$	26868.2	1.03452	0.517261	−	0.855828i	$-0.326952\pi$
$877$	26868.2	1.03452	0.517261	+	0.855828i	$0.326952\pi$
$878$	6792.24	0.261079
$879$	0	0
$880$	−6005.76	−0.230062
$881$	−12933.5	−0.494599	−0.247300	−	0.968939i	$-0.579543\pi$
$881$	−12933.5	−0.494599	−0.247300	+	0.968939i	$0.579543\pi$
$882$	0	0
$883$	−17537.5	−0.668386	−0.334193	−	0.942505i	$-0.608464\pi$
$883$	−17537.5	−0.668386	−0.334193	+	0.942505i	$0.608464\pi$
$884$	−105.336	−0.00400773
$885$	0	0
$886$	−15903.9	−0.603048
$887$	−35294.4	−1.33604	−0.668022	−	0.744142i	$-0.732859\pi$
$887$	−35294.4	−1.33604	−0.668022	+	0.744142i	$0.732859\pi$
$888$	0	0
$889$	0	0
$890$	−17535.2	−0.660428
$891$	0	0
$892$	−1367.88	−0.0513453
$893$	13761.0	0.515670
$894$	0	0
$895$	−115.520	−0.00431444
$896$	0	0
$897$	0	0
$898$	40982.6	1.52295
$899$	62180.2	2.30681
$900$	0	0
$901$	−131.915	−0.00487761
$902$	−13461.4	−0.496914
$903$	0	0
$904$	28189.1	1.03712
$905$	15120.0	0.555367
$906$	0	0
$907$	30615.8	1.12082	0.560408	−	0.828217i	$-0.310644\pi$
$907$	30615.8	1.12082	0.560408	+	0.828217i	$0.310644\pi$
$908$	1146.40	0.0418993
$909$	0	0
$910$	0	0
$911$	−28971.0	−1.05362	−0.526812	−	0.849982i	$-0.676613\pi$
$911$	−28971.0	−1.05362	−0.526812	+	0.849982i	$0.676613\pi$
$912$	0	0
$913$	6375.45	0.231103
$914$	35729.2	1.29302
$915$	0	0
$916$	−4461.57	−0.160933
$917$	0	0
$918$	0	0
$919$	26239.4	0.941847	0.470924	−	0.882174i	$-0.343921\pi$
$919$	26239.4	0.941847	0.470924	+	0.882174i	$0.343921\pi$
$920$	−12107.0	−0.433867
$921$	0	0
$922$	2486.32	0.0888098
$923$	2574.31	0.0918032
$924$	0	0
$925$	517.598	0.0183984
$926$	−1556.14	−0.0552245
$927$	0	0
$928$	−17127.6	−0.605864
$929$	−17943.2	−0.633690	−0.316845	−	0.948477i	$-0.602624\pi$
$929$	−17943.2	−0.633690	−0.316845	+	0.948477i	$0.602624\pi$
$930$	0	0
$931$	0	0
$932$	5762.92	0.202544
$933$	0	0
$934$	2284.13	0.0800205
$935$	−2349.61	−0.0821823
$936$	0	0
$937$	−43418.1	−1.51378	−0.756888	−	0.653545i	$-0.773281\pi$
$937$	−43418.1	−1.51378	−0.756888	+	0.653545i	$0.773281\pi$
$938$	0	0
$939$	0	0
$940$	−3663.38	−0.127113
$941$	30894.9	1.07029	0.535146	−	0.844759i	$-0.320256\pi$
$941$	30894.9	1.07029	0.535146	+	0.844759i	$0.320256\pi$
$942$	0	0
$943$	−20614.3	−0.711870
$944$	−1862.54	−0.0642166
$945$	0	0
$946$	386.272	0.0132757
$947$	−7752.62	−0.266026	−0.133013	−	0.991114i	$-0.542465\pi$
$947$	−7752.62	−0.266026	−0.133013	+	0.991114i	$0.542465\pi$
$948$	0	0
$949$	−2102.15	−0.0719059
$950$	2127.45	0.0726565
$951$	0	0
$952$	0	0
$953$	−21377.5	−0.726638	−0.363319	−	0.931665i	$-0.618356\pi$
$953$	−21377.5	−0.726638	−0.363319	+	0.931665i	$0.618356\pi$
$954$	0	0
$955$	−19484.5	−0.660212
$956$	−10971.6	−0.371179
$957$	0	0
$958$	−38929.7	−1.31290
$959$	0	0
$960$	0	0
$961$	56157.9	1.88506
$962$	−164.879	−0.00552591
$963$	0	0
$964$	10424.9	0.348303
$965$	−23046.4	−0.768798
$966$	0	0
$967$	−47707.3	−1.58652	−0.793259	−	0.608885i	$-0.791617\pi$
$967$	−47707.3	−1.58652	−0.793259	+	0.608885i	$0.791617\pi$
$968$	15915.0	0.528438
$969$	0	0
$970$	11832.9	0.391682
$971$	−43434.6	−1.43551	−0.717756	−	0.696295i	$-0.754831\pi$
$971$	−43434.6	−1.43551	−0.717756	+	0.696295i	$0.754831\pi$
$972$	0	0
$973$	0	0
$974$	35889.6	1.18067
$975$	0	0
$976$	−16523.3	−0.541904
$977$	33487.6	1.09658	0.548292	−	0.836287i	$-0.315278\pi$
$977$	33487.6	1.09658	0.548292	+	0.836287i	$0.315278\pi$
$978$	0	0
$979$	−36772.5	−1.20046
$980$	0	0
$981$	0	0
$982$	48019.0	1.56043
$983$	15767.5	0.511603	0.255802	−	0.966729i	$-0.417661\pi$
$983$	15767.5	0.511603	0.255802	+	0.966729i	$0.417661\pi$
$984$	0	0
$985$	−15595.5	−0.504483
$986$	9505.49	0.307015
$987$	0	0
$988$	200.011	0.00644049
$989$	591.521	0.0190185
$990$	0	0
$991$	−3584.59	−0.114902	−0.0574512	−	0.998348i	$-0.518297\pi$
$991$	−3584.59	−0.114902	−0.0574512	+	0.998348i	$0.518297\pi$
$992$	−23674.7	−0.757735
$993$	0	0
$994$	0	0
$995$	20744.7	0.660955
$996$	0	0
$997$	36013.5	1.14399	0.571995	−	0.820257i	$-0.306170\pi$
$997$	36013.5	1.14399	0.571995	+	0.820257i	$0.306170\pi$
$998$	−11266.6	−0.357351
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bj.1.3		3
3.2	odd	2		735.4.a.s.1.1		3
7.3	odd	6		315.4.j.e.226.1		6
7.5	odd	6		315.4.j.e.46.1		6
7.6	odd	2		2205.4.a.bi.1.3		3
21.5	even	6		105.4.i.c.46.3	yes	6
21.17	even	6		105.4.i.c.16.3	✓	6
21.20	even	2		735.4.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.c.16.3	✓	6	21.17	even	6
105.4.i.c.46.3	yes	6	21.5	even	6
315.4.j.e.46.1		6	7.5	odd	6
315.4.j.e.226.1		6	7.3	odd	6
735.4.a.r.1.1		3	21.20	even	2
735.4.a.s.1.1		3	3.2	odd	2
2205.4.a.bi.1.3		3	7.6	odd	2
2205.4.a.bj.1.3		3	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)

\(f(q)\)	\(=\)	\(q+2.48535 q^{2} -1.82304 q^{4} +5.00000 q^{5} -24.4137 q^{8} +O(q^{10})\)	\(q+2.48535 q^{2} -1.82304 q^{4} +5.00000 q^{5} -24.4137 q^{8} +12.4267 q^{10} +26.0598 q^{11} -3.20425 q^{13} -46.0922 q^{16} -18.0325 q^{17} +34.2399 q^{19} -9.11519 q^{20} +64.7676 q^{22} +99.1824 q^{23} +25.0000 q^{25} -7.96368 q^{26} -212.096 q^{29} -293.170 q^{31} +80.7542 q^{32} -44.8170 q^{34} +20.7039 q^{37} +85.0981 q^{38} -122.068 q^{40} -207.842 q^{41} +5.96397 q^{43} -47.5079 q^{44} +246.503 q^{46} +401.899 q^{47} +62.1337 q^{50} +5.84147 q^{52} +7.31541 q^{53} +130.299 q^{55} -527.132 q^{58} +40.4089 q^{59} +358.484 q^{61} -728.631 q^{62} +569.440 q^{64} -16.0212 q^{65} +587.463 q^{67} +32.8739 q^{68} -803.405 q^{71} +656.051 q^{73} +51.4565 q^{74} -62.4206 q^{76} -1275.56 q^{79} -230.461 q^{80} -516.560 q^{82} +244.647 q^{83} -90.1624 q^{85} +14.8225 q^{86} -636.215 q^{88} -1411.08 q^{89} -180.813 q^{92} +998.859 q^{94} +171.199 q^{95} +952.212 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8	\( 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8} - 15 q^{10} + q^{11} + 79 q^{13} - 79 q^{16} - 72 q^{17} + 29 q^{19} + 5 q^{20} + 143 q^{22} + 63 q^{23} + 75 q^{25} - 339 q^{26} - 220 q^{29} - 136 q^{31} + 155 q^{32} + 220 q^{34} + 43 q^{37} - 21 q^{38} - 45 q^{40} - 599 q^{41} + 170 q^{43} - 135 q^{44} + 265 q^{46} - 3 q^{47} - 75 q^{50} + 701 q^{52} - 331 q^{53} + 5 q^{55} - 472 q^{58} - 1520 q^{59} + 1160 q^{61} - 748 q^{62} + 17 q^{64} + 395 q^{65} + 806 q^{67} - 684 q^{68} + 406 q^{71} + 1192 q^{73} - 959 q^{74} + 591 q^{76} - 2590 q^{79} - 395 q^{80} + 1191 q^{82} - 508 q^{83} - 360 q^{85} + 742 q^{86} - 749 q^{88} + 42 q^{89} + 211 q^{92} + 1167 q^{94} + 145 q^{95} + 1020 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8 - 15 * q^10 + q^11 + 79 * q^13 - 79 * q^16 - 72 * q^17 + 29 * q^19 + 5 * q^20 + 143 * q^22 + 63 * q^23 + 75 * q^25 - 339 * q^26 - 220 * q^29 - 136 * q^31 + 155 * q^32 + 220 * q^34 + 43 * q^37 - 21 * q^38 - 45 * q^40 - 599 * q^41 + 170 * q^43 - 135 * q^44 + 265 * q^46 - 3 * q^47 - 75 * q^50 + 701 * q^52 - 331 * q^53 + 5 * q^55 - 472 * q^58 - 1520 * q^59 + 1160 * q^61 - 748 * q^62 + 17 * q^64 + 395 * q^65 + 806 * q^67 - 684 * q^68 + 406 * q^71 + 1192 * q^73 - 959 * q^74 + 591 * q^76 - 2590 * q^79 - 395 * q^80 + 1191 * q^82 - 508 * q^83 - 360 * q^85 + 742 * q^86 - 749 * q^88 + 42 * q^89 + 211 * q^92 + 1167 * q^94 + 145 * q^95 + 1020 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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